3.3.7: Exercises
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Skills
1) In the following Peruvian counting board, determine how many of each item is represented. Please show all of your calculations along with some kind of explanation of how you got your answer. Note the key at the bottom of the drawing.
2) Draw a quipu with a main cord that has branches (H cords) that show each of the following numbers on them. (You should produce one drawing for this problem with the cord for part a on the left and moving to the right for parts b through d .)
| a. 232 | b. 5065 |
| c. 23451 | d. 3002 |
| 3) 423 in base 5 to base 10 | 4) 3044 in base 5 to base 10 |
| 5) 387 in base 10 to base 5 | 6) 2546 in base 10 to base 5 |
| 7) 110101 in base 2 to base 10 | 8) 11010001 in base 2 to base 10 |
| 9) 100 in base 10 to base 2 | 10) 2933 in base 10 to base 2 |
| 11) Convert 653 in base 7 to base 10 | 12) Convert 653 in base 10 to base 7 |
| 13) 3412 in base 5 to base 2 | 14) 10011011 in base 2 to base 5 |
(Hint: convert first to base 10 then to the final desired base)
The Caidoz System
Suppose you were to discover an ancient base-12 positional system made up twelve symbols. Let’s call this base system the Caidoz system. Here are the symbols for each of the numbers 0 through 12:
Convert each of the following numbers in Caidoz to base 10
C onver t the following base 10 numbers to Caidoz, using the symbols sho wn above.
| 19) 175 | 20) 3030 |
| 21) 10000 | 22) 5507 |
Convert the following numbers to Mayan notation. Show your calculations used to get your answers.
| 23) 135 | 24) 234 |
| 25) 360 | 26) 1215 |
| 27) 10500 | 28) 1100000 |
Convert the following Mayan numbers to decimal (base -10) numbers. Show all calculations.
James Bidwell has suggested that Mayan addition was done by “simply combining bars and dots and carrying to the next higher place.” He goes on to say, “After the combining of dots and bars, the second step is to exchange every five dots for one bar in the same position.” After converting the following base 10 numbers into vertical Maya notation (in base 20, of course), perform the indicated addition:
| 33) 32 + 11 | 34) 82 + 15 |
| 35) 35 + 148 | 36) 2412 + 5000 |
| 37) 450 + 844 | 38) 10000 + 20000 |
| 39) 4500 + 3500 | 40) 130000 + 30000 |
41) Use the fact that the Mayans had a base-20 number system to complete the following multiplication table. The table entries should be in Mayan notation. Remember: Their zero looked like this… . Xerox and then cut out the table below, fill it in, and paste it onto your homework assignment if you do not want to duplicate the table with a ruler.
(To think about but not write up: Bidwell claims that only these entries are needed for “Mayan multiplication.” What does he mean?)
Binary and Hexadecimal Conversions
Modern computers operate in a world of “on” and “off” electronic switches, so use a binary counting system – base 2, consisting of only two digits: 0 and 1.
Convert the following binary numbers to decimal (base -10) numbers.
| 42) 1001 | 43) 1101 |
| 44) 110010 | 45) 101110 |
Convert the following base-10 numbers to binary
| 46) 7 | 47) 12 |
| 48) 36 | 49) 27 |
Four binary digits together can represent any base-10 number from 0 to 15. To create a more human-readable representation of binary-coded numbers, hexadecimal numbers, base 16, are commonly used. Instead of using the 8,13,12 16 notation used earlier, the letter A is used to represent the digit 10, B for 11, up to F for 15, so 8,13,12 16 would be written as 8DC.
Convert the following hexadecimal numbers to decimal (base -10) numbers.
| 50) C3 | 51) 4D |
| 52) 3A6 | 53) BC2 |
Convert the following base-10 numbers to hexadecimal
| 54) 152 | 55) 176 |
| 56) 2034 | 57) 8263 |
Exploration
58) What are the advantages and disadvantages of bases other than ten.
59) Supposed you are charged with creating a base-15 number system. What symbols would you use for your system and why? Explain with at least two specific examples how you would convert between your base-15 system and the decimal system.
60) Describe an interesting aspect of Mayan civilization that we did not discuss in class. Your findings must come from some source such as an encyclopedia article, or internet site and you must provide reference(s) of the materials you used (either the publishing information or Internet address).
61) For a Papuan tribe in southeast New Guinea, it was necessary to translate the bible passage John 5:5 “And a certain man was there, which had an infirmity 30 and 8 years” into “A man lay ill one man, both hands, five and three years.” Based on your own understanding of bases systems (and some common sense), furnish an explanation of the translation . Please use complete sentences to do so. (Hint: To do this problem, I am asking you to think about how base systems work, where they come from, and how they are used. You won’t necessarily find an “answer” in readings or such…you’ll have to think it through and come up with a reasonable response. Just make sure that you clearly explain why the passage was translated the way that it was.)
62) The Mayan calendar was largely discussed leading up to December 2012. Research how the Mayan calendar works, and how the counts are related to the number based they use.


Section 5: Truth in QL
For SL, we split the definition of truth into two parts: a truth value assignment (\(a\)) for sentence letters and a truth function (\(v\)) for all sentences. The truth function covered the way that complex sentences could be built out of sentence letters and connectives.
In the same way that truth for SL is always truth given a truth value assignment , truth for QL is truth in a mode l. The simplest atomic sentence of QL consists of a one-place predicate followed by a constant, like \(Pj\). It is true in a model \(\mathbb{M}\) if and only if the referent of \(j\) is in the extension of \(P\) in \(\mathbb{M}\).
We could go on in this way to define truth for all atomic sentences that contain only predicates and constants: Consider any sentence of the form \(\mathcal{RC}\) 1 ...\(\mathcal{c}\) n where \(\mathcal{R}\) is an n-place predicate and the \(\mathcal{c}\)s are constants. It is true in \(\mathbb{M}\) if and only if hreferent(\(\mathcal{c}\) 1 ),...,referent(\(\mathcal{c}\) n )i is in extension(\(\mathcal{R}\)) in \(\mathbb{M}\).
We could then define truth for sentences built up with sentential connectives in the same way we did for SL. For example, the sentence (\(Pj\) → \(Mda\)) is true in \(\mathbb{M}\) if either \(Pj\) is false in \(\mathbb{M}\) or \(Mda\) is true in \(\mathbb{M}\).
Unfortunately, this approach will fail when we consider sentences containing quantifiers. Consider ∀\(xPx\). When is it true in a model \(\mathbb{M}\)? The answer cannot depend on whether \(Px\) is true or false in \(\mathbb{M}\), because the \(x\) in \(Px\) is a free variable. \(Px\) is not a sentence. It is neither true nor false.
We were able to give a recursive definition of truth for SL because every wellformed formula of SL has a truth value. This is not true in QL, so we cannot define truth by starting with the truth of atomic sentences and building up. We also need to consider the atomic formulae which are not sentences. In order to do this we will define satisfaction; every well-formed formula of QL will be satisfied or not satisfied, even if it does not have a truth value. We will then be able to define truth for sentences of QL in terms of satisfaction.
Satisfaction
The formula \(Px\) says, roughly, that \(x\) is one of the \(Ps\). This cannot be quite right, however, because \(x\) is a variable and not a constant. It does not name any particular member of the UD. Instead, its meaning in a sentence is determined by the quantifier that binds it. The variable \(x\) must stand-in for every member of the UD in the sentence ∀\(xPx\), but it only needs to stand-in for one member in ∃\(xPx\). Since we want the definition of satisfaction to cover \(Px\) without any quantifier whatsoever, we will start by saying how to interpret a free variable like the \(x\) in \(Px\).
We do this by introducing a variable assignment. Formally, this is a function that matches up each variable with a member of the UD. Call this function ‘\(\a\).’ (The ‘\(a\)’ is for ‘assignment’, but this is not the same as the truth value assignment that we used in defining truth for SL.)
The formula \(Px\) is satisfied in a model \(\mathbb{M}\) by a variable assignment \(a\) if and only if \(a\)(\(x\)), the object that \(a\) assigns to \(x\), is in the the extension of \(P\) in \(\mathbb{M}\).
When is ∀\(xPx\) satisfied? It is not enough if \(Px\) is satisfied in \(\mathbb{M}\) by \(a\), because that just means that \(a\)(\(x\)) is in extension(\(P\)). ∀\(xPx\) requires that every other member of the UD be in extension(\(P\)) as well.
So we need another bit of technical notation: For any member Ω of the UD and any variable \(x\), let a[Ω|\(x\)] be the variable assignment that assigns Ω to \(x\) but agrees with \(a\) in all other respects. We have used Ω, the Greek letter Omega, to underscore the fact that it is some member of the UD and not some symbol of QL. Suppose, for example, that the UD is presidents of the United States. The function \(a\)[Grover Cleveland|\(x\)] assigns Grover Cleveland to the variable \(x\), regardless of what \(a\) assigns to \(x\); for any other variable, \(a\)[Grover Cleveland|\(x\)] agrees with \(a\).
We can now say concisely that ∀\(xPx\) is satisfied in a model \(\mathbb{M}\) by a variable assignment \(a\) if and only if, for every object Ω in the UD of \(\mathbb{M}\), \(Px\) is satisfied in \(\mathbb{M}\) by \(a\)[Ω|\(x\)].
You may worry that this is circular, because it gives the satisfaction conditions for the sentence ∀\(xPx\) using the phrase ‘for every object.’ However, it is important to remember the difference between a logical symbol like ‘∀’ and an English language word like ‘every.’ The word is part of the metalanguage that we use in defining satisfaction conditions for object language sentences that contain the symbol.
We can now give a general definition of satisfaction, extending from the cases we have already discussed. We define a function \(s\) (for ‘satisfaction’) in a model \(\mathbb{M}\) such that for any wff \(\mathcal{A}\) and variable assignment \(a\), s(\(\mathcal{A}\),\(a\)) = 1 if \(\mathcal{A}\) is satisfied in \(\mathbb{M}\) by \(a\); otherwise s(\(\mathcal{A}\),\(a\)) = 0.
1. If \(\mathcal{A}\) is an atomic wff of the form \(\mathcal{Pt}\) 1 ...\(\mathcal{t}\) n and Ω i is the object picked out by \(t\) i , then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } \left\langle \Omega _ { 1 } \ldots \Omega _ { n } \right\rangle \text { is in extension } ( \mathcal { P } ) \text { in } \mathbb { M } } \\ { 0 } & { \text { otherwise. } } \end{array} \right. \)
For each term \(t\) i : If \(t\) i is a constant, then Ω i = referent(\(t\) i ). If \(t\) i is a variable, then Ω i = \(a\)(\(t\) i ).
2. If \(\mathcal{A}\) is ¬\(\mathcal{B}\) for some wff \(\mathcal{B}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } s ( \mathcal { B } , a ) = 0 } \\ { 0 } & { \text { otherwise } } \end{array} \right. \)
3. If \(\mathcal{A}\) is (\(\mathcal{B}\) & \(\mathcal{C}\)) for some wffs \(\mathcal{B}\),\(\mathcal{C}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } s ( \mathcal { B } , a ) = 1 \text { and } s ( C , a ) = 1 } \\ { 0 } & { \text { otherwise } } \end{array} \right. \)
4. If \(\mathcal{A}\) is (\(\mathcal{B}\)∨\(\mathcal{C}\)) for some wffs \(\mathcal{B}\),\(\mathcal{C}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 0 } & { \text { if } s ( \mathcal { B } , a ) = 0 \text { and } s ( C , a ) = 0 } \\ { 1 } & { \text { otherwise } } \end{array} \right. \)
5. If \(\mathcal{A}\) is (\(\mathcal{B}\) → \(\mathcal{C}\)) for some wffs \(\mathcal{B}\),\(\mathcal{C}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 0 } & { \text { if } s ( \mathcal { B } , a ) = 1 \text { and } s ( C , a ) = 0 } \\ { 1 } & { \text { otherwise } } \end{array} \right. \)
6. If \(\mathcal{A}\) is (\(\mathcal{B}\) ↔ \(\mathcal{C}\)) for some sentences \(\mathcal{B}\),\(\mathcal{C}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } s ( \mathcal { B } , a ) = s ( C , a ) } \\ { 0 } & { \text { otherwise } } \end{array} \right. \)
7. If \(\mathcal{A}\) is ∀\(\mathcal{xB}\) for some wff \(\mathcal{B}\) and some variable \(\mathcal{x}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } s ( \mathcal { B } , a [ \Omega | x ] ) = 1 \text { for every member } \Omega \text { of the UD, } } \\ { 0 } & { \text { otherwise. } } \end{array} \right. \)
8. If \(\mathcal{A}\) is ∃\(\mathcal{xB}\) for some wff \(\mathcal{B}\) and some variable \(\mathcal{x}\), then
\( s ( \mathcal { A } , a ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } s ( \mathcal { B } , a [ \Omega | x ] ) = 1 \text { for at least one member } \Omega \text { of the UD, } } \\ { 0 } & { \text { otherwise. } } \end{array} \right. \)
This definition follows the same structure as the definition of a wff for QL, so we know that every wff of QL will be covered by this definition. For a model \(\mathbb{M}\) and a variable assignment \(a\), any wff will either be satisfied or not. No wffs are left out or assigned conﬂicting values.
Truth
Consider a simple sentence like∀\(xPx\). By part 7 in the definition of satisfaction, this sentence is satisfied if \(a\)[Ω|\(x\)] satisfies \(Px\) in \(\mathbb{M}\) for every Ω in the UD. By part 1 of the definition, this will be the case if every Ω is in the extension of \(P\). Whether ∀\(xPx\) is satisfied does not depend on the particular variable assignment \(a\). If this sentence is satisfied, then it is true. This is a formalization of what we have said all along: ∀\(xPx\) is true if everything in the UD is in the extension of \(P\).
The same thing holds for any sentence of QL. Because all of the variables are bound, a sentence is satisfied or not regardless of the details of the variable assignment. So we can define truth in this way: A sentence \(\mathcal{A}\) is true in \(\mathbb{M}\) if and only if some variable assignment satisfies \(\mathcal{A}\) in \(M\); \(\mathcal{A}\) is false in \(\mathbb{M}\) otherwise.
Truth in QL is truth in a model . Sentences of QL are not ﬂat-footedly true or false as mere symbols, but only relative to a model. A model provides the meaning of the symbols, insofar as it makes any difference to truth and falsity.
Reasoning about all models (reprise)
At the end of section 5.4, we were stymied when we tried to show that∀\(x\)(\(Rxx\) → \(Rxx\)) is a tautology. Having defined satisfaction, we can now reason in this way:
Consider some arbitrary model \(\mathbb{M}\). Now consider an arbitrary member of the UD; for the sake of convenience, call it Ω. It must be the case either that <Ω,Ω> is in the extension of \(R\) or that it is not. If <Ω,Ω> is in the extension of \(R\), then \(Rxx\) is satisfied by a variable assignment that assigns Ω to \(x\) (by part 1 of the definition of satisfaction); since the consequent of \(Rxx\) → \(Rxx\) is satisfied, the conditional is satisfied (by part 5). If <Ω,Ω> is not in the extension of \(R\), then \(Rxx\) is not satisfied by a variable assignment that assigns Ω to \(x\) (by part 1); since antecedent of \(Rxx\) → \(Rxx\) is not satisfied, the conditional is satisfied (by part 5). In either case, \(Rxx\) → \(Rxx\) is satisfied. This is true for any member of the UD, so ∀\(x\)(\(Rxx\) → \(Rxx\)) is satisfied by any truth value assignment (by part 7). So ∀\(x\)(\(Rxx\) → \(Rxx\)) is true in \(\mathbb{M}\) (by the definition of truth). This argument holds regardless of the exact UD and regardless of the exact extension of \(R\), so ∀\(x\)(\(Rxx\) → \(Rxx\)) is true in any model. Therefore, it is a tautology.
Giving arguments about all possible models typically requires clever combination of two strategies:
1. Divide cases between two possible kinds, such that every case must be one kind or the other. In the argument on p. 92, for example, we distinguished two kinds of models based on whether or not a specific ordered pair was in extension(\(R\)). In the argument above, we distinguished cases in which an ordered pair was in extension(\(R\)) and cases in which it was not.
2. Consider an arbitrary object as a way of showing something more general. In the argument above, it was crucial that Ω was just some arbitrary member of the UD. We did not assume anything special about it. As such, whatever we could show to hold of Ω must hold of every member of the UD— if we could show it for Ω, we could show it for anything. In the same way, we did not assume anything special about \(\mathbb{M}\), and so whatever we could show about \(\mathbb{M}\) must hold for all models.
Consider one more example. The argument ∀\(x\)(\(Hx\)&\(Jx\)) .˙.∀\(xHx\) is obviously valid. We can only show that the argument is valid by considering what must be true in every model in which the premise is true.
Consider an arbitrary model \(\mathbb{M}\) in which the premise ∀\(x\)(\(Hx\)&\(Jx\)) is true. The conjunction \(Hx\)&\(Jx\) is satisfied regardless of what is assigned to \(x\), so \(Hx\) must be also (by part 3 of the definition of satisfaction). As such, (∀\(x\))\(Hx\) is satisfied by any variable assignment (by part 7 of the definition of satisfaction) and true in \(\mathbb{M}\) (by the definition of truth). Since we did not assume anything about \(\mathbb{M}\) besides ∀\(x\)(\(Hx\)&\(Jx\)) being true, (∀\(x\))\(Hx\) must be true in any model in which ∀\(x\)(\(Hx\)&\(Jx\)) is true. So ∀\(x\)(\(Hx\)&\(Jx\)) |= ∀\(xHx\).
Even for a simple argument like this one, the reasoning is somewhat complicated. For longer arguments, the reasoning can be insufferable. The problem arises because talking about an infinity of models requires reasoning things out in English. What are we to do?
We might try to formalize our reasoning about models, codifying the divide-andconquer strategies that we used above. This approach, originally called semantic tableaux , was developed in the 1950s by Evert Beth and Jaakko Hintikka. Their tableaux are now more commonly called truth trees .
A more traditional approach is to consider deductive arguments as proofs. A
proof system
consists of rules that formally distinguish between legitimate and illegitimate arguments— without considering models or the meanings of the symbols. In the next chapter, we develop proof systems for SL and QL.


4.14: Directional Terms Worksheet 2 Answers
Insert the missing directional terms:
- 1. The spleen is ventral to the kidneys
- 2. The hock is distal to the stifle.
- 3. The adrenal gland is cranial to the kidney.
- 4. The thymus is cranial to the heart.
- 5. The thyroid is cranial/ventral to the trachea.
- 6. The stomach is caudal to the heart.
- 7. The withers region is dorsal to the sternum.
- 8. The stifle is proximal to the fetlock
- 9. The lumbar vertebrae are cranial to the sacrum
- 10. The sacrum is medial/dorsal to the pelvic girdle
- 11. The fibula is caudal/lateral to the tibia
- 12. The metacarpals are distal to the carpals
- 13. The pericardium is lateral to the heart
- 14. The diaphragm is caudal to the lungs
- 15. The hair follicle is lateral to the hair shaft
- 16. The eyeball is medial to the orbit
- 17. The mandible is caudal/ventral to the tongue
- 18. The tibia is distal to the femur
- 19. The radius is proximal to the metacarpals
- 20. The uterus is caudal/ventral to the intestines


27.4C: Somite Development
Somites develop from the paraxial mesoderm and participate in the facilitation of multiple developmental processes.
- Describe the functions of somites
Key Points
- The paraxial mesoderm is distinct from the mesoderm found more internally in the embryo.
- Alongside the neural tube, the mesoderm develops distinct paired structures called somites that develop into dermis, skeletal muscle, and vertebrae.
- Each somite has four compartments: the sclerotome, myotome, dermatome, and the syndetome. Each becomes a specific tissue during development.
Key Terms
- neural crest cells : A transient, multipotent, migratory cell population that gives rise to a diverse cell lineage including melanocytes, craniofacial cartilage, bone, smooth muscle, peripheral and enteric neurons, and glia.
- conceptus : The fetus or embryo, including all the surrounding tissues protecting and nourishing it during pregnancy.
- somite : One of the paired masses of mesoderm, distributed along the sides of the neural tube, that will eventually become dermis, skeletal muscle, or vertebrae.
Intraembryonic Coelom Development
In the development of the human embryo the intraembryonic coelom (or somatic coelom) is a portion of the conceptus that forms in the mesoderm. During the second week of development the lateral mesoderm splits into a dorsal somatic mesoderm (somatopleure) and a ventral splanchnic mesoderm (splanchnopleure).
By the third week of development, this process gives rise to a cavity between the somatopleure and splanchnopleure referred to as the intraembryonic celom. This space later gives rise to both the thoracic and abdominal cavities.
Somite Development
In the developing vertebrate embryo, somites are masses of mesoderm that can be found distributed along the two sides of the neural tube. They will eventually become dermis (dermatome), skeletal muscle (myotome), vertebrae (sclerotome), and tendons and cartilage (syndetome).
The mesoderm found lateral to the neural tube is called the paraxial mesoderm. It is separate from the chordamesoderm underneath the neural tube. The paraxial mesoderm is initially called the unsegmented
mesoderm in vertebrates, but is called the segmented mesoderm in chick embryos.
Somites : A dorsal view of a human embryo. The repetitive somites are marked with the older term primitive segments.
As the primitive streak regresses and the neural folds gather preceding the formation of the neural tube, the paraxial mesoderm divides into blocks called somites. Somites play a critical role in early development by participating in the specification of the migration paths of neural crest cells and spinal nerve axons.
Later in development, somites separate into four compartments:
Dermatome
The dermatome is the dorsal portion of the paraxial mesoderm somite. In the human embryo it arises in the third week of embryogenesis.
The dermatomes contribute to the skin, fat, and connective tissue of the neck and of the trunk, though most of the skin is derived from the lateral plate mesoderm.
Myotome
The myotome is that part of a somite that forms the muscles. Each myotome divides into an epaxial part (epimere), at the back, and a hypaxial part (hypomere) at the front.
The myoblasts from the hypaxial division form the muscles of the thoracic and anterior abdominal walls. The epaxial muscle mass loses its segmental character to form the extensor muscles of the neck and trunk of mammals.
Sclerotome
The sclerotome forms the vertebrae and the rib cartilage and part of the occipital bone. It forms the musculature of the back, the ribs, and the limbs.
Syndetome
The syndetome forms the tendons and some blood vessels.


12.5: Binomial Theorem
By the end of this section, you will be able to:
- Use Pascal’s Triangle to expand a binomial
- Evaluate a binomial coefficient
- Use the Binomial Theorem to expand a binomial
Before you get started, take this readiness quiz.
-
Simplify: \(\frac{7 \cdot 6 \cdot 5 \cdot 4}{4 \cdot 3 \cdot 2 \cdot 1}\).
If you missed this problem, review Example 1.25. -
Expand: \((3 x+5)^{2}\).
If you missed this problem, review Example 5.32. -
Expand: \((x-y)^{2}\).
If you missed this problem, review Example 5.32.
Use Pascal's Triangle to Expand a Binomial
In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded \((a+b)^{2}\).
\((a+b)^{2}=a^{2}+2 a b+b^{2}\)
To expand \((a+b)^{3}\), we recognize that this is \((a+b)^{2}(a+b)\) and multiply.
\((a+b)^{3}\)
\((a+b)^{2}(a+b)\)
\(\left(a^{2}+2 a b+b^{2}\right)(a+b)\)
\(a^{3}+2 a^{2} b+a b^{2}+a^{2} b+2 a b^{2}+b^{3}\)
\(a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\)
\((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\)
To find a method that is less tedious that will work for higher expansions like \((a+b)^{7}\), we again look for patterns in some expansions.
| Number of Terms | First Term | Last Term | |
|---|---|---|---|
| \((a+b)^{1}=a+b\) | \(2\) | \(a^{1}\) | \(b^{1}\) |
| \((a+b)^{2}=a^{2}+2 a b+b^{2}\) | \(3\) | \(a^{2}\) | \(b^{2}\) |
| \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) | \(4\) | \(a^{3}\) | \(b^{3}\) |
| \((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}\) | \(5\) | \(a^{4}\) | \(b^{4}\) |
| \((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\) | \(6\) | \(a^{5}\) | \(b^{5}\) |
| \((a+b)^{n}\) | \(n\) | \(a^{n}\) | \(b^{n}\) |
Notice the first and last terms show only one variable. Recall that \(a^{0}=1\), so we could rewrite the first and last terms to include both variables. For example, we could expand \((a+b)^{3}\) to show each term with both variables.
Generally, we don’t show the zero exponents, just as we usually write \(x\) rather than \(1x\).
Patterns in the Expansion of \((a+b)^{n}\)
- The number of terms is \(n+1\).
- The first term is \(a^{n}\) and the last term is \(b^{n}\).
- The exponents on \(a\) decrease by one on each term going left to right.
- The exponents on \(b\) increase by one on each term going left to right.
- The sum of the exponents on any term is \(n\).
Let’s look at an example to highlight the last three patterns.
From the patterns we identified, we see the variables in the expansion of \((a+b)^{n}\), would be
\((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+\ldots+\_\_\_a^{1}b^{n-1}+b^{n}\).
To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.
The array to the right is called Pascal’s Triangle . Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
This triangle gives the coefficients of the terms when we expand binomials.
Pascal's Triangle
In the next example, we will use this triangle and the patterns we recognized to expand the binomial.
Use Pascal's Triangle to expand \((x+y)^{6}\).
Solution :
We know the variables for this expansion will follow the pattern we identified. The nonzero exponents of \(x\) will start at six and decrease to one. The nonzero exponents of \(y\) will start at one and increase to six. The sum of the exponents in each term will be six. In our pattern, \(a=x\) and \(b=y\).
\(\begin{array}{l}{(a+b)^{n}=a^{n}+\_\_\_a^{n-1} b^{1}+\_\_\_a^{n-2} b^{2}+\ldots+\_\_\_a^{1}b^{n-1}+b^{n}} \\ {(x+y)^{6}=x^{6}+\_\_\_x^{5} y^{1}+\_\_\_x^{4} y^{2}+\_\_\_x^{3} y^{3}+\_\_\_x^{2} y^{4}+\_\_\_x^{1} y^{5}+y^{6}}\end{array}\)
Use Pascal’s Triangle to expand \((x+y)^{5}\).
- Answer
-
\(\begin{array}{l}{x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3}} {+5 x y^{4}+y^{5}}\end{array}\)
Use Pascal’s Triangle to expand \((p+q)^{7}\).
- Answer
-
\(\begin{array}{c}{p^{7}+7 p^{6} q+21 p^{5} q^{2}+35 p^{4} q^{3}} {+35 p^{3} q^{4}+21 p^{2} q^{5}+7 p q^{6}+q^{7}}\end{array}\)
In the next example we want to expand a binomial with one variable and one constant. We need to identify the \(a\) and \(b\) to carefully apply the pattern.
Use Pascal’s Triangle to expand \((x+3)^{5}\).
Solution :
We identify the \(a\) and \(b\) of the pattern.
In our pattern, \(a=x\) and \(b=3\).
We know the variables for this expansion will follow the pattern we identified. The sum of the exponents in each term will be five.
\((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+\ldots+\_\_\_a^{1}b^{n-1}+b^{n} \)
\((x+3)^{5}=x^{5}+\_\_\_x^{4}\cdot3^{1}+\_\_\_x^{3}\cdot3^{2}+\_\_\_x^{2}\cdot3^{3}+\_\_\_x^{1}\cdot3^{4}+3^{5}\)
Use Pascal's Triangle to expand \((x+2)^{4}\).
- Answer
-
\(x^{4}+8 x^{3}+24 x^{2}+32 x+16\)
Use Pascal's Triangle to expand \((x+1)^{6}\).
- Answer
-
\(\begin{array}{l}{x^{6}+6 x^{5}+15 x^{4}+20 x^{3}+15 x^{2}} {+6 x+1}\end{array}\)
In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the \(a\) and \(b\) of the pattern, we must once again carefully apply the pattern.
Use Pascal's Triangle to expand \((3x-2)^{4}\).
Solution :
We identify the \(a\) and \(b\) of the pattern.
In our pattern, \(a=3x\) and \(b=-2\).
\((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+\ldots+\_\_\_a^{1}b^{n-1}+b^{n} \)
\((3 x-2 )^{4}=1 \cdot\left(\stackrel{3}{x}+4(3 x)^{3}(-2)^{1}+6(3 x)^{2}(-2)^{2}+4(3 x)^{1}(-2)^{3}+1 \cdot(-2)^{4}\right.\)
\((3 x-2)^{4}=81 x^{4}+4\left(27 x^{3}\right)(-2)+6\left(9 x^{2}\right)(4)+4(3 x)(-8)+1 \cdot 16\)
\((3 x-2 )^{4}=81 x^{4}-216 x^{3}+216 x^{2}-96 x+16\)
Use Pascal's Triangle to expand \((2x-3)^{4}\).
- Answer
-
\(16 x^{4}-96 x^{3}+216 x^{2}-216 x+81\)
Use Pascal's Triangle to expand \((2x-1)^{6}\).
- Answer
-
\(\begin{array}{l}{64 x^{6}-192 x^{5}+240 x^{4}-160 x^{3}} {+60 x^{2}-12 x+1}\end{array}\)
Evaluate a Binomial Coefficient
While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation . This notation is not only used to expand binomials, but also in the study and use of probability.
To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation \(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)\) which is called a binomial coefficient . We read \(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)\) as “\(n\) choose \(r\)” or “\(n\) taken \(r\) at a time”.
A binomial coefficient \(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)\), where \(r\) and \(b\) are integers with \(0 \leq r \leq n\), is defined as
\(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)=\frac{n !}{r !(n-r) !}\)
We read \(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)\) as "\(n\) choose \(r\)" or "\(n\) taken \(r\) at a time."
Evaluate:
- \(\left( \begin{array}{l}{5} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{7} \\ {7}\end{array}\right)\)
- \(\left( \begin{array}{l}{4} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{8} \\ {5}\end{array}\right)\)
Solution :
a. We will use the definition of a binomial coefficient,
\(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)=\frac{n !}{r !(n-r) !}\)
\(\left( \begin{array}{l}{5} \\ {1}\end{array}\right)\)
Use the definition, \(\left( \stackrel{5}{1}\right)=\frac{n !}{r !(n-r) !}\), where \(n=5, r=1\).
\(\frac{5 !}{1 !(5-1) !}\)
Simplify.
\(\frac{5 !}{1 !(4) !}\)
Rewrite \(5!\) as \(5\cdot 4!\)
\(\frac{5 \cdot 4 !}{1 ! \cdot 4 !}\)
Simplify, by removing common factors.
\(\frac{5\cdot \cancel{4 !}}{1 ! \cdot \cancel{4 !}}\)
Simplify.
\(5\)
\(\left( \begin{array}{l}{5} \\ {1}\end{array}\right)=5\)
b. \(\left( \begin{array}{l}{7} \\ {7}\end{array}\right)\)
Use the definition, \(\left( \stackrel{5}{1}\right)=\frac{n !}{r !(n-r) !}\), where \(n=7, r=7\).
\(\frac{7 !}{7 !(7-7) !}\)
Simplify.
\(\frac{7 !}{7 !(0) !}\)
Simplify. Remember \(0!=1\).
\(1\)
\(\left( \begin{array}{l}{7} \\ {7}\end{array}\right)=1\)
c. \(\left( \begin{array}{l}{4} \\ {0}\end{array}\right)\)
Use the definition, \(\left( \stackrel{5}{1}\right)=\frac{n !}{r !(n-r) !}\), where \(n=4, r=0\).
\(\frac{4 !}{0 !(4-0) !}\)
Simplify.
\(\frac{4 !}{0 !(4) !}\)
Simplify.
\(1\)
\(\left( \begin{array}{l}{4} \\ {0}\end{array}\right)=1\)
d. \(\left( \begin{array}{l}{8} \\ {5}\end{array}\right)\)
Use the definition, \(\left( \stackrel{5}{1}\right)=\frac{n !}{r !(n-r) !}\), where \(n=8, r=5\).
\(\frac{8 !}{5 !(8-5) !}\)
Simplify.
\(\frac{8 !}{5 !(3) !}\)
Rewrite \(8!\) as \(8\cdot 7\cdot 6\cdot 5!\) and remove common factors.
\(\frac{8\cdot7\cdot\cancel{6}\cdot\cancel{5!}}{\cancel{5!}\cdot\cancel{3}\cdot\cancel{2}\cdot1}\)
Simplify.
\(56\)
\(\left( \begin{array}{l}{8} \\ {5}\end{array}\right)=56\)
Evaluate each binomial coefficient:
- \(\left( \begin{array}{l}{6} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{8} \\ {8}\end{array}\right)\)
- \(\left( \begin{array}{l}{5} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{7} \\ {3}\end{array}\right)\)
- Answer
-
- \(6\)
- \(1\)
- \(1\)
- \(35\)
Evaluate each binomial coefficient:
- \(\left( \begin{array}{l}{2} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{11} \\ {11}\end{array}\right)\)
- \(\left( \begin{array}{l}{9} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{6} \\ {5}\end{array}\right)\)
- Answer
-
- \(2\)
- \(1\)
- \(1\)
- \(6\)
In the previous example, \((a)\), \((b)\), \((c)\) demonstrate some special properties of binomial coefficients.
Properties of Binomial Coefficients
\(\left( \begin{array}{l}{n} \\ {1}\end{array}\right)=n \quad \left( \begin{array}{l}{n} \\ {n}\end{array}\right)=1 \quad \left( \begin{array}{l}{n} \\ {0}\end{array}\right)=1\)
Use the Binomial Theorem to Expand a Binomial
We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.
Binomial Theorem
For any real numbers \(a\) and \(b\), and positive integer \(n\),
\((a+b)^{n}=\left( \begin{array}{c}{n} \\ {0}\end{array}\right) a^{n}+\left( \begin{array}{c}{n} \\ {1}\end{array}\right) a^{n-1} b^{1}+\left( \begin{array}{c}{n} \\ {2}\end{array}\right) a^{n-2} b^{2}+\ldots+\left( \begin{array}{c}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}+\ldots+\left( \begin{array}{c}{n} \\ {n}\end{array}\right) b^{n}\)
Use the Binomial Theorem to expand \((p+q)^{4}\).
Solution :
We identify the \(a\) and \(b\) of the pattern.
In our pattern, \(a=p\) and \(b=q\).
We use the Binomial Theorem.
\((a+b)^{n}=\left( \begin{array}{c}{n} \\ {0}\end{array}\right) a^{n}+\left( \begin{array}{c}{n} \\ {1}\end{array}\right) a^{n-1} b^{1}+\left( \begin{array}{c}{n} \\ {2}\end{array}\right) a^{n-2} b^{2}+\ldots+\left( \begin{array}{c}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}+\ldots+\left( \begin{array}{c}{n} \\ {n}\end{array}\right) b^{n}\)
Substitute in the values \(a=p, b=q\) and \(n=4\).
\((p+q)^{4}=\left( \begin{array}{c}{4} \\ {0}\end{array}\right) p^{4}+\left( \begin{array}{c}{4} \\ {1}\end{array}\right) p^{4-1} q^{1}+\left( \begin{array}{c}{4} \\ {2}\end{array}\right) p^{4-2} q^{2}+\left( \begin{array}{c}{4} \\ {3}\end{array}\right) p^{4-3} q^{3}+\left( \begin{array}{c}{4} \\ {4}\end{array}\right) q^{4}\)
Simplify the exponents.
\((p+q)^{4}=\left( \begin{array}{l}{4} \\ {0}\end{array}\right) p^{4}+\left( \begin{array}{c}{4} \\ {1}\end{array}\right) p^{3} q+\left( \begin{array}{c}{4} \\ {2}\end{array}\right) p^{2} q^{2}+\left( \begin{array}{c}{4} \\ {3}\end{array}\right) p q^{3}+\left( \begin{array}{c}{4} \\ {4}\end{array}\right) q^{4}\)
Evaluate the coefficients, remember, \(\left( \begin{array}{l}{n} \\ {1}\end{array}\right)=n, \left( \begin{array}{l}{n} \\ {n}\end{array}\right)=1, \left( \begin{array}{l}{n} \\ {0}\end{array}\right)=1\)
\((p+q)^{4}=1 p^{4}+4 p^{3} q^{1}+\frac{4 !}{2 !(2) !} p^{2} q^{2}+\frac{4 !}{3 !(4-3) !} p^{1} q^{3}+1 q^{4}\)
\((p+q)^{4}=p^{4}+4 p^{3} q+6 p^{2} q^{2}+4 p q^{3}+q^{4}\)
Use the Binomial Theorem to expand \((x+y)^{5}\).
- Answer
-
\(\begin{array}{l}{x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3}} {+5 x y^{4}+y^{5}}\end{array}\)
Use the Binomial Theorem to expand \((m+n)^{6}\).
- Answer
-
\(\begin{array}{l}{m^{6}+6 m^{5} n+15 m^{4} n^{2}+20 m^{3} n^{3}} {+15 m^{2} n^{4}+6 m n^{5}+n^{6}}\end{array}\)
Notice that when we expanded \((p+q)^{4}\) in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal's Triangle .
The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.
Use the Binomial Theorem to expand \((x-2)^{5}\).
Solution :
We identify the \(a\) and \(b\) of the pattern.
In our pattern, \(a=x\) and \(b=-2\).
We use the Binomial Theorem.
\((a+b)^{n}=\left( \begin{array}{c}{n} \\ {0}\end{array}\right) a^{n}+\left( \begin{array}{c}{n} \\ {1}\end{array}\right) a^{n-1} b^{1}+\left( \begin{array}{c}{n} \\ {2}\end{array}\right) a^{n-2} b^{2}+\ldots+\left( \begin{array}{c}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}+\ldots+\left( \begin{array}{c}{n} \\ {n}\end{array}\right) b^{n}\)
Substitute in the values \(a=x, b=-2\), and \(n=5\).
\((x-2)^{5}=\left( \begin{array}{l}{5} \\ {0}\end{array}\right) x^{5}+\left( \begin{array}{c}{5} \\ {1}\end{array}\right) x^{5-1}(-2)^{1}+\left( \begin{array}{c}{5} \\ {2}\end{array}\right) x^{5-2}(-2)^{2}+\left( \begin{array}{c}{5} \\ {3}\end{array}\right) x^{5-3}(-2)^{3}+\left( \begin{array}{c}{5} \\ {4}\end{array}\right) x^{5-4}(-2)^{4}+\left( \begin{array}{c}{5} \\ {5}\end{array}\right)(-2)^{5}\)
Simplify the coefficients. Remember, \(\left( \begin{array}{l}{n} \\ {1}\end{array}\right)=n, \left( \begin{array}{l}{n} \\ {n}\end{array}\right)=1, \left( \begin{array}{l}{n} \\ {0}\end{array}\right)=1\).
\((x-2)^{5}=\left( \begin{array}{l}{5} \\ {0}\end{array}\right) x^{5}+\left( \begin{array}{c}{5} \\ {1}\end{array}\right) x^{4}(-2)+\left( \begin{array}{c}{5} \\ {2}\end{array}\right) x^{3}(-2)^{2}+\left( \begin{array}{c}{5} \\ {3}\end{array}\right) x^{2}(-2)^{3}+\left( \begin{array}{c}{5} \\ {4}\end{array}\right) x(-2)^{4}+\left( \begin{array}{c}{5} \\ {5}\end{array}\right)(-2)^{5}\)
\((x-2)^{5}=1 x^{5}+5(-2) x^{4}+\frac{5 !}{2 ! \cdot 3 !}(-2)^{2} x^{3}+\frac{5 !}{3 ! 2 !}(-2)^{3} x^{2}+\frac{5 !}{4 !1 !}(-2)^{4} x+1(-2)^{5}\)
\((x-2)^{5}=x^{5}+5(-2) x^{4}+10 \cdot 4 \cdot x^{3}+10(-8) x^{2}+5 \cdot 16 \cdot x+1(-32)\)
\((x-2)^{5}=x^{5}-10 x^{4}+40 x^{3}-80 x^{2}+80 x-32\)
Use the Binomial Theorem to expand \((x-3)^{5}\).
- Answer
-
\(\begin{array}{l}{x^{5}-15 x^{4}+90 x^{3}-270 x^{2}} {+405 x-243}\end{array}\)
Use the Binomial Theorem to expand \((y-1)^{6}\).
- Answer
-
\(\begin{array}{l}{y^{6}-6 y^{5}+15 y^{4}-20 y^{3}+15 y^{2}} {-6 y+1}\end{array}\)
Things can get messy when both terms have a coefficient and a variable.
Use the Binomial Theorem to expand \((2x-3y)^{4}\).
Solution :
We identify the \(a\) and \(b\) of the pattern.
In our pattern, \(a=2x\) and \(b=-3y\).
We use the Binomial Theorem.
\((a+b)^{n}=\left( \begin{array}{c}{n} \\ {0}\end{array}\right) a^{n}+\left( \begin{array}{c}{n} \\ {1}\end{array}\right) a^{n-1} b^{1}+\left( \begin{array}{c}{n} \\ {2}\end{array}\right) a^{n-2} b^{2}+\ldots+\left( \begin{array}{c}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}+\ldots+\left( \begin{array}{c}{n} \\ {n}\end{array}\right) b^{n}\)
Substitute in the values \(a=2x, b=-3y\) and \(n=4\).
\((2 x-3 y)^{4}=\left( \begin{array}{l}{4} \\ {0}\end{array}\right)(2 x)^{4}+\left( \begin{array}{c}{4} \\ {1}\end{array}\right)(2 x)^{4-1}(-3 y)^{1}+\left( \begin{array}{c}{4} \\ {2}\end{array}\right)(2 x)^{4-2}(-3 y)^{2}+\left( \begin{array}{c}{4} \\ {3}\end{array}\right)(2 x)^{4-3}(-3 y)^{3}+\left( \begin{array}{c}{4} \\ {4}\end{array}\right) (-3y)^{4}\)
Simplify the exponents.
\((2 x-3 y)^{4}=\left( \begin{array}{l}{4} \\ {0}\end{array}\right)(2 x)^{4}+\left( \begin{array}{c}{4} \\ {1}\end{array}\right)(2 x)^{3}(-3 y)^{1}+\left( \begin{array}{c}{4} \\ {2}\end{array}\right)(2 x)^{2}(-3 y)^{2}+\left( \begin{array}{c}{4} \\ {3}\end{array}\right)(2 x)^{1}(-3 y)^{3}+\left( \begin{array}{c}{4} \\ {4}\end{array}\right)(-3 y)^{4}\)
Evaluate the coefficients. Remember, \(\left( \begin{array}{l}{n} \\ {1}\end{array}\right)=n, \left( \begin{array}{l}{n} \\ {n}\end{array}\right)=1, \left( \begin{array}{l}{n} \\ {0}\end{array}\right)=1\)
\((2 x-3 y)^{4}=1(2 x)^{4}+4(2 x)^{3}(-3 y)^{1}+\frac{4 !}{2 !(2 x) !}(2 x)^{2}+\frac{4 !}{3 !(4-3) !}(2 x)^{3}(-3 y)^{3}+1(-3 y)^{4}\)
\((2 x-3 y)^{4}=16 x^{4}+4 \cdot 8 x^{3}(-3 y)+6\left(4 x^{2}\right)\left(9 y^{2}\right)+4(2 x)\left(-27 y^{3}\right)+81 y^{4}\)
\((2 x-3 y)^{4}=16 x^{4}-96 x^{3} y+216 x^{2} y^{2}-216 x y^{3}+81 y^{4}\)
Use the Binomial Theorem to expand \((3x-2y)^{5}\).
- Answer
-
\(\begin{array}{l}{243 x^{5}-810 x^{4} y+1080 x^{3} y^{2}} {-720 x^{2} y^{3}+240 x y^{4}-32 y^{5}}\end{array}\)
Use the Binomial Theorem to expand \((4x-3y)^{4}\).
- Answer
-
\(\begin{array}{l}{256 x^{4}-768 x^{3} y+864 x^{2} y^{2}} {-432 x y^{3}+81 y^{4}}\end{array}\)
The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem .
Notice, that in each case the exponent on the \(b\) is one less than the number of the term. The \((r+1)^{st}\) term is the term where the exponent of \(b\) is \(r\). So we can use the format of the \((r+1)^{st}\) term to find the value of a specific term.
Find a Specific Term in a Binomial Expansion
The \((r+1)^{s t}\) term in the expansion of \((a+b)^{n}\) is
\(\left( \begin{array}{l}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}\)
Find the fourth term of \((x+y)^{7}\).
Solution :
| In our pattern, \(n=7, a=x\) and \(b=y\). | |
|
We are looking for the fourth term. Since \(r+1=4\), then \(r=3\). |
|
| Write the formula | |
| Substitute in the values, \(n=7, r=3, a=x\), and \(b=y\). | |
| Simplify. | |
| Simplify. |
Find the third term of \((x+y)^{6}\).
- Answer
-
\(15x^{4}y^{2}\)
Find the fifth term of \((a+b)^{8}\).
- Answer
-
\(8ab^{7}\)
Find the coefficient of the \(x^{6}\) term of \((x+3)^{9}\).
Solution :
| In our pattern, then \(n=9, a=x\), and \(b=3\). | |
| We are looking for the coefficient of the \(x^{6}\) term. Since \(a=x\), and \(x^{9-r}=x^{6}\), we know \(r=3\). | |
| Write the formula. | |
| Substitute in the values, \(n=9, 4=3, a=x\), and \(b=3\). | |
| Simplify. | |
| Simplify. | |
| Simplify. |
Find the coefficient of the \(x^{5}\) term of \((x+4)^{8}\).
- Answer
-
\(7,168\)
Find the coefficient of the \(x^{4}\) term of \((x+2)^{7}\).
- Answer
-
\(280\)
Access these online resources for additional instruction and practice with sequences.
Key Concepts
-
Patterns in the expansion of
\((a+b)^{n}\(
- The number of terms is \(n+1\).
- The first term is \(a^{n}\) and the last term is \(b^{n}\).
- The exponents on \(a\) decrease by one on each term going left to right.
- The exponents on \(b\) increase by one on each term going left to right.
- The sum of the exponents on any term is \(n\).
- Pascal’s Triangle
- Binomial Coefficient \(\left( \begin{array}{l}{\mathbf{n}} \\ {\mathbf{r}}\end{array}\right)\) : A binomial coefficient \(\left( \begin{array}{l}{\mathbf{n}} \\ {\mathbf{r}}\end{array}\right)\), where \(r\) and \(n\) are integers with \(0≤r≤n\), is defined as
\(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)=\frac{n !}{r !(n-r) !}\)
We read \(\left( \begin{array}{l}{n} \\ {r}\end{array}\right)\) as “\(n\) choose \(r\)” or “\(n\) taken \(r\) at a time”.
- Properties of Binomial Coefficients
\(\left( \begin{array}{l}{n} \\ {1}\end{array}\right)=n \quad \left( \begin{array}{l}{n} \\ {n}\end{array}\right)=1 \quad \left( \begin{array}{l}{n} \\ {0}\end{array}\right)=1\)
- Binomial Theorem:
For any real numbers \(a\), \(b\), and positive integer \(n\),
\((a+b)^{n}=\left( \begin{array}{c}{n} \\ {0}\end{array}\right) a^{n}+\left( \begin{array}{c}{n} \\ {1}\end{array}\right) a^{n-1} b^{1}+\left( \begin{array}{c}{n} \\ {2}\end{array}\right) a^{n-2} b^{2}+\ldots+\left( \begin{array}{c}{n} \\ {r}\end{array}\right) a^{n-r} b^{r}+\ldots+\left( \begin{array}{c}{n} \\ {n}\end{array}\right) b^{n}\)


8.3: Using Sparse Matrices
Sparse matrix formats should be used, instead of conventional 2D arrays, when dealing sparse matrices of size \(10^3\times 10^3\) or larger. (For matrices of size \(100\times 100\) or less, the differences in performance and memory usage are usually negligible.)
Usually, it is good to choose either the CSR or CSC format, depending on what mathematical operations you intend to perform. You can construct the matrix either (i) directly, by means of a
(data, idx)
tuple as described above, or (ii) by creating an LIL matrix, filling in the desired elements, and then converting it to CSR or CSC format.
If you are dealing with a sparse matrix that is "strongly diagonal" (i.e., the non-zero elements occupy a very small number of diagonals, such as a tridiagonal matrix), then you can consider using the DIA format. The main advantage of the DIA format is that it is very easy to construct, by supplying a
(data, offsets)
input to the
dia_matrix
function, as described above. However, the format usually does not perform significantly better than CSR/CSC; and if the matrix is not strongly diagonal, its performance is much worse.
Another common way to construct a sparse matrix is to use the
scipy.sparse.diags
or
scipy.sparse.spdiags
functions. These functions let you specify the contents of the matrix in terms of its diagonals, as well as which sparse format to use. The two functions have slightly different calling conventions; see the documentation for details.
8.3.1 The
dot
method
Each sparse matrix has a
dot
method, which calculates the product of the matrix with an input (in the form of a 1D or 2D array, or sparse matrix), and returns the result. For sparse matrix products, this method should be used instead of the stand-alone function, similarly named
dot
, which is used for non-sparse matrix products. The sparse matrix method makes use of matrix sparsity to speed up the calculation of the product. Typically, the CSR format is faster at this than the other sparse formats.
For example, suppose we want to calculate the product \(\mathbf{A} \vec{x}\), where
\[\mathbf{A} = \begin{bmatrix}0 & 1.0 & 0 & 0 & 0 \\ 0 & 2.0 & -1.0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 6.6 & 0 & 0 & 0 & 1.4\end{bmatrix}, \quad \vec{x} = \begin{bmatrix}1\\1\\2\\3 \\5\end{bmatrix}.\]
This could be accomplished with the following program:
from scipy import * import scipy.sparse as sp data = [1.0, 2.0, -1.0, 6.6, 1.4] rows = [0, 1, 1, 3, 3] cols = [1, 1, 2, 0, 4] A = sp.csr_matrix((data, [rows, cols]), shape=(4,5)) x = array([1.,1.,2.,3.,5.]) y = A.dot(x) print(y)
Running this program gives the expected result:
[ 1. 0. 0. 13.6]
8.3.2
spsolve
The
spsolve
function, provided in the
scipy.sparse.linalg
module, is a solver for a sparse linear system of equations. It takes two inputs, \(\mathbf{A}\) and \(\mathbf{b}\), where \(\mathbf{A}\) should be a sparse matrix; \(\mathbf{b}\) can be either sparse, or an ordinary 1D or 2D array. It returns the \(\mathbf{x}\) which solves the linear equations \(\mathbf{x}\). The return value can be either a conventional array or a sparse matrix, depending on whether \(\mathbf{b}\) is sparse.
For sparse problems, you should always use
spsolve
instead of
scipy.linalg.solve
(the usual solver for non-sparse problems). Here is an example program showing
spsolve
in action:
from scipy import * import scipy.sparse as sp import scipy.sparse.linalg as spl ## Make a sparse matrix A and a vector x data = [1.0, 1.0, 1.0, 1.0, 1.0] rows = [0, 1, 1, 3, 2] cols = [1, 1, 2, 0, 3] A = sp.csr_matrix((data, [rows, cols]), shape=(4,4)) b = array([1.0, 5.0, 3.0, 4.0]) ## Solve Ax = b x = spl.spsolve(A, b) print(" x = ", x) ## Verify the solution: print("Ax = ", A.dot(x)) print(" b = ", b)
Running the program gives:
x = [ 4. 1. 4. 3.] Ax = [ 1. 5. 3. 4.] b = [ 1. 5. 3. 4.]
8.3.3
eigs
For eigenvalue problems involving sparse matrices, one typically does not attempt to find all the eigenvalues (and eigenvectors). Sparse matrices are often so huge that solving the full eigenvalue problem would take an impractically long time, even if we receive a speedup from sparse matrix arithmetic. Luckily, in most situations we only need to find a subset of the eigenvalues (and eigenvectors). For example, after discretizing the 1D Schrödinger wave equation, we are normally only interested in the several lowest energy eigenvalues.
The
eigs
function, provided in the
scipy.sparse.linalg
module, is an eigensolver for sparse matrices. Unlike the eigensolvers we have previously discussed, such as
scipy.linalg.eig
, the
eigs
function only returns a specified subset of the eigenvalues and eigenvectors.
The
eigsh
function is similar to
eigs
, except that it is specialized for Hermitian matrices. Both functions make use of a low-level numerical eigensolver library named
ARPACK
, which is also used by GNU Octave, Matlab, and many other numerical tools. We will not discuss how the algorithm works.
The first input to
eigs
or
eigsh
is the matrix for which we want to find the eigenvalues. Several other optional inputs are also accepted. Here are the most commonly-used ones:
-
The optional parameter named
k
specifies the number of eigenvalues to find (the default is 6). -
The optional parameter named
M
, if supplied, specifies a right-hand matrix for a generalized eigenvalue problem. -
The optional parameter named
sigma
, if supplied, should be a number; it means to find thek
eigenvalues which are closest to that number. -
The optional parameter named
which
specifies which eigenvalues to find, using a criteria different fromsigma
:'LM'
means to find the eigenvalues with the largest magnitudes,'SM'
means to find those with the smallest magnitudes, etc. You cannot simultaneously specify bothsigma
andwhich
. When finding small eigenvalues, it is usually better to usesigma
instead ofwhich
(see the discussion in the next section). -
The optional parameter named
return_eigenvectors
, ifTrue
(the default), means to return both eigenvalues and eigenvectors. IfFalse
, the function returns the eigenvalues only.
For the full list of inputs, see the full function documentation for
eigs
and
eigsh
.
By default,
eigs
and
eigsh
return two values: a 1D array (which is complex for
eigs
and real for
eigsh
) containing the found eigenvalues, and a 2D array where each column is one of the corresponding eigenvectors. If the optional parameter named
return_eigenvectors
is set to
False
, then only the 1D array of eigenvalues is returned.
In the next section, we will see an example of using
eigsh
.


 You would be appalled. Yet sleepiness can be worse than drunkenness in terms of its likelihood of causing an accident (Figure 7.1). Researchers have shown that sleep-deprived individuals drive more recklessly (hit more cones in driving courses) and have worse coordination and reaction time than those who are drunk. Sleepiness in fact causes as many deaths and injuries from car accidents as drunk driving. Those numbers are probably lower than they should be, since highway patrol officers do not have a test for sleep debt and also because people are often unaware of their degree of sleepiness. People can experience four seconds of sleep while driving, performing surgery, flying a plane—you name it—and not realize they are asleep. It is chilling to combine this information with the fact that one in three Americans admits that at least once in the previous month, they have put themselves in the driver’s seat even though they were finding it challenging to keep their eyes open. More than 40 percent of adults report that they rarely or never get enough sleep on weeknights. If legislators could see the deadly effects of drowsiness the same way they see those of drunk driving, perhaps we could motivate them to support an effective educational and health-care movement to address our national sleep debt emergency.
Economics
An effective approach may be to talk to people about the financial cost of sleep debt. One of my mentors in social justice and antiracism work told me, “We do this work because we know it is the right thing to do, but if we can show leaders how making these changes is a way for them to save or make money, then we get their attention.” What is the financial cost of sleep debt? $411 billion annually for the US. This comes from a RAND Corporation 2016 report that also listed the annual cost of insufficient sleep for Japan ($138 billion), Germany ($60 billion), the United Kingdom ($50 billion), and Canada ($21 billion) (Figure 7.2).1 If loss of life is not enough reason to justify the allocation of resources for sleep wellness education, saving hundreds of billions of dollars each year should do it.
Valiant efforts have been made to help change attitudes toward sleep in the US. William Dement, known as the father of sleep medicine, dedicated decades to the cause, in particular from 1991–94, when he served as chair of the US Congress–mandated National Commission on Sleep Disorders Research. Yet still, we find our country to be, in the words of US senator Mark Hatfield, a “vast reservoir of ignorance about sleep, sleep deprivation and sleep disorders” (Figure 7.3). It may surprise you to know that Hatfield made this remark all the way back in 1993, and yet sleep debt–related tragedies have been multiplying ever since.
Antiracism
As we consider the need for action to address the issue of sleep debt, we should keep in mind race-associated inequities in sleep wellness. Is healthy sleep a luxury, afforded only to “non-Hispanic whites”?
Before moving on, it is important to clarify that race is a social construct. There is no biological or anthropological evidence that humans come from different races. We are one race: the human race (Figure 7.4). But race labels, such as Black, are a part of this discussion due to the research, in which they are used to create groups for data analysis. Sometimes, these groups have to do with ancestry, such as in the case of Alaska Natives, so some of these terms are mixed into this section, depending on the studies being cited.
The US Centers for Disease Control and Prevention analyzed data from over four hundred thousand adults and found the prevalence of healthy sleep duration to be significantly lower in Native Hawaiians / Pacific Islanders, non-Hispanic Black people, multiracial non-Hispanics, and American Indians / Alaska Natives compared to non-Hispanic whites, Hispanics, and Asians. This study is just one of several that have provided evidence that there is racial inequality in sleep wellness. Harvard researchers reported that Black participants are five times more likely to have insufficient sleep compared to other groups. Even when socioeconomic status is factored out, the Black participants still get less sleep.2 This has enormous implications when we consider which groups have the highest rates of diabetes, obesity, high blood pressure, and other sleep debt–related disorders. For example, if Blacks and Native Hawaiians, two groups with higher rates of those disorders, are getting poor sleep, and we know poor sleep can cause these disorders, we have an extra layer of responsibility to address the racial inequalities around sleep health.
It is important to point out that the scientific community agrees that there are no innate biological reasons for the sleep differences based on race. Researchers suggest the experience of racism, even in its subtlest forms, impacts a person’s ability to sleep well and, in particular, to enter the deep and restorative sleep of NREM 3. This likely plays a role in the poor sleep reported by those experiencing discrimination based on sexual orientation as well, so consideration for sleep equity must go beyond race, to all groups experiencing discrimination and oppression. It makes sense that sleeping deeply would require the mind to be in a state of ease, knowing we are safe and free. The situation is exacerbated by the reported connection between lack of sleep and reduced opportunity for civic engagement, such as being able to safely and conveniently vote. Insufficient sleep is associated with reduced political participation and decreases in other measures of social capital (Figure 7.5).
Thus sleep inequality research adds one more justification, on top of the mountain of reasons, for fighting racism. It also illustrates the importance of developing targeted sleep wellness education and health services for these groups.
Business
Company leaders are in a strong position to make their mark, and increase profits, by addressing employee sleep debt. One study of four large companies in the US determined sleepiness was costing them—in lost productivity alone—around $3,000 annually per employee. For the four companies in the study, the yearly capital loss was over $50 million. On a national level, poor sleep causes on average, per person, eleven days of lost productivity in the US. In the United Kingdom, one in five workers report that they had recently arrived late to work or skipped work due to insufficient sleep. More than one in four employees in Canada take sick days because of sleepiness. Sometimes the reason we don’t get enough sleep is because we are staying at our jobs late into the evening in hopes of completing more work. The irony is that if we are low on sleep, it will take us longer to finish the work because of decreased cognitive and physical functioning. We, and our companies, would be better served to call it a day, get a good night’s sleep, and start new in the morning. But first, a company must develop a prosleep culture that supports this wise decision-making.
In Japan, as part of a response to survey results indicating that 90 percent of adults do not get enough sleep, some companies are paying their employees to sleep. One Japanese company uses a phone app to record hours of sleep, and if the employee reaches the target, they earn points to use for cafeteria purchases. In the US, Ben and Jerry’s, Google, Huffington Post, and Nike have places where staff can sleep while at work (Figure 7.7). Reboot, a marketing company in London, provides a peaceful room for napping. Many companies around the world are seeing the benefit of allowing their employees to work the hours better matched to their chronotype: for example, letting the night owls start their shift later in the morning. Considering the impact of poor sleep on cognitive function, productivity, accidents, and illness, companies could get an enormous return on their investment by supporting healthy sleep for their employees.
High Schools and Colleges
An international comparison found that among the fifty countries studied, the US has the most sleep-deprived students. One in three high school students fall asleep in class, and although teenagers need nine hours of sleep each night, most are sleeping around seven or fewer; less than 10 percent of them are getting enough sleep (Figure 7.8). African American and Hispanic students, as well as those from low-income households, get even less. In Japan, half of high school students are sleeping six or fewer hours on weeknights.
Adolescent sleep deprivation is an alarming epidemic. The American Academy of Pediatrics, the American Association of Sleep Medicine, and the American Medical Association have all identified insufficient sleep in adolescents as a serious public health issue and recommend that high schools should not start before 8:30 a.m., even though most of them still begin much earlier. Consider the short- and long-term impact of insufficient sleep on teenage mental and physical health, such as increased rates of depression, anxiety, high blood pressure, obesity, and diabetes. Research suggests teen suicide, violence, and accidents are reduced if teens are given the opportunity to get a healthy amount of sleep. In addition to educating families about the importance of sleep, convincing school districts to move to later start times would start a revolution with tremendous and far-reaching impact. Along with higher academic achievement, school officials could boast about reductions in their students’ rates of illness, depression, tardiness, and suicide.
If the traditional school start time is 8:00 a.m. and a student awakens at 6:30 a.m. to get ready and catch a bus, it is almost impossible that the teen could have gotten enough sleep: to get the nine hours most teens need, they would have to be sleeping by 9:30 p.m. Add to the equation their delayed circadian rhythm, a normal physiological part of being a teen, and it is even more unlikely they would be able to pull this off, even under the best of circumstances. For their bodies, the experience of getting up at 6:30 a.m. would be like an adult getting up at 4:30 a.m. every day for work. So it comes as no surprise that schools that shift to a later start time report a reduction in mental and physical health problems, alcohol and drug use, and traffic accidents, as well as increased academic success.
In Japan, Australia, New Zealand, England, and Finland, they have had later school start times for decades, and each of these countries has higher achievement rates than the US on standardized exams. In the fall of 2019, California became the first US state to mandate later high school start times, reflecting its value for its children’s health. Since studies have shown that bus scheduling, after-school programs, student jobs, and sports activities are not affected by later start times, hopefully more states will follow California’s lead (Figure 7.9).
Get involved in your community by having discussions with local school administrators about the American Academy of Pediatrics 2014 policy statement 3 and the Society of Behavioral Medicine position statement ,4 which are calls to action, with compelling scientific evidence, for delaying school start times. You can also contact wise politicians such as US congresswoman Zoe Lofgren, who in 2017 introduced the ZZZ’s to A’s Act as a House Bill to “direct the Secretary of Education to conduct a study to determine the relationship between school start times and adolescent health, well-being, and performance.” An easy step for getting involved, and a way to find a range of resources, would be to visit startschoollater.net .
When you were in school, do you remember having lessons about healthy foods and sex education, as well as classes emphasizing the importance of physical fitness? Most people in the US would answer yes. However, what about lessons on the importance of sleep? Let’s encourage our teachers and school administrators to incorporate lessons on the importance of healthy sleep for academic and athletic performance, stable mood, safe driving, and physical health. Getting children and teens motivated to sleep well is a wise place to build momentum for this much-needed sleep revolution.
Students fortunate enough to make it to college are faced with further challenges. With the high cost of tuition and textbooks, there is considerable pressure on college students to work long hours and take too many credits at once to finish school early so they can get a job, leaving only a small amount of time for sleep.  In Korea, college students sleep on average 6.7 hours per night, and I imagine many college students reading this book wish they could get six hours. The connection between depression and poor sleep, along with the high rates of depression and suicide on college campuses, adds more urgency to the issue. Surveying college students about their sleep is one way to start conversations and increase awareness about sleep debt. This opens the door for us to share resources about how to improve sleep.  Stanford University has a Refresh program that has been modified and implemented on many other campuses as well, including Dartmouth, the University of Chicago, and the University of Iowa. These programs teach students about the importance of sleep health and provide successful strategies for getting healthy sleep. Several campuses in the US, the United Kingdom, and Japan have also created napping spaces for students. Some have beanbags and others have cots in areas where students sign up for a napping timeslot. Students can reach out to their student government organizations and student health centers for opportunities to provide sleep wellness education activities and find resources to create napping spaces.
Health-Care Providers
The issue around sleep debt and health-care providers has three components. The first one is foundational: the lack of education on sleep wellness and sleep disorders provided to our doctors and nurses. Studies report that the total amount of time dedicated to sleep education in our doctors’ preclinical training is only fifteen minutes. If they received more education, we could expect a decrease in the current rate of sleep disorders that are left undiagnosed (95 percent).
The second component is the lack of sleep-health education and screening provided by health-care practitioners (kahuna lapaʻau in Hawaiian) to their patients.
Primary care physicians should administer a sleep-quality questionnaire and screen patients for sleep problems like how they screen everyone for high blood pressure (Figure 7.12). If a physician sees a patient for something as minor as a splinter, they still have the medical assistant slap on a blood pressure cuff to screen for hypertension (high blood pressure). We need to approach sleep-health screening in the same manner. Every patient should get surveyed; then the survey data should be used as talking points to emphasize the importance of sleep and address any problem areas. Drowsiness should be discussed and pursued. Patients should be asked to keep ten-day sleep diaries and submit those in follow-up appointments. A primary care clinic in Idaho surveyed a little over 1,200 patients who were coming to the clinic for a variety of reasons (besides sleep disorders) and found over 60 percent of them also had sleep disorder symptoms. At that point, all but two of the patients had not been diagnosed. Imagine if we could generalize this type of care and reduce illness, accidents, and deaths related to sleep debt and sleep disorders.
The third and final component is the demanding shift work required of our health-care providers and hospital workers. We must change the guidelines for this because there are too many deaths and accidents clearly documented and linked to health-care provider sleep debt. For example, physicians in their residency (the two to seven years they practice while learning their specialty) are working with such high sleep debt that one in twenty report that they have killed a patient due to errors they made because they had not gotten enough sleep (Figure 7.13). In a survey of residents in a San Francisco hospital, over 40 percent of residents disclosed killing at least one patient due to sleepiness. Stanford University researchers have used the multiple sleep latency test for years on numerous residents and nurses, and according to them, of all those respondents, only one person “was not in the twilight zone of extreme sleepiness.”5 Johns Hopkins released a study in 2016 stating medical errors are the third-highest cause of death in the US, making medical errors the reason for 10 percent of all US deaths.6 Knowing how sleep-deprived medical workers are, it is not a leap to consider lack of sleep playing a part in those medical errors and therefore deaths.
The medical establishment needs to be held accountable and revise the work schedules of our health-care providers. The National Academies of Science, Engineering, and Medicine gathered a group of medical and scientific experts to examine evidence and propose revised work schedules for medical residents. For example, with these revisions, they would get a five-hour break for sleeping after working sixteen of their thirty hours in a shift. However, the Accreditation Council for Graduate Medical Education (ACGME) has done too little to have much impact, and way too many sleep debt–related accidents and deaths continue to occur. To put it in perspective, in the US, the ACGME mandates that the maximum number of hours a resident can work per week is eighty, but many European countries, whose medical programs still have excellent success rates and train physicians in a similar number of years, set the maximum at forty-eight.7 We need to increase awareness of the tragic number of preventable deaths and injuries associated with the sleep deprivation imposed on our health-care providers and pressure the medical establishment to change.
Your Next Steps
The World Health Organization says we are in the midst of a “global epidemic of sleeplessness.” The Centers for Disease Control and Prevention report that over 40 percent of adults said they had fallen asleep during the day unintentionally at least once in the past month. In the US and Japan, more than 65 percent of adults are not getting enough sleep. The problem is not limited to industrialized societies. A study of people living in rural, low-income communities without the trappings of industry in eight African and Asian countries found that a large number of adults were not getting enough sleep. The authors used their study’s results to urge people to see the global nature of the sleep debt epidemic.8
There are many approaches to resolving this problem and decreasing its associated catastrophes. One place to start is to address the lack of awareness about sleep debt and the dearth of public policies promoting healthy sleep.
Let’s take a glance at previously successful campaigns that had impacts on public health. Thanks to scientific evidence about the dangers of cigarette smoking, we saw the rise of consumer warnings added to packaging as well as designated nonsmoking areas. After learning more about automobile accidents, we went from cars not even having seat belts to passing laws requiring that all passengers wear them. Vaccine awareness and access helped eradicate smallpox and almost eradicated polio and other diseases. Research on death in infants led to the Back to Sleep campaign to reduce the incidence of sudden infant death syndrome. Widespread distribution of posters provided education about reducing the spread of disease through handwashing. We know we can have an impact, and now is the time to act to increase sleep-health education.
After completing my course on the science of sleep, or simply reading this book, you are likely a sleep expert compared to most of the people in your community, so I ask you to take that knowledge and use it to make an impact on your community. You can read the previous sections for ideas, but here are a few more:
- • Choose some sleep wellness and sleep disorders information (for potential content, see chapters 1 and 6 in this book) and put it in a format you like—a flyer, brochure, poster, or sheet of talking points—and go with a friend to do targeted sleep-health education in underserved neighborhoods. You might consider visiting a beauty salon, barbershop, church, or school to share your expertise (Figure 7.14). A good way to start the conversation is by asking people to tell you about their sleep and their early evening routines. People usually like to share their stories. Your first step is to encourage dialogue about sleep.
- • Get resources from, or provide support to, a nonprofit such as Pajama Program 9 and help children get sleep.
- • Reach out to educate leaders in occupations known to have increased levels of sleep debt: health-care workers, airline employees, bus drivers, truck drivers, police officers, first responders, and military.
- • Visit your campus health center and ask them to consider providing sleep wellness screenings and to discuss snoring, insomnia, apnea, and daytime drowsiness.
- • Talk to colleagues at work about their sleep. Identify things at your workplace that could change to support healthy sleep. Approach an ally in a leadership role in your company and discuss the financial gains likely achieved if they adopted a prosleep culture. Ask them to consider creating a safe napping space, providing sleep disorder and insomnia screening, starting a healthy sleep awareness program, and adjusting shift hours based on chronotype.
- • Start a petition or grassroots effort to eliminate daylight savings time.
Consider the successful business leaders, school administrators, and politicians mentioned earlier in this chapter, who have chosen to make healthy sleep a priority for large groups of people and achieved much along the way. Please find an arena where you have a natural interest—perhaps a school, a local political group, a veterans club, an eldercare facility, a health clinic for the underserved, your workplace or college campus—and begin a conversation with someone about how to raise consciousness about sleep wellness. Let’s work together to help people get the sleep they deserve so we can bring more equanimity, health, and peace to our communities and beyond.
1 Marco Hafner et al., “Why Sleep Matters—the Economic Costs of Insufficient Sleep: A Cross-Country Comparative Analysis,” Rand Health Quarterly 6, no. 4 (2017): 11, https://doi.org/10.7249/RR1791 .
2 Yong Liu et al., “Prevalence of Healthy Sleep Duration among Adults—United States, 2014,” Morbidity and Mortality Weekly Report 65, no. 6 (February 2016): 137–41, http://dx.doi.org/10.15585/mmwr.mm6506a1 .
3 Au, Rhoda, et al. “School Start Times for Adolescents.” Pediatrics 134, no. 3 (2014): 642-649, https://publications.aap.org/pediatrics/article/134/3/642/74175/School-Start-Times-for-Adolescents .
4 Trevorrow, T., E.S. Zhou, J.R. Dietch, B.D. Gonzalez. “Start Middle and High Schools 8:30 a.m. or Later to Promote Student Health and Learning.” Society of Behavioral Medicine , (November 2017): https://www.sbm.org/UserFiles/file/late-school-start-statement-FINAL.pdf .
5 Rafael Pelayo, C. William Dement, and Krystle Singh, Dement’s Sleep and Dreaming (self-published, 2016), 430.
6 Johns Hopkins Medicine, “Study Suggests Medical Errors Now Third Leading Cause of Death in the U.S.—05/03/2016,” Johns Hopkins Medicine-News and Publications , May 2016, https://www.hopkinsmedicine.org/news/media/releases/study_suggests_medical_errors_now_third_leading_cause_of_death_in_the_us .
7 Pelayo, Dement, and Singh, Dement’s Sleep and Dreaming , 428.
8 Saverio Stranges et al., “Sleep Problems: An Emerging Global Epidemic? Findings from the INDEPTH WHO-SAGE Study among More Than 40,000 Older Adults from 8 Countries across Africa and Asia,” Sleep 35, no. 8 (August 2012): 1173–81, https://doi.org/10.5665/sleep.2012 .
9 “Pajama Program,” accessed on December 3, 2021, https://pajamaprogram.org/our-programs/ .


8.8: Saturn’s Rings
Saturn’s rings are made up of particles that range from dust-size to meters in diameter. The rings are mostly water and ice in composition, with some rocky silicate materials. Galileo Galilei first observed them in 1610; he could not discern them as rings and referred to them as the ears of Saturn . In 1655, astronomer Christian Huygens hypothesized that Saturn had a ring, based on his telescopic observations. Giovanni Cassini suggested in 1655 that Saturn’s ring was actually made up of numerous smaller rings with gaps or divisions between these smaller rings. Saturn exhibits seven major rings with multiple divisions and gaps in the rings. The outer F ring is twisted and knotted like a piece of taffy; this is due to interactions of the F ring with Saturn’s moon Prometheus. There are small moons, called shepherd or sheepherder moons , which orbit near the outer edges of a ring or within gaps in the rings.
These sheepherder moons maintain a sharply defined edge to the ring. Radial features in the B ring, called spokes, were suggested by Steve O’Meara (Earth observations) and confirmed by spacecraft. These spokes , seasonal in nature, appear to be caused by electrostatic forces. A quite-distant ring of dust was discovered in 2009, called the Phoebe ring, and the Saturnian moon Phoebe that is not visible in Earth-based telescopes.


Career Development
See if you can remember a time in your childhood when you noticed somebody doing professional work. Maybe a nurse or doctor, dressed in a lab coat, was listening to your heartbeat. Maybe a worker at a construction site, decked in a hard hat, was operating noisy machinery. Maybe a cashier at the checkout line in a grocery store was busily scanning bar codes. Each day in your young life you could have seen a hundred people doing various jobs. Surely some of the experiences drew your interest and appealed to your imagination.
If you can recall any such times, those are moments from the beginning stage of your career development.
What exactly is career development? It’s a lifelong process in which we become aware of, interested in, knowledgeable about, and skilled in a career. It’s a key part of human development as our identity forms and our life unfolds.
Stages of Career Development
There are five main stages of career development (developed by Donald Super). Each stage correlates with attitudes, behaviors, and relationships we all tend to have at that point and age. As we progress through each stage and reach the milestones identified, we prepare to move on to the next one.
Which stage of career development do you feel you are in currently? Think about each stage. What challenges are you facing now? Where are you headed?
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#
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STAGE
|
DESCRIPTION
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1
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GROWING
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This is a time in early years (4–13 years old) when you begin to have a sense about the future. You begin to realize that your participation in the world is related to being able to do certain tasks and accomplish certain goals.
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2
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EXPLORING
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This period begins when you are a teenager, extends into your mid-twenties, and may extend later. In this stage you find that you have specific interests and aptitudes. You are aware of your inclinations to perform and learn about some subjects more than others. You may try out jobs in your community or at your school. You may begin to explore a specific career. At this stage, you have some detailed “data points” about careers, which will guide you in certain directions.
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3
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ESTABLISHING
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This period covers your mid-twenties through mid-forties. By now you are selecting or entering a field you consider suitable, and you are exploring job opportunities that will be stable. You are also looking for upward growth, so you may be thinking about an advanced degree.
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4
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MAINTAINING
|
This stage is typical for people in their mid-forties to mid-sixties. You may be in an upward pattern of learning new skills and staying engaged. But you might also be merely “coasting and cruising” or even feeling stagnant. You may be taking stock of what you’ve accomplished and where you still want to go.
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5
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REINVENTING
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In your mid-sixties, you are likely transitioning into retirement. But retirement in our technologically advanced world can be just the beginning of a new career or pursuit—a time when you can reinvent yourself. There are many new interests to pursue, including teaching others what you’ve learned, volunteering, starting online businesses, consulting, etc.
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Keep in mind that your career development path is personal to you, and you may not fit neatly into the categories described above. It’s more common than it has been in the past for people to change careers in their thirties, forties, fifties, and even sixties. Perhaps your socioeconomic background changes how you fit into the schema. Perhaps your physical and mental abilities affect how you define the idea of a “career.” And for everyone, too, there are factors of chance that can’t be predicted or anticipated. You are unique, and your career path can only be developed by you.
Career Development Resources in Your College, Community, and Beyond
Career experts say that people will change careers (not to mention jobs) five to seven times in a lifetime. So your career will likely not be a straight and narrow path. Be sure to set goals and assess your interests, skills and values often. Seek opportunities for career growth and enrichment. And take advantage of the rich set of resources available to you. Below are just a few.
Career Development Office on Campus
Whether you are a student, a graduate, or even an employer, you can obtain invaluable career development assistance at your college or university. Campus career centers can support, guide, and empower you in every step of the career development process, from initial planning to achieving lifelong career satisfaction.
Books on Career Development
Going to college is one of the best steps you can take to prepare for a career. But soon-to-be or recently graduated students are not necessarily guaranteed jobs. Staying educated about strategies for developing your career and finding new jobs will help you manage ongoing transitions. The book
The Secret to Getting a Job After College: Marketing Tactics to Turn Degrees into Dollars
, by author Larry Chiagouris, was written specifically to help recent grads increase their chances of finding a job right after college. It speaks to students in all majors and provides tips and tactics to attract the attention of an employer and successfully compete with other candidates to get the job you want.
The following video provides an introduction to the book. You can download a transcript of the video here.
Video:
The Secret to Getting a Job After College
A YouTube element has been excluded from this version of the text. You can view it online here:
https://press.rebus.community/blueprint2/?p=59
Career Roadmap
You can use the Career Roadmap, from DePaul University, to evaluate where you are and where you want to be in your career. It can help you decide if you want to change career paths and can guide you in searching for a new job. The road map identifies the following four cyclical steps:
-
Know yourself
-
Explore and choose options
-
Gain knowledge and experience
-
Put it all together: the job search process
Plan, Do, Check, Act
PDCA (plan–do–check–act)
is a four-step strategy for carrying out change. You can use it to evaluate where you are in the career development process and to identify your next steps. The strategy is typically used in the business arena as a framework for improving processes and services. But you can think of your career as a personal product you are offering or selling.
-
PLAN
: What are your goals and objectives? What process will you use to get to your targets? You might want to plan smaller to begin with and test out possible effects. For instance, if you are thinking of getting into a certain career, you might plan to try it out first as an intern or volunteer or on a part-time basis. When you start on a small scale, you can test possible outcomes.
-
DO
: Implement your plan.  Collect data as you go along; you will need it for charting and analyzing in the Check and Act steps ahead.
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CHECK
: Look at your results so far. Are you happy with your job or wherever you are in the career development process? How is your actual accomplishment measuring up next to your intentions and wishes? Look for where you may have deviated in your intended steps. For example, did you take a job in another city when your initial plans were for working closer to friends and family? What are the pros and cons? If you like, create a chart that shows you all the factors. With a chart, it will be easier to see trends over several PDCA cycles.
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ACT
: How should you act going forward? What changes in planning, doing, and checking do you want to take? The PDCA framework is an ongoing process. Keep planning, doing, checking, and acting. The goal is continuous improvement.
Internet Sites for Career Planning
Visit the Internet Sites for Career Planning Web site at the National Career Development Association’s site. You will find extensive, definitive, and frequently updated information on the following topics:
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Online Employment
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Self-Assessment
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Career Development Process
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Occupational Information
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Employment Trends
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Resources and Services for Ex-Offenders
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Resources and Services the Older Client
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Researching Employers
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Military
Activity: Campus to Career
Objective
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Examine two critical questions about developing your career while still in college: How do I prepare myself for a career while I’m in college? How do I position myself to get ahead?
Instructions
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Review the Campus to Career Web site called “Top College Career Tips from Freshman to Senior Year.”
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Visit the section for each year of college: Freshman Year, Sophomore Year, Junior Year, and Senior Year. You may need to return to the main page of the site to access the sophomore, junior, and senior year pages of content.
Licenses and Attributions:
CC licensed content, Original:
CC licensed content, Shared previously
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Campus to Career Quiz. Authored by: Ronda Dorsey Neugebauer. Provided by: Chadron State College. Project: Kaleidoscope Open Course Initiative. License: CC BY: Attribution.
All rights reserved content:
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The Secret to Getting a Job After College. Authored by: CT STYLE. Located at:
https://youtu.be/OpeIqQ5qTjc
. License: All Rights Reserved. License Terms: Standard YouTube License.
Adaptions: slight formatting changes, removed image of people in lab, removed image of plan-do-check-act, removed quote. Relocated Learning Objectives.


16.1: Administration of Medications Via Other Routes Introduction
- Safely administer medications and irrigations for the eye, ear, inhalation, and vaginal routes
- Select the appropriate equipment
- Calculate correct amount to administer
- Select appropriate site
- Modify the procedure to reflect variations across the life span
- Document actions and observations
- Recognize and report significant deviations from norms
This chapter will review specific information related to administering medications via the topical, transdermal, eye, ear, inhalation, and vaginal routes. Read additional information about administering medications in the “ Administration of Enteral Medications ” and “ Administration of Parenteral Medications ” chapters.


4.4: Team Activity
Overview
Student teams continue to build a description of the operating environment for their sector-based organization. They select an appropriate risk management framework for the sector-based organization. In the absence of one required by the industry, teams should begin to apply the NIST Cybersecurity Framework to the selected organization. Each team’s work should be reviewed by the instructor.
Team Activity Objectives
- Identify various risk management frameworks and standards, such as the NIST Cybersecurity Critical Infrastructure Framework (“NIST Cybersecurity Framework”) and North American Electricity Reliability Council (NERC).
- Describe how to use the framework core process.
Assignment
Below are some of the risk management frameworks available. Please select one of them to ensure your team can complete the Team Assignment in Module 7.
- NIST Framework for Improving Critical Infrastructure Cybersecurity (“NIST Cybersecurity Framework”)
- NIST Special Publication 800-53 Rev 3 and NIST Special Publication 800-53 Rev 3 App l
- NIST Special Publication 800-53 Rev 4 and NIST Special Publication 800-53 Rev 4 App l
- Consensus Audit Guidelines (CAG)
- Cyber Resilience Review (CRR): Questions Set with Guidance
- CFATS Risk-Based Performance Standards (RBPS): Chemical Facilities Anti-Terrorism Standard , “RBPS 8 – Cyber,” pp. 71-81
- Committee on National Security Systems (CNSS) Instruction No. 1253 , Baseline Security Categorization Method
- Committee on National Security Systems Instruction (CNSSI) No. 1253, Security Control Overlays for Industrial Control System (ICS), Volume 1
- DHS Catalog of Control Systems Security: Recommendations for Standards Developers, Revisions 6 and 7
- TSA Pipeline Security and Incident Recovery Protocol Plan
- Information Assurance Implementation, Department of Defense , DODI 8500.2, February 6, 2003.
- ISO/IEC 15408 revision 3.1: Common Criteria for Information Technology Security Evaluation, Revision 3.1
- NERC Reliability Standards CIP-002-009 Revisions 3 and 4
- NIST Special Publication 800-82 Guide to Industrial Control Systems Security, June 2011
- NIST Special Publication 800-82 Rev 1
- NIST Special Publication 800-82 Rev 2 (Draft)
- NIST Special Publication 800-53, Recommended Security Controls for Federal Information Systems, Rev 2
- NRC Regulatory Guide 5.71, Cyber Security Programs for Nuclear Facilities, January 2010
- Nuclear Energy Institute (NEI) 08-09 Cyber Security Plan for Nuclear Power Reactors
- TSA Pipeline Security Guidelines, April 2011
Assignment Options
Option 1: Write a 2-page abstract summarizing why your team chose your selected risk management framework for your sector-based organization.
Option 2: Prepare 2–3 presentation slides on your justification for selecting this risk management framework.
Grading Criteria Rubric
- Content
- Evidence of teamwork
- References
- Use of American Psychological Association (APA) style in writing the assignment
Total Points: 100


7.3: The Pigeonhole Principle
The word “pigeonhole” can refer to a hole in which a pigeon roosts (i.e. pretty much what it sounds like) or a series of roughly square recesses in a desk in which one could sort correspondence (see Figure \(7.3.1\)).
Whether you prefer to think of roosting birds or letters being sorted, the first and easiest version of the pigeonhole principle is that if you have more “things” than you have “containers” there must be a container holding at least two things.
If we have \(6\) pigeons who are trying to roost in a coop with \(5\) pigeonholes, two birds will have to share.
If we have \(7\) letters to sort and there are \(6\) pigeonholes in our desk, we will have to put two letters in the same compartment.
The “things” and the “containers” don’t necessarily have to be interpreted in the strict sense that the “things” go into the “containers.” For instance, a nice application of the pigeonhole principle is that if there are at least \(13\) people present in a room, some pair of people will have been born in the same month. In this example, the things are the people and the containers are the months of the year.
The abstract way to phrase the pigeonhole principle is:
If f is a function such that \(|\text{Dom}(f)| > |\text{Rng}(f)|\) then \(f\) is not injective.
The proof of this statement is an easy example of proof by contradiction so we’ll include it here.
- Proof
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Suppose to the contrary that \(f\) is a function with \(|\text{Dom}(f)| > |\text{Rng}(f)|\) and that \(f\) is injective. Of course \(f\) is onto its range, so since we are presuming that \(f\) is injective it follows that \(f\) is a bijection between \(\text{Dom}(f)\) and \(\text{Rng}(f)\). Therefore (since \(f\) provides a one-to-one correspondence) \(|\text{Dom}(f)| = |\text{Rng}(f)|\). This clearly contradicts the statement that \(|\text{Dom}(f)| > |\text{Rng}(f)|\).
Q.E.D.
For a statement with an almost trivial proof the pigeonhole principle is very powerful. We can use it to prove a host of existential results – some are fairly silly, others very deep. Here are a few examples:
There are two people (who are not bald) in New York City having exactly the same number of hairs on their heads.
There are two books in (insert your favorite library) that have the same number of pages.
Given \(n\) married couples (so \(2n\) people) if we choose \(n + 1\) people we will be forced to choose both members of some couple.
Suppose we select \(n + 1\) numbers from the set \(\{1, 2, 3, . . . , 2^n\}\), we will be forced to have chosen two numbers such that one is divisible by the other.
We can come up with stronger forms of the pigeonhole principle by considering pigeonholes with capacities. Suppose we have \(6\) pigeonholes in a desk, each of which can hold \(10\) letters. What number of letters will guarantee that one of the pigeonholes is full? The largest number of letters we could have without having \(10\) in some pigeonhole is \(9 · 6 = 54\), so if there are \(55\) letters we must have \(10\) letters in some pigeonhole.
More generally, if we have \(n\) containers, each capable of holding m objects, than if there are \(n · (m − 1) + 1\) objects placed in the containers, we will be assured that one of the containers is at capacity.
The ordinary pigeonhole principle is the special case \(m = 2\) of this stronger version.
There is an even stronger version, which ordinarily is known as the “strong form of the pigeonhole principle.” In the strong form, we have pigeonholes with an assortment of capacities.
If there are \(n\) containers having capacities \(m_1, m_2, m_3, . . . , m_n\), and there are \(1 + \sum^{n}_{i=1}(m_i − 1)\) objects placed in them, then for some \(i\), container \(i\) has (at least) \(m_i\) objects in it.
- Proof
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If no container holds its full capacity, then the largest the total of the objects could be is \(\sum^{n}_{i=1}(m_i − 1)\).
Q.E.D.
Exercises:
The statement that there are two non-bald New Yorkers with the same number of hairs on their heads requires some careful estimates to justify it. Please justify it.
A mathematician, who always rises earlier than her spouse, has developed a scheme – using the pigeonhole principle – to ensure that she always has a matching pair of socks.  So as not to wake her husband she must select some number of socks from her drawer in the early morning dark and take them with her to the adjacent bathroom where she dresses. What number of socks does she choose?
If we select \(1001\) numbers from the set \(\{1, 2, 3, . . . , 2000\}\) it is certain that there will be two numbers selected such that one divides the other. We can prove this fact by noting that every number in the given set can be expressed in the form \(2^k · m\) where \(m\) is an odd number and using the pigeonhole principle. Write-up this proof.
Given any set of \(53\) integers, show that there are two of them having the property that either their sum or their difference is evenly divisible by \(103\).
Prove that if \(10\) points are placed inside a square of side length \(3\), there will be \(2\) points within \(\sqrt{2}\) of one another.
Prove that if \(10\) points are placed inside an equilateral triangle of side length \(3\), there will be \(2\) points within \(1\) of one another.
Prove that in a simple graph (an undirected graph with no loops or parallel edges) having \(n\) nodes, there must be two nodes having the same degree.