28.8: Patterns of Inheritance
By the end of this section, you will be able to:
- Differentiate between genotype and phenotype
- Describe how alleles determine a person’s traits
- Summarize Mendel’s experiments and relate them to human genetics
- Explain the inheritance of autosomal dominant and recessive and sex-linked genetic disorders
We have discussed the events that lead to the development of a newborn. But what makes each newborn unique? The answer lies, of course, in the DNA in the sperm and oocyte that combined to produce that first diploid cell, the human zygote.
From Genotype to Phenotype
Each human body cell has a full complement of DNA stored in 23 pairs of chromosomes. Figure 28.24 shows the pairs in a systematic arrangement called a karyotype . Among these is one pair of chromosomes, called the sex chromosomes , that determines the sex of the individual (XX in females, XY in males). The remaining 22 chromosome pairs are called autosomal chromosomes . Each of these chromosomes carries hundreds or even thousands of genes, each of which codes for the assembly of a particular protein—that is, genes are “expressed” as proteins. An individual’s complete genetic makeup is referred to as their genotype . The characteristics that the genes express, whether they are physical, behavioral, or biochemical, are a person’s phenotype .
In genetics and reproduction, "parent" is often used to describe the individual organisms that contribute genetic material to offspring, usually in the form of gamete cells and their chromosomes. The concept of a genetic parent is distinct from social and legal concepts of parenthood, and may differ from those whom people consider their parents.
You inherit one chromosome in each pair—a full complement of 23—from each parent. This occurs when the sperm and oocyte combine at the moment of your conception. Homologous chromosomes—those that make up a complementary pair—have genes for the same characteristics in the same location on the chromosome. Because one copy of a gene, an allele , is inherited from each parent, the alleles in these complementary pairs may vary. Take for example an allele that encodes for dimples. A child may inherit the allele encoding for dimples on the chromosome from the one parent and the allele that encodes for smooth skin (no dimples) on the chromosome from the other parent.
Although a person can have two identical alleles for a single gene (a homozygous state), it is also possible for a person to have two different alleles (a heterozygous state). The two alleles can interact in several different ways. The expression of an allele can be dominant, for which the activity of this gene will mask the expression of a nondominant, or recessive, allele. Sometimes dominance is complete; at other times, it is incomplete. In some cases, both alleles are expressed at the same time in a form of expression known as codominance.
In the simplest scenario, a single pair of genes will determine a single heritable characteristic. However, it is quite common for multiple genes to interact to confer a feature. For instance, eight or more genes—each with their own alleles—determine eye color in humans. Moreover, although any one person can only have two alleles corresponding to a given gene, more than two alleles commonly exist in a population. This phenomenon is called multiple alleles. For example, there are three different alleles that encode ABO blood type; these are designated I A , I B , and i.
Over 100 years of theoretical and experimental genetics studies, and the more recent sequencing and annotation of the human genome, have helped scientists to develop a better understanding of how an individual’s genotype is expressed as their phenotype. This body of knowledge can help scientists and medical professionals to predict, or at least estimate, some of the features that an offspring will inherit by examining the genotypes or phenotypes of the parents. One important application of this knowledge is to identify an individual’s risk for certain heritable genetic disorders. However, most diseases have a multigenic pattern of inheritance and can also be affected by the environment, so examining the genotypes or phenotypes of a person’s parents will provide only limited information about the risk of inheriting a disease. Only for a handful of single-gene disorders can genetic testing allow clinicians to calculate the probability with which a child born to the two parents tested may inherit a specific disease.
Mendel’s Theory of Inheritance
Our contemporary understanding of genetics rests on the work of a nineteenth-century monk. Working in the mid-1800s, long before anyone knew about genes or chromosomes, Gregor Mendel discovered that garden peas transmit their physical characteristics to subsequent generations in a discrete and predictable fashion. When he mated, or crossed, two pure-breeding pea plants that differed by a certain characteristic, the first-generation offspring all looked like one of the parents. For instance, when he crossed tall and dwarf pure-breeding pea plants, all of the offspring were tall. Mendel called tallness dominant because it was expressed in offspring when it was present in a purebred parent. He called dwarfism recessive because it was masked in the offspring if one of the purebred parents possessed the dominant characteristic. Note that tallness and dwarfism are variations on the characteristic of height. Mendel called such a variation a trait . We now know that these traits are the expression of different alleles of the gene encoding height.
Mendel performed thousands of crosses in pea plants with differing traits for a variety of characteristics. And he repeatedly came up with the same results—among the traits he studied, one was always dominant, and the other was always recessive. (Remember, however, that this dominant–recessive relationship between alleles is not always the case; some alleles are codominant, and sometimes dominance is incomplete.)
Using his understanding of dominant and recessive traits, Mendel tested whether a recessive trait could be lost altogether in a pea lineage or whether it would resurface in a later generation. By crossing the second-generation offspring of purebred parents with each other, he showed that the latter was true: recessive traits reappeared in third-generation plants in a ratio of 3:1 (three offspring having the dominant trait and one having the recessive trait). Mendel then proposed that characteristics such as height were determined by heritable “factors” that were transmitted, one from each parent, and inherited in pairs by offspring.
In the language of genetics, Mendel’s theory applied to humans says that if an individual receives two dominant alleles, one from each parent, the individual’s phenotype will express the dominant trait. If an individual receives two recessive alleles, then the recessive trait will be expressed in the phenotype. Individuals who have two identical alleles for a given gene, whether dominant or recessive, are said to be homozygous for that gene (homo- = “same”). Conversely, an individual who has one dominant allele and one recessive allele is said to be heterozygous for that gene (hetero- = “different” or “other”). In this case, the dominant trait will be expressed, and the individual will be phenotypically identical to an individual who possesses two dominant alleles for the trait.
It is common practice in genetics to use capital and lowercase letters to represent dominant and recessive alleles. Using Mendel’s pea plants as an example, if a tall pea plant is homozygous, it will possess two tall alleles ( TT ). A dwarf pea plant must be homozygous because its dwarfism can only be expressed when two recessive alleles are present ( tt ). A heterozygous pea plant ( Tt ) would be tall and phenotypically indistinguishable from a tall homozygous pea plant because of the dominant tall allele. Mendel deduced that a 3:1 ratio of dominant to recessive would be produced by the random segregation of heritable factors (genes) when crossing two heterozygous pea plants. In other words, for any given gene, parents are equally likely to pass down either one of their alleles to their offspring in a haploid gamete, and the result will be expressed in a dominant–recessive pattern if both parents are heterozygous for the trait.
Because of the random segregation of gametes, the laws of chance and probability come into play when predicting the likelihood of a given phenotype. Consider a cross between an individual with two dominant alleles for a trait ( AA ) and an individual with two recessive alleles for the same trait ( aa ). All of the parental gametes from the dominant individual would be A , and all of the parental gametes from the recessive individual would be a (Figure 28.25). All of the offspring of that second generation, inheriting one allele from each parent, would have the genotype Aa , and the probability of expressing the phenotype of the dominant allele would be 4 out of 4, or 100 percent.
This seems simple enough, but the inheritance pattern gets interesting when the second-generation Aa individuals are crossed. In this generation, 50 percent of each parent’s gametes are A and the other 50 percent are a . By Mendel’s principle of random segregation, the possible combinations of gametes that the offspring can receive are AA , Aa , aA (which is the same as Aa ), and aa . Because segregation and fertilization are random, each offspring has a 25 percent chance of receiving any of these combinations. Therefore, if an Aa × Aa cross were performed 1000 times, approximately 250 (25 percent) of the offspring would be AA ; 500 (50 percent) would be Aa (that is, Aa plus aA ); and 250 (25 percent) would be aa . The genotypic ratio for this inheritance pattern is 1:2:1. However, we have already established that AA and Aa (and aA ) individuals all express the dominant trait (i.e., share the same phenotype), and can therefore be combined into one group. The result is Mendel’s third-generation phenotype ratio of 3:1.
Mendel’s observation of pea plants also included many crosses that involved multiple traits, which prompted him to formulate the principle of independent assortment. The law states that the members of one pair of genes (alleles) from a parent will sort independently from other pairs of genes during the formation of gametes. Applied to pea plants, that means that the alleles associated with the different traits of the plant, such as color, height, or seed type, will sort independently of one another. This holds true except when two alleles happen to be located close to one other on the same chromosome. Independent assortment provides for a great degree of diversity in offspring.
Mendelian genetics represent the fundamentals of inheritance, but there are two important qualifiers to consider when applying Mendel’s findings to inheritance studies in humans. First, as we’ve already noted, not all genes are inherited in a dominant–recessive pattern. Although all diploid individuals have two alleles for every gene, allele pairs may interact to create several types of inheritance patterns, including incomplete dominance and codominance.
Secondly, Mendel performed his studies using thousands of pea plants. He was able to identify a 3:1 phenotypic ratio in second-generation offspring because his large sample size overcame the influence of variability resulting from chance. In contrast, no human couple has ever had thousands of children. If we know that parents are both heterozygous for a recessive genetic disorder, we would predict that one in every four of their children would be affected by the disease. In real life, however, the influence of chance could change that ratio significantly. For example, if parents are both heterozygous for cystic fibrosis, a recessive genetic disorder that is expressed only when the individual has two defective alleles, we would expect one in four of their children to have cystic fibrosis. However, it is entirely possible for them to have seven children, none of whom is affected, or for them to have two children, both of whom are affected. For each individual child, the presence or absence of a single gene disorder depends on which alleles that child inherits from their parents.
Autosomal Dominant Inheritance
In the case of cystic fibrosis, the disorder is recessive to the normal phenotype. However, a genetic abnormality may be dominant to the normal phenotype. When the dominant allele is located on one of the 22 pairs of autosomes (non-sex chromosomes), we refer to its inheritance pattern as autosomal dominant . An example of an autosomal dominant disorder is neurofibromatosis type I, a disease that induces tumor formation within the nervous system that leads to skin and skeletal deformities. Consider a couple in which one parent is heterozygous for this disorder (and who therefore has neurofibromatosis), Nn , and one parent is homozygous for the normal gene, nn . The heterozygous parent would have a 50 percent chance of passing the dominant allele for this disorder to their offspring, and the homozygous parent would always pass the normal allele. Therefore, four possible offspring genotypes are equally likely to occur: Nn , Nn , nn , and nn . That is, every child of this couple would have a 50 percent chance of inheriting neurofibromatosis. This inheritance pattern is shown in Figure \(\PageIndex{3}\), in a form called a Punnett square , named after its creator, the British geneticist Reginald Punnett.
Other genetic diseases that are inherited in this pattern are achondroplastic dwarfism, Marfan syndrome, and Huntington’s disease. Because autosomal dominant disorders are expressed by the presence of just one gene, an individual with the disorder will know that they have at least one faulty gene. The expression of the disease may manifest later in life, after the childbearing years, which is the case in Huntington’s disease (discussed in more detail later in this section).
Autosomal Recessive Inheritance
When a genetic disorder is inherited in an autosomal recessive pattern, the disorder corresponds to the recessive phenotype. Heterozygous individuals will not display symptoms of this disorder, because their unaffected gene will compensate. Such an individual is called a carrier . Carriers for an autosomal recessive disorder may never know their genotype unless they have a child with the disorder.
An example of an autosomal recessive disorder is cystic fibrosis (CF), which we introduced earlier. CF is characterized by the chronic accumulation of a thick, tenacious mucus in the lungs and digestive tract. Decades ago, children with CF rarely lived to adulthood. With advances in medical technology, the average lifespan in developed countries has increased into middle adulthood. CF is a relatively common disorder that occurs in approximately 30,000 people in the United States. A child born to two CF carriers would have a 25 percent chance of inheriting the disease. This is the same 3:1 dominant:recessive ratio that Mendel observed in his pea plants would apply here. The pattern is shown in Figure \(\PageIndex{4}\), using a diagram that tracks the likely incidence of an autosomal recessive disorder on the basis of parental genotypes.
On the other hand, a child born to a CF carrier and someone with two unaffected alleles would have a 0 percent probability of inheriting CF, but would have a 50 percent chance of being a carrier. Other examples of autosome recessive genetic illnesses include the blood disorder sickle-cell anemia, the fatal neurological disorder Tay–Sachs disease, and the metabolic disorder phenylketonuria.
X-linked Dominant or Recessive Inheritance
An X-linked transmission pattern involves genes located on the X chromosome of the 23rd pair (Figure 28.28). Recall that a male has one X and one Y chromosome.
When an abnormal allele for a gene that occurs on the X chromosome is dominant over the normal allele, the pattern is described as X-linked dominant . This is the case with vitamin D–resistant rickets: an affected male would pass the disease gene to all of the female offspring, but none of the male offspring, because the male transmits only the Y chromosome to male offspring (see Figure \(\PageIndex{6}\) a ). If it is the female parent who is affected, all of the offspring—male or female—would have a 50 percent chance of inheriting the disorder because the female parent can only pass an X chromosome on to children (see Figure \(\PageIndex{6}\) b ). For an affected female, the inheritance pattern would be identical to that of an autosomal dominant inheritance pattern in which one parent is heterozygous and the other is homozygous for the normal gene.
X-linked recessive inheritance is much more common because females can be carriers of the disease yet still have a normal phenotype. Diseases transmitted by X-linked recessive inheritance include color blindness, the blood-clotting disorder hemophilia, and some forms of muscular dystrophy. For an example of X-linked recessive inheritance, consider parents in which the female is an unaffected carrier and the male is normal. None of the female offspring would have the disease because they receive a normal gene from their male parent. However, they have a 50 percent chance of receiving the disease gene from their female parent and becoming a carrier. In contrast, 50 percent of the male offspring would be affected (Figure \(\PageIndex{7}\)).
With X-linked recessive diseases, males either have the disease or are genotypically normal—they cannot be carriers. Females, however, can be genotypically normal, a carrier who is phenotypically normal, or affected with the disease. A female can inherit the gene for an X-linked recessive illness when the female parent is a carrier or affected, or the male parent is affected. Female offspring will be affected by the disease only if they inherit an X-linked recessive gene from both parents. As you can imagine, X-linked recessive disorders affect many more males than females. For example, color blindness affects at least 1 in 20 males, but only about 1 in 400 females.
Other Inheritance Patterns: Incomplete Dominance, Codominance, and Lethal Alleles
Not all genetic disorders are inherited in a dominant–recessive pattern. In incomplete dominance , the offspring express a heterozygous phenotype that is intermediate between one parent’s homozygous dominant trait and the other parent’s homozygous recessive trait. An example of this can be seen in snapdragons when red-flowered plants and white-flowered plants are crossed to produce pink-flowered plants. In humans, incomplete dominance occurs with one of the genes for hair texture. When one parent passes a curly hair allele (the incompletely dominant allele) and the other parent passes a straight-hair allele, the effect on the offspring will be intermediate, resulting in hair that is wavy.
Codominance is characterized by the equal, distinct, and simultaneous expression of both parents’ different alleles. This pattern differs from the intermediate, blended features seen in incomplete dominance. A classic example of codominance in humans is ABO blood type. People are blood type A if they have an allele for an enzyme that facilitates the production of surface antigen A on their erythrocytes. This allele is designated I A . In the same manner, people are blood type B if they express an enzyme for the production of surface antigen B. People who have alleles for both enzymes ( I A and I B ) produce both surface antigens A and B. As a result, they are blood type AB. Because the effect of both alleles (or enzymes) is observed, we say that the I A and I B alleles are codominant. There is also a third allele that determines blood type. This allele ( i ) produces a nonfunctional enzyme.  If a person has I A and i alleles, the person will have blood type A. Notice that it does not make any difference whether a person has two I A alleles or one I A and one i allele. In both cases, the person is blood type A. Because I A masks i , we say that I A is dominant to i . Table \(\PageIndex{1}\) summarizes the expression of blood type.
| Blood type | Genotype | Pattern of inheritance |
|---|---|---|
| A | I A I A or I A i | I A is dominant to i |
| B | I B I B or I B i | I B is dominant to i |
| AB | I A I B | I A is co-dominant to I B |
| O | ii | Two recessive alleles |
Certain combinations of alleles can be lethal, meaning they prevent the individual from developing in utero, or cause a shortened life span. In recessive lethal inheritance patterns, a child who is born to two heterozygous (carrier) parents and who inherited the faulty allele from both would not survive. An example of this is Tay–Sachs, a fatal disorder of the nervous system. In this disorder, parents with one copy of the allele for the disorder are carriers. If they both transmit their abnormal allele, their offspring will develop the disease and will die in childhood, usually before age 5.
Dominant lethal inheritance patterns are much more rare because neither heterozygotes nor homozygotes survive. Of course, dominant lethal alleles that arise naturally through mutation and cause miscarriages or stillbirths are never transmitted to subsequent generations. However, some dominant lethal alleles, such as the allele for Huntington’s disease, cause a shortened life span but may not be identified until after the person reaches reproductive age and has children. Huntington’s disease causes irreversible nerve cell degeneration and death in 100 percent of affected individuals, but it may not be expressed until the individual reaches middle age. In this way, dominant lethal alleles can be maintained in the human population. Individuals with a family history of Huntington’s disease are typically offered genetic counseling, which can help them decide whether or not they wish to be tested for the faulty gene.
Mutations
A mutation is a change in the sequence of DNA nucleotides that may or may not affect a person’s phenotype. Mutations can arise spontaneously from errors during DNA replication, or they can result from environmental insults such as radiation, certain viruses, or exposure to tobacco smoke or other toxic chemicals. Because genes encode for the assembly of proteins, a mutation in the nucleotide sequence of a gene can change amino acid sequence and, consequently, a protein’s structure and function. Spontaneous mutations occurring during meiosis are thought to account for many spontaneous abortions (miscarriages).
Chromosomal Disorders
Sometimes a genetic disease is not caused by a mutation in a gene, but by the presence of an incorrect number of chromosomes. For example, Down syndrome is caused by having three copies of chromosome 21. This is known as trisomy 21. The most common cause of trisomy 21 is chromosomal nondisjunction during meiosis. The frequency of nondisjunction events appears to increase with age, so the frequency of bearing a child with Down syndrome increases in females over 36. The age of the male parent matters less because nondisjunction is much less likely to occur in a sperm than in an egg.
Whereas Down syndrome is caused by having three copies of a chromosome, Turner syndrome is caused by having just one copy of the X chromosome. This is known as monosomy. The affected child is always female.
Career Connection
Genetic Counselor
Given the intricate orchestration of gene expression, cell migration, and cell differentiation during prenatal development, it is amazing that the vast majority of newborns are healthy and free of major birth defects. When a person over 35 is pregnant or intends to become pregnant, or their partner is over 55, or if there is a family history of a genetic disorder, they may want to speak to a genetic counselor to discuss the likelihood that their child may be affected by a genetic or chromosomal disorder. A genetic counselor can interpret a couple’s family history and estimate the risks to their future offspring.
For many genetic diseases, a DNA test can determine whether a person is a carrier. For instance, carrier status for Fragile X, an X-linked disorder associated with intellectual disabilites, or for cystic fibrosis can be determined with a simple blood draw to obtain DNA for testing. A genetic counselor can educate a couple about the implications of such a test and help them decide whether to undergo testing. For chromosomal disorders, the available testing options include a blood test, amniocentesis (in which amniotic fluid is tested), and chorionic villus sampling (in which tissue from the placenta is tested). Each of these has advantages and drawbacks. A genetic counselor can also help a couple cope with the news that either one or both partners is a carrier of a genetic illness, or that their unborn child has been diagnosed with a chromosomal disorder or other birth defect.
To become a genetic counselor, one needs to complete a 4-year undergraduate program and then obtain a Master of Science in Genetic Counseling from an accredited university. Board certification is attained after passing examinations by the American Board of Genetic Counseling. Genetic counselors are essential professionals in many branches of medicine, but there is a particular demand for preconception and prenatal genetic counselors.
Interactive Link
Visit the National Society of Genetic Counselors website for more information about genetic counselors.
Interactive Link
Visit the American Board of Genetic Counselors, Inc., website for more information about genetic counselors.


35.2: Neurons and Glial Cells - Neurons
- Describe the functions of the structural components of a neuron
The nervous system of the common laboratory fly, Drosophila melanogaster , contains around 100,000 neurons, the same number as a lobster. This number compares to 75 million in the mouse and 300 million in the octopus.  Despite these very different numbers, the nervous systems of these animals control many of the same behaviors, from basic reflexes to more complicated behaviors such as finding food and courting mates. The ability of neurons to communicate with each other, as well as with other types of cells, underlies all of these behaviors.
Most neurons share the same cellular components. But neurons are also highly specialized: different types of neurons have different sizes and shapes that relate to their functional roles.
Parts of a Neuron
Each neuron has a cell body (or soma) that contains a nucleus, smooth and rough endoplasmic reticulum, Golgi apparatus, mitochondria, and other cellular components. Neurons also contain unique structures, relative to most cells, which are required for receiving and sending the electrical signals that make neuronal communication possible. Dendrites are tree-like structures that extend away from the cell body to receive messages from other neurons at specialized junctions called synapses. While some neurons have no dendrites, other types of neurons have multiple dendrites. Dendrites can have small protrusions called dendritic spines, which further increase surface area for possible synaptic connections.
Once a signal is received by the dendrite, it then travels passively to the cell body. The cell body contains a specialized structure, the axon hillock, that integrates signals from multiple synapses and serves as a junction between the cell body and an axon: a tube-like structure that propagates the integrated signal to specialized endings called axon terminals. These terminals, in turn, synapse on other neurons, muscles, or target organs. Chemicals released at axon terminals allow signals to be communicated to these other cells. Neurons usually have one or two axons, but some neurons, like amacrine cells in the retina, do not contain any axons. Some axons are covered with myelin, which acts as an insulator to minimize dissipation of the electrical signal as it travels down the axon, greatly increasing the speed on conduction. This insulation is important as the axon from a human motor neuron can be as long as a meter: from the base of the spine to the toes. The myelin sheath is not actually part of the neuron. Myelin is produced by glial cells. Along these types of axons, there are periodic gaps in the myelin sheath. These gaps, called “nodes of Ranvier,” are sites where the signal is “recharged” as it travels along the axon.
It is important to note that a single neuron does not act alone. Neuronal communication depends on the connections that neurons make with one another (as well as with other cells, such as muscle cells). Dendrites from a single neuron may receive synaptic contact from many other neurons. For example, dendrites from a Purkinje cell in the cerebellum are thought to receive contact from as many as 200,000 other neurons.
Types of Neurons
There are different types of neurons; the functional role of a given neuron is intimately dependent on its structure. There is an amazing diversity of neuron shapes and sizes found in different parts of the nervous system (and across species).
While there are many defined neuron cell subtypes, neurons are broadly divided into four basic types: unipolar, bipolar, multipolar, and pseudounipolar. Unipolar neurons have only one structure that extends away from the soma. These neurons are not found in vertebrates, but are found in insects where they stimulate muscles or glands. A bipolar neuron has one axon and one dendrite extending from the soma. An example of a bipolar neuron is a retinal bipolar cell, which receives signals from photoreceptor cells that are sensitive to light and transmits these signals to ganglion cells that carry the signal to the brain. Multipolar neurons are the most common type of neuron. Each multipolar neuron contains one axon and multiple dendrites. Multipolar neurons can be found in the central nervous system (brain and spinal cord). The Purkinje cell, a multipolar neuron in the cerebellum, has many branching dendrites, but only one axon. Pseudounipolar cells share characteristics with both unipolar and bipolar cells. A pseudounipolar cell has a single structure that extends from the soma (like a unipolar cell), which later branches into two distinct structures (like a bipolar cell). Most sensory neurons are pseudounipolar and have an axon that branches into two extensions: one connected to dendrites that receives sensory information and another that transmits this information to the spinal cord.
Key Points
- Dendrites are the tree-like structures in neurons that extend away from the cell body to receive messages from other neurons at synapses; not all neurons have dendrites.
- Synapses enable the dendrites from a single neuron to interact and receive signals from many other neurons.
- Axons are tube-like structures that send signals to other neurons, muscles, or organs; not all neurons have axons.
- Neurons are divided into four major types: unipolar, bipolar, multipolar, and pseudounipolar.
- Unipolar neurons have only one structure extending from the soma; bipolar neurons have one axon and one dendrite extending from the soma.
- Multipolar neurons contain one axon and many dendrites; pseudounipolar neurons have a single structure that extends from the soma, which later branches into two distinct structures.
Key Terms
- dendrite : branched projections of a neuron that conduct the impulses received from other neural cells to the cell body
- axon : long slender projection of a nerve cell that conducts nerve impulses away from the cell body to other neurons, muscles, and organs
- synapse : the junction between the terminal of a neuron and either another neuron or a muscle or gland cell, over which nerve impulses pass


1.1: Identifying Word Parts in Medical Terms
Welcome to Medical Terminology . Medical terminology is a language that is used in health care settings. Medical terms are built from Greek and Latin word parts and in addition include acronyms, such as “laser” (light amplification by stimulated emission of radiation), eponyms, such as West Nile Virus (named after a geographical location where the virus was identified) or Alzheimer disease (named after the person who discovered it) and modern-day language terms, such as “nuclear medicine scanner”, which is derived from the English language.
Word Parts
Medical terms are built from four word parts. Those word parts are prefix, word root, suffix, and combining vowel. When a word root is combined with a combining vowel, the word part is referred to as a combining form.
Word Root
The word root contains the fundamental meaning of the word. It is the core part of the word. Each medical term contains at least one word root.
Examples:
- In the word: play/er ( play is the word root)
- In the medical term: arthr/itis ( arthr – meaning “joint” – is the word root)
Prefixes
Prefixes are located at the beginning of a medical term. The prefix alters the meaning of the medical term. It is important to spell and pronounce prefixes correctly.
Examples
- In the word: re/play (“re” is the prefix)
- In the medical term: intra/ven/ous (“intra” is the prefix)
Many prefixes that you find in medical terms are common to English language prefixes. A good technique to help with memorization is the following:
- Start by reviewing the most common prefixes.
- Consider common English language words that begin with the same prefixes.
- Compare them to the examples of use in medical terms.
Prefixes often indicate:
|
Suffixes
Suffixes are word parts that are located at the end of words. Suffixes can alter the meaning of medical terms. In order to properly spell and pronounce medical terms, it is helpful to learn the suffixes.
ExamplesF
- In the word: king/dom (“-dom” is the suffix)
- In the medical term: hepat/itis (“-itis” is the suffix)
When defining a medical term you often begin with the meaning of the suffix. For example: hepat/itis would be defined as “inflammation of the liver.”
Suffixes often indicate:
|
Combining Vowel
The combining vowel is a word part – most often an o – that helps pronunciation.
The combining vowel is placed to connect two word roots or to connect a word root and a suffix. Do NOT place a combining vowel to connect a prefix and a word root. Not all medical terms will have combining vowels.
Table 1.1 Combining Vowel Guidelines
| Guideline | Example |
| 1. When connecting a word root and a suffix, a combining vowel is used if the suffix DOES NOT begin with a vowel | arthr/o/pathy |
| 2. When connecting a word root and a suffix, a combining vowel is usually NOT USED if the suffix BEGINS with a vowel | hepat/ic |
| 3. When connecting two word roots, a combining vowel is usually used even if vowels are present at the junction | oste/o/arthr/itis |
| 4. When connecting a prefix and a word root, a combining vowel is NOT USED | sub/hepat/ic |
Categories of Medical Terms
All medical terms are divided into two basic categories:
- terms built from word parts
- terms NOT built from word parts
Table: 1.2 Categories of Medical Terms
| Category | Origin | Example | Learning Method |
| Terms Built from Word Parts | Word parts of Greek and Latin origin combined to create literal translations |
1. cytogenic
2. dysplasia |
1. Analyzing terms
2. Defining terms 3. Building terms |
| Terms NOT built from Word Parts |
1. Eponyms
2. Acronyms 3. Modern Language 4. Terms from Greek and Latin word parts that cannot be easily translated to find their meanings |
1. Parkinson’s Disease
2. AIDS (acquired immunodeficiency syndrome) 3. posttraumatic stress disorder 4. orthopedics |
1. Recalling terms
2. Matching terms 3. Defining terms |
Medical terms are also further divided into:
- Disease and Disorder – These are terms that describe any harmful deviation from the normal structural or functional state of an organism, generally associated with certain signs and symptoms. It does not include physical injury. An example of a disease is bronchopneumonia, which is a diseased state of the bronchi and lungs.
- Diagnostic/Procedural – These are terms related to the process of identifying a disease, condition, or injury from its signs and symptoms. An example of a diagnostic term is transrectal ultrasound, which is an ultrasound procedure used to diagnose prostate cancer.
- Therapeutic – These are terms regarding treating or curing of diseases. An example of a therapeutic term is nebulizer, which is a device that creates a mist used to deliver medication for respiratory treatment.
- Anatomical – These are terms used to describe specific areas and movement of the body as well as the relation of body parts to each other. An example of an anatomical term is medial, which describes the middle or direction toward the middle of the body.
Language Rules
Language rules are a good place to start when building a medical terminology foundation. Many medical terms are built from word parts and can be translated literally. At first, literal translations sound awkward. Once you build a medical vocabulary and become proficient at using it, the awkwardness will slip away. For example, suffixes will no longer be stated and will be assumed. The definition of intravenous then becomes within the vein .
Since you are at the beginning of building your medical terminology foundation stay literal when applicable. It should be noted that as with all language rules there are always exceptions and we refer to those as rebels.
By the end of this resource, you will have identified hundreds of word parts within medical terms. Here are some common medical terms that many non-medically trained people may be familiar with.
Examples
Osteoarthritis
Oste/o/arthr/itis – Inflammation of bone and joint.
Oste/o is a combining form that means bone
arthr/o is a combining form that means joint
-itis is a suffix that means inflammation
Intravenous
Intra/ven/ous – Pertaining to within a vein.
Intra- is a prefix that means within
ven/o – is a combining form that means vein
-ous is a suffix that means pertaining to
When breaking down words place slashes between word parts and a slash on each side of a combining vowel. Notice how the term is defined by beginning with the meaning of the suffix, then shifts to the beginning of the term with the meaning of the word parts in the order they appear.
Pronunciation
Special marking above vowels indicate the proper pronunciation of the vowel. When you see a macron (or straight line) above the vowel, that means the vowel sound is long. A u-shaped symbol above a vowel indicates a short vowel sound. CAPITAL letters indicate where to place the emphasis when pronouncing a word. The table below provides examples, try pronouncing them aloud.
Table 1.3 Combining Vowels Tips
| Guidelines | Examples |
| 1. Words are distorted minimally to indicate the proper phonetic sound. | doctor (dŏk-tŏr) |
| 2. The macron (line above the letter) indicated a long vowel sound. |
prorate (prō-rāt)
As in the: ā in play ē in be ī in wine ō in go ū in mule |
| 3. A u-shaped mark above the vowel indicates a short sound. |
medical (mĕd-ĭ-căl)
As in the: ă in mad ĕ in bet ĭ in tip ŏ in mop ŭ in cup |
| 4. Primary emphasis is indicated by capital letters. |
debride (di-BRĒD)
dehydration (dē-hī-DRĀ-shŏn) |
Review
Table 1.4 Combining Vowels Tips
| Combining Vowel Tip | Example |
| 1. Use a combining vowel when connecting a word root to a suffix that begins with a consonant. | arthr/o/ p athy |
| 2. DO NOT use a combining vowel if the suffix begins with a vowel. | hepat/ic |
| 3. Use a combining vowel when connection two word roots, even if vowels are present at the conjunction. | oste/o/arthr/itis |
| 4. DO NOT use a combining vowel when connect a prefix and a word root. | sub/hepat/ic |
Table 1.5 Word Parts and Combining Forms
| Part | Definition | Example |
| Word Root | Core of the word | hepat/ itis |
| Suffix | Attached to the end of a word root to alter its meaning | hepat/ itis |
| Prefix | Attached to the beginning of the word root to alter its meaning | sub/ hepatic |
| Combining Vowel | Typically an “o” used to assist pronunciation | hepat/ o /magaly |
| Combining Form | Word root with a combining vowel | hepat/o |
Table 1.6 Medical Terminology Learning Techniques
Technique |
Instructions |
Example |
| Analyze |
1. Divide into word parts
2. Label the word parts 3. Underline and label each combining form |
1. oste/o/arthr/o/pathy
2. oste =WR/ o= CV/ arthr = WR/ o = CV/ pathy = S 3 . oste/o / arthr/o /pathy |
| Define |
1. Define each word part in the term
2. Begin defining the suffix meaning and then move to the beginning of the term |
1. oste = bone, arthr = joint, pathy= disease
2. Disease of the bone and joint |
| Build |
1. Place word parts together to add terms
2. Add combining vowels |
1.
Disease
of the
bone
and
joint =
oste/ /arthr/ /pathy 2. oste/ o /arthr/ o /pathy |


5.9.5: Chapter Review
-
- Last updated
- Save as PDF
Statements and Quantifiers
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{1}\)
The ______________ of a logical statement has the opposite truth value of the original statement.
Exercise \(\PageIndex{2}\)
_______________ are logical statements presented as the facts used to support the conclusion of a logical argument.
Determine whether each of the following sentences represents a logical statement, also called a proposition. If it is a logical statement, determine whether it is true or false.
Exercise \(\PageIndex{3}\)
Where is the restroom?
Exercise \(\PageIndex{4}\)
No even numbers are odd numbers.
Exercise \(\PageIndex{5}\)
4 + 3 = 8.
Write the negation of each following statement symbolically and in words.
Exercise \(\PageIndex{6}\)
\(\text{~}p{:}\) Pink Floyd’s album The Wall is not a rock opera.
Exercise \(\PageIndex{7}\)
\(q{:}\) Some dogs are Labrador retrievers.
Exercise \(\PageIndex{8}\)
\(\text{~}r{:}\) Some universities are not expensive.
Draw a logical conclusion to the following arguments, and include in both one of the following quantifiers: all, some, or none.
Exercise \(\PageIndex{9}\)
Spaghetti noodles are made with wheat, ramen noodles are made with wheat, and lo mein noodles are made with wheat.
Exercise \(\PageIndex{10}\)
A Porsche Boxster does not have four doors, a Volkswagen Beetle does not have four doors, and a Mazda Miata does not have four doors.
Compound Statements
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{11}\)
___________________ are words or symbols used to join two or more logical statement together to from a compound statement.
Exercise \(\PageIndex{12}\)
__________________ and __________________ have equal dominance and are evaluated from left to right when no parentheses are present in a compound logical statement.
Translate each compound statement below into symbolic form.
Exercise \(\PageIndex{13}\)
If Tweety Bird is a bird, then Sylvester will not chase him.
Exercise \(\PageIndex{14}\)
Tweety Bird is a bird and Sylvester chases him if and only if Bugs says, “What’s up Doc?”
Translate the symbolic form of each compound logical statement below into words.

Exercise \(\PageIndex{15}\)
\(\sim q \vee p \rightarrow \sim s\)
Exercise \(\PageIndex{16}\)
\(\sim(p \wedge \sim s) \leftrightarrow q \rightarrow r\)
For each of the following compound logical statements, apply the proper dominance of connectives by adding parentheses to indicate the order to evaluate the statement.
Exercise \(\PageIndex{17}\)
\(p \vee q \wedge r \rightarrow \sim s \wedge t\)
Exercise \(\PageIndex{18}\)
\(\sim p \rightarrow q \vee r \leftrightarrow p \wedge s \rightarrow \sim t\)
Constructing Truth Tables
Fill in the blanks to complete the sentences.
Exercise \(\PageIndex{19}\)
A ______________ is true if at least one of its component statements is true.
Exercise \(\PageIndex{20}\)
For a ____________________ to be true, all of its component statements must be true.
Given the statements, \(p{:}\) “No fish are mammals,” \(q{:}\) “All lions are cats,” and \(\text{~}r{:}\) “Some birds do not lay eggs,” construct a truth table to determine the truth value of each compound statement below.
Exercise \(\PageIndex{21}\)
\(p \wedge \sim r\)
Exercise \(\PageIndex{22}\)
\(\sim(p \vee \sim r)\)
Exercise \(\PageIndex{23}\)
\(\sim p \vee q \wedge \sim(\sim r)\)
Construct a truth table to analyze all the possible outcomes of the following statements, and determine whether the statements are valid.
Exercise \(\PageIndex{24}\)
\(\sim p \vee q \wedge p\)
Exercise \(\PageIndex{25}\)
\(\sim p \vee q \vee \sim q\)
Truth Tables for the Conditional and Biconditional
Fill in the blanks to complete the following sentences.
Exercise \(\PageIndex{26}\)
If the , \(p\), of a conditional statement is true, then the conclusion, \(q\), must also be true for the conditional statement \(p \rightarrow q\) to be true.
Exercise \(\PageIndex{27}\)
The biconditional statement \(p \leftrightarrow q\) is whenever the truth value of \(p\) matches the truth value of \(q\), otherwise it is
Complete the truth tables below to determine the truth value of the proposition in the last column.
Exercise \(\PageIndex{28}\)
| \(p\) | \(q\) | \(r\) | \(p \vee q\) | \( \sim(p \vee q) \) | \(\text{~}{r}\) | \(\sim(p \vee q) \rightarrow \sim r\) |
|---|---|---|---|---|---|---|
| F | F | T |
Exercise \(\PageIndex{29}\)
| \(p\) | \(q\) | \(\text{~}{q}\) | \(p \rightarrow q \) | \(\sim(p \rightarrow q)\) | \(p \wedge \sim q \) | \(\sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)\) |
|---|---|---|---|---|---|---|
| T | F |
Assume the following statements are true. \(p{:}\) “Poof is a baby fairy,” \(q{:}\) “Timmy Turner has fairly odd parents,” \(r{:}\) “Cosmo and Wanda will grant Timmy’s wishes,” and \(t{:}\) “Timmy Turner is 10 years old.” Translate each of the following statements into symbolic form, then determine its truth value.
Exercise \(\PageIndex{30}\)
If Timmy Turner is 10 years old and Poof is not a baby fairy, then Timmy Turner has fairly odd parents.
Exercise \(\PageIndex{31}\)
Cosmos and Wanda will not grant Timmy’s wishes if and only if Timmy Turner is 10 years old or he does not have fairly odd parents.
Exercise \(\PageIndex{32}\)
Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument.
\(\text{~}p \vee q{\text{ }} \leftrightarrow {\text{ }}\text{~}q \to {\text{ }}\text{~}p\)
Equivalent Statements
Fill in the blanks to complete the sentences below. 33. The _________________ is logically equivalent to the inverse \( \sim p \rightarrow \sim q \)\(\text{~}p \to\text{~}q.\)
Exercise \(\PageIndex{34}\)
The _________________ is logically equivalent to the conditional \(p \rightarrow q \)\(p \to q.\)
Use the conditional statement, \(p \rightarrow q \)\(p \to q{:}\) “If Novak makes the basket, then Novak’s team will win the game," to answer the following questions. 35. Write the conclusion of the conditional statement in words and label it appropriately.
Exercise \(\PageIndex{36}\)
Write the hypothesis of the conditional statement in words and label it appropriately.
Exercise \(\PageIndex{37}\)
Identify the following statement as the converse, inverse, or contrapositive: “If Novak does not make the basket, then his team will not win the game.”
Exercise \(\PageIndex{38}\)
Identify the following statement as the converse, inverse, or contrapositive: “If Novak’s team wins the game, then he made the basket.”
De Morgan’s Laws
Fill in the blanks to complete the sentences.
Exercise \(\PageIndex{39}\)
De Morgan’s Law for the negation of a disjunction states that \( \(\text{~}(p \vee q) \equiv \)____________.
Exercise \(\PageIndex{40}\)
De Morgan’s Law for the negation of a conjunctions states that _____________\( \(\equiv {\text{ }}\text{~}p{\text{ }} \vee \text{~}q.\)
Exercise \(\PageIndex{41}\)
Apply De Morgan’s Law to write the statement without parentheses: \( \(\text{~}\left( {\text{~}p \wedge q} \right)\).
Exercise \(\PageIndex{42}\)
Apply the property for the negation of a conditional to write the statement as a conjunction or disjunction: \( \(\text{~}(\text{~}p \wedge q \to\text{~}r)\).
Exercise \(\PageIndex{43}\)
Write the negation of the conditional statement in words: If Thomas Edison invented the phonograph, then albums are made of vinyl, or the transistor radio was the first portable music device.
Exercise \(\PageIndex{44}\)
Construct a truth table to verify that the logical property is valid: \(\text{~}\left( {\text{~}p \to\text{~}q} \right) \equiv \text{~}p \wedge q\).
Logical Arguments
Fill in the blanks to complete the sentences below.
Exercise \(\PageIndex{45}\)
The _____ ___ __________ is a valid logical argument with premises, \(p \to q\) and \(p\), used to support the conclusion, \(q.\)
Exercise \(\PageIndex{46}\)
The chain rule for conditional arguments states that the ___________________ property applies to conditional arguments, so that: \( \(\left( {p \to q} \right) \wedge \left( {q \to r} \right) \to \left( {p \to r} \right).\)
Assume each pair of statements represents true premises in a logical argument. Based on these premises, state a valid conclusion that is consistent with the form of the argument.
Exercise \(\PageIndex{47}\)
If the Tampa Bay Buccaneers did not win Super Bowl LV, then Tom Brady was not their quarterback. Tom Brady was the Tampa Bay Buccaneers quarterback.
Exercise \(\PageIndex{48}\)
If \(\text{~}q\), then \(p\) and if \(r\), then \(\text{~}q\).
Exercise \(\PageIndex{49}\)
If Kamala Harris is the vice president of the United States, then Kamala Harris is the president of the U.S. Senate. Kamala Harris is the vice president of the United States.
Exercise \(\PageIndex{50}\)
Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, determine whether it is sound.
If all frogs are brown, then Kermit is not a frog. Kermit is a frog. Therefore, some frogs are not brown.


3.3.5: Seamus Heaney (1939-2013)
Selected Poems Irish Contemporary Literature Born in 1939, Heaney grew up in a large Roman Catholic family on a farm in County Derry, near Belfast, in Northern Ireland. His rural upbringing provided him with an appreciation for the small details of rural life and for the land; these qualities would come to mark his poetry vividly. While he lived in a largely Protestant area that experienced the violent "troubles" between Catholic and Protestant militants, Heaney never advocated strongly for the Catholic cause in his poetry, an omission for which he was sometimes criticized. Heaney began publishing poetry as a student at Queen's University in Belfast, but his career as a poet really began with the 1966 publication of his first book of poems, Death of a Naturalist . Over the course of Heaney's career, he taught English at a number of Irish colleges, was Poet in Residence at Harvard University, and was Professor of Poetry at Oxford. After moving to Dublin, in The Republic of Ireland, in 1972, Heaney wrote his two arguably most political volumes of poetry, North (1975) and Field Work (1979). One of his best known and loved volumes, The Haw Lantern, was published after his mother's death in 1987. Heaney won the Nobel Prize for Literature in 1995, cited by the committee for his "works of lyrical beauty and ethical depth, which exalt everyday miracles and the living past." In 1999, he published a critically lauded translation of Beowulf . In 2006, he suffered a minor stroke; he documented this experience in his 2010 collection of poems Human Chain . "The Tollund Man," published in Heaney's 1972 collection Wintering Out , is the first of Heaney's famous "bog poems" inspired by the mummified bodies found by archeologists in Jutland. Heaney felt an affinity with the bodies as Northern Ireland is characterized by a number of bogs. This volume of poems is seen as one in which Heaney is attempting to work out the significance in his own life of the political troubles in his native Northern Ireland. "The Haw Lantern" is the titular poem of Heaney's 1987 collection published after his mother's death. Consider while reading:
- In Ireland, the haw fruit is a symbol of endurance and defiance of harsh winters. How does the poet use the fruit as a political symbol or a comment on the Catholic-Protestant violence?
- What is the story of Diogenes, and how does Heaney employ it in "The Haw Lantern"?
- In "The Tollund Man," how does Heaney draw a parallel between the ritualized killing of the man in the bog with victims of violence in Northern Ireland?
- How does Heaney use the poem "Tollund Man" to explore themes of fate and martyrdom?
Written by Anita Turlington


1.5: Dashes
selected template will load here
This action is not available.
The dash is kind of like the superhero of punctuation. Not only does it have a cool, sleek name, it’s often referred to as the “super comma.” But you can’t just go around using dashes because you want super commas. There are some rules you should follow, and you don’t want to overuse any one type of punctuation.
The following two rules should help you make good decisions regarding that dashing dash:
example
His ideas regarding an evacuation in case of a zombie attack are certainly controversial—even radical.
I think the reason those books became so popular—so much so that they became a cultural phenomenon—is that the world was ready for a little bit of magic.
2. You can also use dashes to set apart an appositive phrase or extra information that contains commas.
example
The changes that came with the arrival of the eleventh doctor—the screwdriver, the TARDIS, and all of the side characters—certainly surprised many fans.
Of course, while the rules related to using dashes are relatively simple, there seems to be great confusion over how the dash is actually created in word processing programs.
To be sure, the dash is no hyphen. The hyphen is smaller and comes between words like mother-in-law. The dash is longer and is created by placing two hyphens in a row. Then, after you space after the word after the dash, most word processing programs will turn those two little hyphens -- into a dash—like so.


2.4: Properties of the Real Numbers
Overview
- The Closure Properties
- The Commutative Properties
- The Associative Properties
- The Distributive Properties
- The Identity Properties
- The Inverse Properties
A property of a collection of objects is a characteristic that describes the collection. We shall now examine some of the properties of the collection of real numbers. The properties we will examine are expressed in terms of addition and multiplication.
The Closure Properties
If \(a\) and \(b\) are real numbers, then \(a + b\) is a unique real number, and \(a \cdot b\) is a unique real number.
For example, 3 and 11 are real numbers; \(3 + 11 = 14\) and \(3 \cdot 11 = 33\), and both 14 and 33 are real numbers. Although this property seems obvious, some collections are not closed under certain operations. For example,
- The real numbers are not closed under division since, although 5 and 0 are real numbers, \(5/0\) and \(0/0\) are not real numbers.
- The natural numbers are not closed under subtraction since, although 8 is a natural number, \(8 - 8\) is not. (\(8 - 8 = 0\) and 0 is not a natural number).
The Commutative Properties
Let \(a\) and \(b\) represent real numbers.
Commutative Property of Addition:
\(a + b = b + a\)
Commutative Property of Multiplication:
\(a \cdot b = b \cdot a\)
The commutative properties tell us that two numbers can be added or multiplied in any order without affecting the result.
Sample Set A
The following are examples of the commutative properties.
\(3 + 4 = 4 + 3\) Both equal 7
\(5 + x = x + 5\) Both represent the same sum.
\(4 \cdot 8 = 8 \cdot 4\) Both equal 32
\(y7 = 7y\) Both represent the same product
\(5(a + 1) = (a + 1)5\) Both represent the same product
\((x+4)(y+2) = (y+2)(x+4)\) Both represent the same product
Practice Set A
Fill in the ( ) with the proper number or letter so as to make the statement true. Use the commutative properties.
\(6 + 5 + ( ) + 6\)
- Answer
-
5
\(m + 12 = 12 + ( )\)
- Answer
-
\(m\)
\(9 \cdot 7 = ( ) \cdot 9\)
- Answer
-
7
\(6a = a( )\)
- Answer
-
6
\(4(k - 5) = ( )4\)
- Answer
-
\(k - 5\)
\((9a - 1)( ) = (2b + 7)(9a - 1)\)
- Answer
-
\(2b + 7\)
The Associative Properties
Let \(a\), \(b\), and \(c\) represent real numbers.
Associative Property of Addition:
\((a + b) + c = a + (b + c)\)
Associative Property of Multiplication:
\((ab)c = a(bc)\)
The associative properties tell us that we may group together the quantities as we please without affecting the result.
Sample Set B
The following examples show how the associative properties can be used.
\((2 + 6) + 1 = 2 + (6 + 1)\)
\(8 + 1 = 2 + 7\)
\(9 = 9\)
Both equal 9
\((3 + x) + 17 = 3 + (x + 17)\)
Both represent the same sum.
\((2 \cdot 3) \cdot 5 = 2 \cdot (3 \cdot 5)\)
\(6 \cdot 5 = 2 \cdot 15\)
\(30 = 30\)
Both equal 30
\((y)4 = 9(y4)\)
Both represent the same product.
Practice Set B
Fill in the ( ) to make each statement true. Use the associative properties.
\((9 + 2) + 5 = 9 + ( )\)
Solution
\(2 + 5\)
\(x + (5 + y) = ( ) + y\)
Solution
\(x + 5\)
\((11a)6 = 11( )\)
Solution
\(a \cdot 6\)
\([(7m - 2)(m + 3)](m + 4) = (7m - 2)[( ) ( )]\)
Solution
\((m + 3)(m + 4)\)
Sample Set C
Simplify (rearrange into a simpler form): \(5x6b8ac4\).
According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.
\(5 \cdot 6 \cdot 8 \cdot 4 \cdot x \cdot b \cdot a \cdot c\)
\(960xbac\) Multiply the numbers.
\(960abcx\) By convention, we will, when possible, write all letters in alphabetical order.
Practice Set C
Simplify each of the following quantities.
\(3a7y9d\)
Solution
\(189ady\)
\(6b8acz4\cdot5\)
Solution
\(960abcz\)
\(4p6qr3(a + b)\)
Solution
\(72pqr(a + b)\)
The Distributive Properties
When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.
\(4 + 4 + 4 = 3 \cdot 4\)
Notice that there are three 4's, that is, 4 appears 3 times. Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.
When a number \(a\) is added repeatedly \(n\) times, we have:
\(\underbrace{a+a+a+\cdots+a}_{a \text { appears } n \text { times }}\)
Then, using multiplication as a description for repeated addition, we can replace:
\(\underbrace{a+a+a+\cdots+a}_{n \text { times }}\) with \(na\)
For example:
\(x + x + x + x\) can be wrirten as \(4x\) since \(x\) is repeated added \(4\) times.
\(x + x + x + x = 4x\)
\(r + r\) can be written as \(2r\) since \(r\) is repeatedly added \(2\) times.
\(r + r = 2r\)
The distributive property involved both multiplication and addition. Let's rewrite \(4(a + b)\). We proceed by reading \(4(a + b)\) as a multiplication: 4 times the quantity \((a + b)\). This directs us to write:
$$
\begin{aligned}
4(a+b) &=(a+b)+(a+b)+(a+b)+(a+b) \\
&=a+b+a+b+a+b+a+b
\end{aligned}
\]
Now we use the commutative property of addition to collect all the \(a\)'s together and all the \(b\)'s together.
$$
4(a+b)=\underbrace{a+a+a+a}_{4 a^{\prime} s}+\underbrace{b+b+b+b}_{4 b^{\prime} s}
$$
Now, using multiplication as a description for repeated addition, we have
$$
4(a+b)=4 a+4 b
$$
We have distributed the 4 over the sum to both \(a\) and \(b\)
\(4(a+b)=4a+4b\)
\(a(b + c) = a\cdot b + a \cdot c\)
\((b + c)a = a \cdot b + a \cdot c\)
The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.
Sample Set D
Use the distributive property to rewrite each of the following quantities.
\(2(5+7)=2 \cdot 5+2 \cdot 7 \quad \text { Both equal } 24\)
\(6(x + 3) = 6 \cdot x + 6 \cdot 3\) Both represent the same number.
= \(6x + 18\)
\((z + 5)y = zy + 5y = yz + 5y\)
Practice Set D
What property of real numbers justifies:
\(a(b + c) = (b + c)a\)?
- Answer
-
the commutative property of multiplication
Use the distributive property to rewrite each of the following quantities.
\(3(2 + 1)\)
- Answer
-
6 + 3
\((x+6)7\)
- Answer
-
\(7x + 42\)
\(4(a + y)\)
- Answer
-
\(4a + 4y\)
\((9 + 2)a\)
- Answer
-
\(9a + 2a\)
\(a(x + 5)\)
- Answer
-
\(ax + 5a\)
\(1(x + y)\)
- Answer
-
\(x + y\)
The Identity Properties
Additive Identity
The number 0 is called the
additive identity
since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, \(6 + 0 = 6\)
Multiplicative Identity
The number 1 is called the
multiplicative identity
since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example \(6 \cdot 1 = 6\).
We summarize the identity properties as follows:
If \(a\) is a real number, then:
\(a + 0 = a\) and \(0 + a = a\)
If \(a\) is a real number, then:
\(a \cdot 1 = a\) and \(1 \cdot a = a\)
The Inverse Properties
Additive Inverses
When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to −3 the result is 0, that is, \(3+(−3)=0\). The numbers 3 and −3 are additive inverses of each other.
Multiplicative Inverses
When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when \(6\) and \(\dfrac{1}{6}\) are multiplied together, the result is 1, that is, \(6 \cdot 16 = 1\). The numbers 6 and \(\dfrac{1}{6}\) are multiplicative inverses of each other.
We summarize the inverse properties as follows:
The Inverse Properties:
1. If \(a\) is any real number, then there is a unique real number \(-a\), such that
\(a + (-a) = 0\) and \(-a + a = 0\)
The numbers \(a\) and \(-a\) are called
additive inverses
of each other.
2. If \(a\) is any nonzero real number, then there is a unique real number \(\dfrac{1}{a}\) such that
\(a \cdot \dfrac{1}{a} = 1\) and \(\dfrac{1}{a} \cdot a = 1\).
The numbers \(a\) and \(\dfrac{1}{a}\) are called
multiplicative inverses
of each other.
Expanding Quantities:
When we perform operations such as \(6(a + 3) = 6a + 18\), we say we are expanding the quantity \(6(a + 3)\).
Exercises
Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.
\(x + 3\)
- Answer
-
\(3+x\)
\(5 + y\)
\(10x\)
- Answer
-
\(x10\)
\(18z\)
\(r6\)
- Answer
-
\(6r\)
\(ax\)
\(xc\)
- Answer
-
\(cx\)
\(7(2 + b)\)
\(6(s + 1)\)
- Answer
-
\((s + 1)6\)
\((8 + a)(x + 6)\)
\((x + 16)(a + 7)\)
- Answer
-
\((a + 7)(x + 16)\)
\((x + y)(x - y)\)
\(0.06m\)
- Answer
-
\(m(0.06)\)
\(x + 3\)
- Answer
-
\(3+x\)
\(5(6h + 1)\)
- Answer
-
\((6h + 1)5\)
\(m(a + 2b)\)
\(k(10a - b)\)
- Answer
-
\((10a - b)k\)
\((21c)(0.008)\)
\((-16)(4)\)
- Answer
-
\((4)(-16)\)
\((5)(b - 6)\)
Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.
\(9x2y\)
- Answer
-
\(18xy\)
\(5a6b\)
\(2a3b4c\)
- Answer
-
\(24abc\)
\(5x10y5z\)
\(1u3r2z5m1n\)
- Answer
-
\(30mnruz\)
\(6d4e1f2(g + 2h)\)
\((\dfrac{1}{2})d(\dfrac{1}{4})e(\dfrac{1}{2})a\)
- Answer
-
\(\dfrac{1}{16}ade\)
\(3(a + 6)2(a - 9)6b\)
\(1(x + 2y)(6 + z)9(3x + 5y)\)
- Answer
-
\(9(x + 2y)(6 + z)(3x + 5y)\)
For the following problems, use the distributive property to expand the quantities.
\(2(y + 9)\)
\(b(r + 5)\)
- Answer
-
\(br + 5b\)
\(m(u + a)\)
\(k(j + 1)\)
- Answer
-
\(jk + k\)
\(x(2y + 5)\)
\(z(x + 9w)\)
- Answer
-
\(xz + 9wz\)
\((1 + d)e\)
\((8 + 2f)g\)
- Answer
-
\(8g + 2fg\)
\(c(2a + 10b)\)
\(15x(2y + 3z)\)
- Answer
-
\(30xy + 45xz\)
\(8y(12a + b)\)
\(z(x + y + m)\)
- Answer
-
\(xz + yz + mz\)
\((a + 6)(x + y)\)
\((x + 10)(a + b + c)\)
- Answer
-
\(zx + bx + cx + 10a + 10b + 10c\)
\(1(x + y)\)
\(1(a + 16)\)
- Answer
-
\(a + 16\)
\(0.48(0.34a + 0.61)\)
\(21.5(16.2a + 3.8b + 0.7c)\)
- Answer
-
\(348.3a + 81.7b + 15.05c\)
Exercises for Review
Find the value of \(4 \cdot 2 + 5(2 \cdot 4 - 6 \div 3) - 2 \cdot 5\).
Is the statement \(3(5 \cdot 3 - 3 \cdot 5) + 6 \cdot 2 - 3 \cdot 4 < 0\) true or false?
- Answer
-
false
Draw a number line that extends from \(-2\) to \(2\) and place points at all integers between and including \(-2\) and \(3\).
Replace the \(*\) with the appropriate relation symbol \((<,>)\). \(-7 * -3\).
- Answer
-
\(<\)
What whole numbers can replace \(x\) so that the statement \(-2 \le x < 2\) is true?


11.4: Relative Motion Analysis
Relative motion analysis is one method used to analyze bodies undergoing general planar motion . General planar motion is motion where bodies can both translate and rotate at the same time. Besides relative motion analysis, the alternative is absolute motion analysis. Either method can be used for any general planar motion problem, however one method may be significantly easier to apply in certain situations.
Absolute motion analysis will require calculus and is generally faster for simple problems and problems where only the velocities (and not accelerations) are required. Relative motion analysis will not require calculus, but does necessitate using multiple coordinate systems and is generally easier to use for more complex problems and problems where velocities and accelerations are being analyzed.
Utilizing Relative Motion Analysis
To start our discussion on relative motion analysis, we are going to imagine a simple robotic arm such as the one below. In this arm, we have two arm sections of fixed length, with motors causing rotations at joint A and joint B.
The first step in relative motion analysis is to break the motion down into simple steps and assign a coordinate system (with \(r\) and \(\theta\) directions) to each step in the chain of motion. We will always start at a fixed point and move step by step from there. In the case of our robotic arm, Joint A is the only point that will not be moving so we start there; then we have two cases of rotation without extension as we move from A to B, then B to C.
Since there are two steps to the motion, there will be two coordinate systems. The first coordinate system will be attached to member AB, with the \(r\) direction going from point A to point B. The \(\theta\) direction will then be ninety degrees counterclockwise from the \(r\) direction. The second coordinate system will be attached to member BC, with the \(r\) direction going from point B to point C. Again, the \(\theta\) direction is ninety degrees counterclockwise from the \(r\) direction. At this point it is usually good to identify the angles of each of the \(r\) and \(\theta\) directions with respect to ground. Below is a picture of the robotic arm with both coordinate systems drawn in.
For relative motion analysis, we can identify the velocity or acceleration of the end point of the arm (C) with respect to the ground (A) by finding the velocity acceleration of B with respect to A and adding the velocity or acceleration of C with respect to B, just as we did with particles.
\begin{align} \text{Velocity:} \quad &\, \vec{v}_{C / A} = \vec{v}_{B / A} + \vec{v}_{C / B} \\[4pt] \text{Acceleration:} \quad &\, \vec{a}_{C / A} = \vec{a}_{B / A} + \vec{a}_{B / C} \end{align}
The relative motion analysis equations above are for a two-part motion (as there are two sections to the arm in our example), but we can easily expand the above equation into three, four, or even more pieces for more complex mechanisms by adding more steps to the left side of our equation.
To use the above equations, we will need to plug in the information we know. Plugging in velocities or accelerations that are given as part of the problem for any particular points is a good place to start. If the velocity of points B or C were given, for example, we would plug that in for \(\vec{v}_{B / A}\) or \(\vec{v}_{C / A}\) respectively. Do remember that point A is our fixed ground point, so \(\vec{v}_{C / A}\) is the velocity of point C relative to the ground while \(\vec{v}_{C / B}\) is the velocity of point C with respect to point B. If any point is fixed (other than our original ground point, which will always be fixed), we can also plug in zeros for both velocity and acceleration of that point. Remember that this equation is a vector equation, so these velocities have both a magnitude and a direction.
To take into account rotation or extension of any individual pieces, we will need to look back to our kinematics equations for polar coordinator systems. Below are the equations we had for velocity and acceleration.
\begin{align} \text{Velocity:} \quad &\, v = \dot{r} \hat{u}_r + r \dot{\theta} \hat{u}_{\theta} \\[4pt] \text{Acceleration:} \quad &\, a = (\ddot{r} - r \dot{\theta}^2) \hat{u}_r + (2 \dot{r} \dot{\theta} + r \ddot{\theta}) \hat{u}_{\theta} \end{align}
In the above equations, \(r\) represents the length of the respective arm piece (the length of member AB or BC), \(\dot{r}\) represents the rate at which that length is increasing, and \(\ddot{r}\) represents the rate at which that rate is increasing. The \(\theta\) term represents current angle of the arm piece, \(\dot{\theta}\) is the angular velocity of that arm piece, and \(\ddot{\theta}\) is the angular acceleration of that arm piece. The \(u_r\) and \(u_{\theta}\) directions are the \(r\) and \(\theta\) directions that are attached to that particular section of the arm (in our earlier drawing, \(\hat{u}_{r_1}\) and \(\hat{u}_{\theta_1}\) were attached to member AB and \(\hat{u}_{r_2}\) and \(\hat{u}_{\theta_2}\) were attached to member BC).
Though it's certainly possible to have a mechanism that is rotating and extending at the same time, we will often have either just simple rotation, or just simple extension along a fixed direction. With simple rotation , the \(\dot{r}\) and \(\ddot{r}\) terms are zero, and our equations reduce down to the following.
\begin{align} \text{Velocity:} \quad &\, v = r \dot{\theta} \hat{u}_{\theta} \\[4pt] \text{Acceleration:} \quad &\, a = - r \dot{\theta}^2 \hat{u}_{r} + r \ddot{\theta} \hat{u}_{\theta} \end{align}
If we have simple extension , where the length of the piece is changing but there is no rotation, the \(\dot{\theta}\) and \(\ddot{\theta}\) terms would be zero and our original equations would reduce down to the following.
\begin{align} \text{Velocity:} \quad &\, v = \dot{r} \hat{u}_r \\ \text{Acceleration:} \quad &\, a = \ddot{r} \hat{u}_r \end{align}
If we use the appropriate set of equations for the type of motion in each step and plug in known quantities for angular velocities, angular accelerations, and rates or extension for each piece, we can add these pieces to our relative motion analysis equation from earlier. Again, these are vectors so be sure to indicate both the magnitudes and directions when we put them into the equation.
Once we have everything in our vector equation, we will break the vector equation into \(x\) and \(y\) components in order to solve for any unknowns. Simply find the angles or each of the \(r\) and \(\theta\) directions using your diagram and then use sines and cosines to break the individual vectors down into \(x\) and \(y\) components. Once you have everything in component form, you should be able to solve for any unknowns in your equations. As a note, it is often necessary to start with the velocity equations and solve for some unknowns there before moving on to the acceleration equations.
Alternate Notation for Rigid Body Relative Motion
In some cases, we want to analyze multiple points on the same or several rigid bodies. Another notation used for this is:
\begin{align} \text{Position:} \quad &\, \vec{r}_B = \vec{r}_A + \vec{r}_{B / A} \\[5pt] \text{Velocity:} \quad &\, \vec{v}_B = \vec{v}_A + \vec{v}_{B / A} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B / A} \\[5pt] \text{Acceleration:} \quad &\, \vec{a}_B = \vec{a}_A + \vec{a}_{B / A} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B / A} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{B / A}) \end{align}In this notation, A and B are two points on the same rigid body. When we use only one subscript, that indicates the position/velocity/acceleration of that point with respect to the fixed coordinate system. Because the body is rigid, the two points will not change the distance between them, but the position vector between them can change orientation (which is where the relative velocity between the two points comes from). \(\vec{\omega}\) is the angular velocity of that rigid body (analogous to \(\dot{\theta}\)), and \(\vec{\alpha}\) is the angular acceleration of that rigid body (analogous to \(\ddot{\theta}\)).
For planar motion, where the angular velocity vector (out of plane) is always perpendicular to the position vector (in the plane), the acceleration can be simplified to:
\[ \text{Acceleration:} \quad \, \vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B / A} - \omega^2 \vec{r}_{B / A} \]
If we consider the same robotic arm, we can translate our previous notation to the new rigid body notation:
\begin{align} \dot{\theta} \hat{k} &= \vec{\omega}_{AB} \\[4pt] \dot{\phi} \hat{k} &= \vec{\omega}_{BC} \\[4pt] \ddot{\theta} \hat{k} &= \vec{\alpha}_{AB} \\[4pt] \ddot{\phi} \hat{k} &= \vec{\alpha}_{BC} \end{align}
All vectors will be defined with respect to the \(x\) and \(y\) coordinates in the coordinate system shown, rather than the radial and theta directions as above.
If we want to find the velocity and acceleration of the end point, C, knowing the angular velocities and accelerations, we will find the following relationships:
\begin{align} \text{Velocity at point B:} \quad &\, \vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B / A} \\[5pt] \text{Velocity at point C:} \quad &\, \vec{v}_C = \vec{v}_B + \vec{\omega} \times \vec{r}_{C / B} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B / A} + \vec{\omega} \times \vec{r}_{C / B} \\[5pt] \text{Acceleration at point B:} \quad &\, \vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B / A} - \omega^2 \vec{r}_{B / A} \\[5pt] \text{Acceleration at point C:} \quad &\, \vec{a}_C = \vec{a}_B + \vec{\alpha} \times \vec{r}_{C / B} - \omega^2 \vec{r}_{C / B} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B / A} - \omega^2 \vec{r}_{B / A} + \vec{\alpha} \times \vec{r}_{C / B} - \omega^2 \vec{r}_{C / B} \end{align}
Example \(\PageIndex{1}\)
The robotic arm shown below has a fixed orange base at A and fixed length members AB and BC. Motors at A and B allow for rotational motion at the joints. Based on the angular velocities and accelerations shown at each joint, determine the velocity and the acceleration of the end effector at C.
- Solution
Example \(\PageIndex{2}\)
The robotic arm from the previous problem is in the configuration shown below. Assume that \(\theta\) is currently 30 degrees and that point C currently lies along the \(x\) axis. If we want the end effector at C to travel 1 ft/s in the negative \(x\)-direction, what should the angular velocities be at joints A and B?
- Solution
Example \(\PageIndex{3}\)
A ladder is propped up against a wall as shown below. If the base of the ladder is sliding out at a speed of 2 m/s, what is the speed of the top of the ladder?
- Solution
Example \(\PageIndex{4}\)
The crank-rocker mechanism as shown below consists of a crank with a radius of 0.5 meters rotating about its fixed center at C, at a constant rate of 2 rad/s clockwise. Rocker AB is fixed at its base at A and connects to point B along the edge of the crank. The pin at point B can slide along a frictionless slot in AB. In its current state, what is the angular velocity of rocker AB?
- Solution


5.5: Acceleration
Similarly to the way the velocity vector is determined from the position vector, the acceleration vector can be determined from the velocity vector (see Figure 5.5).
The (instantaneous) acceleration vector \(\overrightarrow{\boldsymbol{a}}_{i}\) is defined as the time derivative of the velocity vector:
\[\overrightarrow{\boldsymbol{a}}_{i}(t) \equiv \frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t} \equiv \lim _{\mathrm{d} t \rightarrow 0} \frac{\overrightarrow{\boldsymbol{v}}_{i}(t+\mathrm{d} t)-\overrightarrow{\boldsymbol{v}}_{i}(t)}{\mathrm{d} t} \tag{5.23} \label{5.23}\]
The absolute acceleration is defined as: \(a_{i}=\left|\overrightarrow{\boldsymbol{a}}_{i}\right|\). By taking the time derivative of equation (5.21) using the product rule it is found that:
\[\overrightarrow{\boldsymbol{a}}_{i}(t)=\frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t}=\frac{\mathrm{d} v_{s, i}}{\mathrm{~d} t} \hat{\boldsymbol{s}}+v_{s, i} \frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t} \tag{5.24} \label{5.24}\]
In contrast to the velocity vector, the acceleration vector is not tangential to the path curve \(\hat{\boldsymbol{s}}\), except if the path is straight \(\left(\frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t}=\overrightarrow{\mathbf{0}}\right)\), or if the speed is zero. Since the magnitude of \(\hat{\boldsymbol{s}}\) is constant, the vector \(\frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t}\) is perpendicular to \(\hat{\boldsymbol{s}}\). For that reason we have for the acceleration component tangential to the path curve \(a_{i, s, t}\) :
\[a_{s, i, t}=\overrightarrow{\boldsymbol{a}}_{i} \cdot \hat{\boldsymbol{s}}=\frac{\mathrm{d} v_{s, i}}{\mathrm{~d} t} \tag{5.25} \label{5.25}\]
This tangential component of the acceleration vector is also called the path acceleration \(a_{s, i}\) and can be different from the absolute value of the acceleration vector \(\left(\left|a_{s, i}\right| \neq\left|\overrightarrow{\boldsymbol{a}}_{i}\right|\right)\). We will discuss later how the acceleration vector in various coordinate systems can be determined, which will be very important for applying Newton’s second law.


3.1: Hildegard of Bingen
In this chapter you will
- learn about the medieval musical form known as plainsong or plainchant
- understand how rare and valuable it is to have such an extensive literary as well as musical output from a composer in the Middle Ages
Hildegard of Bingen played a leadership role in Christian monasticism; authored commentaries on biblical texts, books of her visions, and more; and composed a great deal of music that continues to be performed today. Although today she is well known for her music, historically her music was neglected and she was more famous as a mystic. She provides an important example of how composers drew on the Bible during the Middle Ages, when the form known as “plainsong” predominated. It is called that because voices sing in unison, in contrast with later choral music that often involves singers creating harmony by singing different notes at the same time. In the process of examining Hildegard’s music, we learn a great deal about how she and others in her time understood, interpreted, and otherwise interacted with the Bible.
Since there were already standard ways of chanting biblical texts, it is unsurprising that she focused her work as a composer on creating not only new music but new words. Her song lyrics are her own compositions and, at the same time, abound in allusions to the Bible. [1] As one example, we may consider a work such as “Ave Maria O Auctrix Vite,” or “Hail Mary, Authoress of Life.” The lyrics allude to the Hail Mary / Ave Maria, a prayer that has often been set to music and in turn is based on the angel’s greeting to Mary in the Gospel of Luke ( Luke 1:28 ). They also allude to the promise in Genesis 3 that the offspring of Eve would bruise the head of the serpent, interpreted as predicting the birth of Jesus and the defeat of Satan thereby. In songs such as these, we encounter biblical words interwoven with the composer’s own and learn a great deal about how the Bible was understood in that time. Elsewhere in the book we have explored works that combine or draw on multiple biblical texts in a single work of music and the way music often reflects not only the Bible but also the way it has been interpreted. Hildegard provides a distinctive perspective on this because of something that sets her apart from the other composers and songwriters included in this book and would still make her stand out even if somehow we managed to include all the world’s composers from every time and place. It is rare if not unique to have from the same author an extensive output of music on biblical themes, lengthy books of homilies and commentaries that indicate in prose how that same individual interpreted the Bible, letters, an autobiographical work, and much more. Hildegard, however, has left us precisely that, which is all the more striking given the distance in time that separates her from our own era in which print and, more recently, digital means of literary expression have made it possible for us to have much more from a composer than just their music. The illustrated manuscripts of the sermons she delivered to the nuns in her monasteries offer a glimpse of how the arts and theology were connected in the life of those medieval monastic communities. In addition to her many remarkable accomplishments, Hildegard is remarkable simply because of how much we know about her life when compared to other women, or indeed people in general, in the Middle Ages. [2]
Honey Meconi encapsulates just how distinctive Hildegard not only was but is:
Many know Hildegard today foremost or even exclusively as a composer, and her achievements in this area are indeed noteworthy: the most prolific composer of plainchant; one of the earliest—male or female—that we know by name; creator of the first musical “morality play” (and the only one for whom we have a named composer); and composer of seventy-seven songs, all but one set to her own idiosyncratic poetry in a distinctive and glorious musical style. But just as her music, all plainchant , distinguishes her from the many other feisty and creative spiritual women of the twelfth century, so, too, do her nonmusical accomplishments separate her from virtually all other major composers (whether of the Middle Ages or later times), few of whom are known for anything but composition, and almost none of whom are of equal significance in another field. Yet during her lifetime and until very recently it was not Hildegard’s music that led to her fame; rather, it was her spirituality. Indeed, Hildegard’s was a holistic life, and her music can only be understood as one facet of a creativity that mirrored and was generated by her religious beliefs. [3]
Works like O gloriosissimi lux vivens angeli (O most glorious living light of the angels) and O vos angeli (O you angels), while not settings of biblical texts, show how the influence of the Bible as understood in Hildegard’s Roman Catholic tradition, her visions as a mystic, and her compositions are connected. The melody of the former she attributed to a vision of angels that she had, and what she has to say about angels reflects the way Pope Gregory the Great interpreted certain parables in the Gospel of Luke. [4]
In addition to engaging with the Bible in her musical compositions and in her writings about the Bible, Hildegard also indicated (in response to an order from a superior that she and her community cease celebrating the chants of the Divine Offices) that she understood the prophetic Spirit whose voice we hear through the Bible to not merely permit but order that music and song be offered to God. [5]
Have a listen to more of Hildegard’s music , and perhaps read a bit more about her fascinating life while you do so! Try to at least take a glimpse at some of her impressive literary output. Works like her Solutions to Thirty-Eight Questions illustrate just what a learned and insightful interpreter of the Bible she was. [6]
For Further Reading
Bain, Jennifer. Hildegard of Bingen and Musical Reception: The Modern Revival of a Medieval Composer . Cambridge: Cambridge University Press, 2015.
Coakley, John W. Women, Men, and Spiritual Power: Female Saints and Their Male Collaborators . New York: Columbia University Press, 2006.
Fassler, Margot. “Composer and Dramatist: ‘Melodious Singing and the Freshness of Remorse.’” In Voice of the Living Light: Hildegard of Bingen and Her World , edited by Barbara Newman, 149–75. Berkeley: University of California Press, 1998.
Flanagan, Sabina. Hildegard of Bingen: A Visionary Life . London: Routledge, 1998.
Hildegard of Bingen. Homilies on the Gospels . Collegeville: Liturgical Press, 2011.
———. Symphonia: A Critical Edition of the “Symphonia Armonie Celestium Revelationum” (Symphony of the Harmony of Celestial Revelations) . New York: Cornell University Press, 1998.
Kienze, Beverly Mayne. Hildegard of Bingen and Her Gospel Homilies: Speaking New Mysteries . Turnhout, Belgium: Brepols, 2009.
———. Hildegard of Bingen, Gospel Interpreter . Lanham, MD: Lexington/Fortress, 2020.
Maddocks, Fiona. Hildegard of Bingen: The Woman of Her Age . New York: Doubleday, 2001.
Meconi, Honey. Hildegard of Bingen . Urbana: University of Illinois Press, 2018.
- In chapter 20, we focus on the subject of allusions . ↵
- Andrea Janelle Dickens, The Female Mystic: Great Women Thinkers of the Middle Ages (New York: I. B. Tauris, 2009), 26. ↵
- Honey Meconi, Hildegard of Bingen (Urbana: University of Illinois Press, 2018), 1–2. ↵
- On this, see Tova Leigh-Choate, William T. Flynn, and Margot E. Fassler, “Hearing the Heavenly Symphony: An Overview of Hildegard’s Musical Oeuvre with Case Studies,” in A Companion to Hildegard of Bingen , ed. Debra Stoudt, George Ferzoco, and Beverly Kienzle (Leiden: E. J. Brill, 2013), 176–77. Video of the recording by Sequentia released by deutsche harmonia mundi, on whose behalf it is licensed to YouTube by SME. ↵
- See the excerpt from a letter she sent to the prelate of Mainz around the year 1178, provided in Josiah Fisk, ed., Composers on Music: Eight Centuries of Writings (Boston: Northeastern University Press, 1997), 2–3. ↵
- On Hildegard’s educational background and her acceptance and authority as a mystic, see Gerda Lerner, The Creation of Feminist Consciousness: From the Middle Ages to Eighteen-Seventy (New York: Oxford University Press, 1993), 52–54. ↵


4.25: Working in Microsoft Excel
Create workbooks and format data in Microsoft Excel
Imagine that you have a lot of business data. Perhaps you have names and addresses for a mailing list. Maybe you have inventory data or quarterly sales values. All this information could be kept in a Word document, but Microsoft Office actually has an extremely useful program for organizing, storing, and even manipulating data: Microsoft Excel.
Learning to use Microsoft Excel is one of the most helpful and versatile workplace skills you can acquire, and creating a worksheet in a workbook is the first step. Many of the skills you learned for Microsoft Word can also be applied to Microsoft Excel, such as basic text formatting and file extensions. The file extension for a Microsoft Excel workbook is .xlsx , although pre-2003 versions of Excel might use .xls .
In this page, we’ll focus on the manipulation of data, rather than the appearance of the worksheet. Additionally, while this page only provides one method of completing each task, there can be multiple ways to accomplish a single goal. For more in-depth instruction, check out this online course covering the basics of Microsoft Excel.
Using Excel
Before using a workbook, it is helpful to know a few key terms.
- Cell. This is the area where you will enter data.
- Row. Rows are cells aligned horizontally.
- Column. Columns are cells aligned vertically.
- Worksheet. A worksheet is a single page within a workbook. Like the tabs in an internet browser, the tabs in an Excel workbook show different pages, or worksheets. A workbook may have many worksheets included in it. In this screenshot, the workbook only has one worksheet and one tab, which is labeled Sheet1. The selected tab shows the selected worksheet. Clicking the + button will add another worksheet. When you save a workbook in Excel, all of the worksheets in that workbook are saved.
Comma Styles
At times, you may also wish to use a specific comma style with numbers entered into an Excel worksheet. For example, you may wish “1234” to display as typed or with a comma like “1,234.”
Comma styles are easy to change in Excel using a quick select option in the Number group in the ribbon. Simply to click on the Comma Style button in the Number group.
When clicking the comma style button, the comma style default is to display numbers with a comma in the thousands place and include two decimal places (Ex: “1200” becomes “1,200.00). This will also change the visible cell styles in the Style” area of the ribbon so you can easily select different options for comma and display format.
Listed below are the three most common options for comma and display format.
- Comma : Comma with two decimal points (e.g., 1,234.00)
- Comma [0] : Comma with no decimal points (e.g., 1,234)
- Currency : Comma with two decimal points and a dollar sign (e.g., $1,234.00)
Cell Format
As mentioned previously, Excel will default to certain styles when you create a new worksheet. In particular, this includes the way that numbers are displayed and whether or not commas are automatically included. In this section, we will take a look at changing these defaults.
When you type numbers into an Excel workbook, it will often default to a specific format. For example, if you type “12/15/17,” Excel will convert this to read “12/15/2017,” assuming you were entering month, day, and abbreviated year. Similarly, “3/4” will display at “4-Mar,” the fourth day of March. However, it is possible that you may have been entering fractions, so “3/4” was meant to indicate three-quarters instead.
If this is the case, you will need to format your cells to properly display the information you are entering. When possible, consider formatting your cells before you enter the data. Otherwise, Excel may convert some of the entries and you will need to re-enter that information.
- Begin by highlighting the cells you plan to use.
- Select the Format dropdown from the Cells group of the ribbon.
- Select the Format cells option at the bottom of the dropdown menu.
Flash Fill
Like many modern software programs, Excel is designed to recognize certain patterns. For example, perhaps you are creating a table that lists the last and first names of attendees at a company training session. After all the names have been entered into two separate columns, you realize you would like a single column to correctly display the full name. An easy way to achieve this without having to manually retype the entire list is to use Flash Fill.
- Create a new column for the combined information you wish to display.
- In the first cell, type the name as you wish it to display. In our screenshots, this would be “John Smith.”
- Begin typing the next piece of data in the next cell. Excel should automatically suggest a Flash Fill option.
- If the Flash Fill suggestion matches how you would like the information displayed, simply hit the Enter key and the rest of your column should fill in automatically.
Flash Fill is especially helpful if your data is initially in different forms but you want the final information to display in the same fashion. For example, in our attendee list, some of the names were capitalized, in all caps, or had no capitalization. Sometimes you may need to manually update more than one option but Excel will detect your pattern.
Flash Fill should automatically be turned on in Excel but if it is not, you can turn it on using the File > Options > Advanced menus. You can also turn Flash Fill on or off using the shortcut Ctrl+E. Be aware that the Mac version of Excel does not have Flash Fill.
SUM Data
One of the main uses for Excel is to organize and manipulate numerical data. Often you may wish to add up all the numbers in a column or row. Excel has formulas and commands to automatically add your data, and the easiest way to use this feature is the AutoSum button.
- Once your numbers are organized in either a row or column, click on the cell where you would like the total sum to display. In the screenshot below this was A13.
- Click on the AutoSum button from the Editing group of the ribbon.
- Excel will highlight the cells that it is adding up and will apply the SUM formula.
- Hit Enter to accept the highlighted cells and see the total value of your data.
Note that it is possible to SUM several columns (or rows) at once. Select all the cells you wish to display a SUM and click AutoSum. Excel will individually add up the columns.
Sorting Data
- Select the column or row you wish to sort.
- From the Sort & Filter button in the Editing group in the ribbon, click the Sort button.
- From the menu, choose how you would like to sort the data. For example, A to Z or Z to A. Note that A to Z is equivalent to Smallest to Largest and Z to A is equivalent to Largest to Smallest.
Filtering Data
After entering data in Excel, it is also possible to filter, or hide some parts of the data, based on user-indicated categories. When using the Filter option, no data is lost; it is just hidden from view.
- Select the column or row you wish to sort.
- From the Sort & Filter button in the Editing group in the ribbon, click the Filter button.
- When the Filter menu appears, you can choose which categories of data to hide and deselect the appropriate buttons. For example, you can deselect the button next to large and you will no longer see the large cells in your table.
Functions
Excel can perform a variety of really nice data analysis features for you. We’ve already touched upon how you can filter data. But you can also look for other connections, or screen large numbers of cells to determine how often something occurs.
COUNTIF
COUNTIF is a way for you to ask Excel to count how many times a certain piece of information appears in your worksheet. For example, perhaps you want to know how often “shirt” appear in an inventory list. All you need to do is ask Excel to count the number of cells that contain the word “shirt.”
- Determine which cells you want Excel to look at. In our example, we will look at A2 though A13.
- Click on the cell you wish your count to be displayed in.
-
Type the formula for a count
=COUNTIF(A2:A13, “shirt”)
Here you are telling Excel which cells to examine—A2 through A13—and what to look for: “shirt.” Note that your text must match exactly what is typed in the cells, and if you are looking for a specific word it needs to be enclosed in quotation marks (so “shirt” instead of shirt).
- Hit enter and your results will appear.
IF
Another commonly used function in Excel is the “IF” function. In this case, you are asking Excel to look for something and then tell you if that something occurred. For example, perhaps you want to compare whether your monthly expenses were under your monthly budget. That is the scenario we will look at in our example.
In this case, let us just ask for a simple “yes” or “no” answer. Looking at the screenshot below, you can see how the worksheet has all the data at hand. We are looking for whether the information in the C column is less than the information in the B column. We would like the D column to display the answer (yes or no).
-
Click on D2 and enter the “IF” function for what you want Excel to compare and do.
=IF(C2<B2, “Yes”,”No”)
- You do not have to manually reenter the formula into the other cells in D. Instead you can copy and paste the formula from D2 into D3, D4, and so on. Each time you do this, the formula should automatically update with the correct cell number to compare.
As you can see, the D cells begin to display “Yes” or “No.” “Yes” means that the expenses in the C column were less than the monthly budget entered into the B column. “No” means that expenses were higher than the budget. Just as in the COUNTIF function, you need to enclose text in quotation marks.
Clustered Column Charts
Excel is not just used for organizing and processing data and formulas. It also can be used to visually represent data in the form of charts and graphs. In this page, we will work on creating a basic chart, the clustered column chart, and then modifying a chart style.
A clustered column chart is sometimes called a bar graph, because it shows data organized in solid shapes like pillars. A clustered column chart organizes these pillars up and down, so they are “columns.” On the other hand, a clustered bar graph organizes these pillars left to right, so they are “bars.” Bar graphs are useful charts when looking at changes from month to month or across employees.
The first step to creating any chart is to organize your data. It is definitely a good idea to include headers in the first cell of each column. By default, a clustered column chart will cluster the data by the columns in your table, so try to keep that in mind when setting up the worksheet.
- After organizing your data, select the cells you wish to include in the chart. This should be at least two columns.
- Click on the Insert tab and find the Charts group of the ribbon.
- “Clustered column chart” is actually a recommended chart. Click on that chart.
- When you select the chart, you will see colored boxes surrounding the data that connect to the different categories of the chart.
Chart Styles
Once you have created a chart, or if you are given a worksheet that contains a chart, it is very easy to change the chart style.
- Click on the chart you wish to change. The Design tab should appear in the ribbon area.
- Click on Change Chart Type button
- Click on the type of chart you would like.
From this same window, you can also switch the data that is being charted. For example, you can switch which data from a row to a column or change which data is arranged on the x- or y-axis.
Conditional Formatting
As we have learned so far, Excel has a wide variety of easy to use tools for organizing, sorting, and otherwise marking information. Think back to when we applied styles to a cell to indicate good information or information that needs to be verified. Excel also has the ability to automatically apply such markings through conditional formatting.
With conditional formatting, you provide Excel with a rule, such as “less than 10,” and the program will scan through your data and highlight all the cells that meet that rule. There are several rules already available, but you can also create and apply your own rules and visual clues.
- Select the cells, rows, or columns you wish to have conditional formatting.
- From the Styles group, click on the Conditional Formatting button
- Select the style of formatting you would like. Here we have Highlight Cell Rules.
- Select the specific type of rule you would like to use and then apply your target value. Here we have selected Less Than.
- The formatting will appear automatically so you can see what it will look like. Note that Excel will automatically provide a value, but you can manually change it.
- Hit OK if you wish to apply the formatting. Otherwise, when you leave the formatting menu, it will disappear.
One important tool to keep in mind is the ability to enter your own rules. This can include applying formatting to specific date ranges, to specific text (like names), or even cells that are blank. In this case, you also set the format, so instead of highlighting cells you can choose to strikethrough text or change the font, change the size, or bold the text.