11.9: Image Formation by Lenses
Learning Objectives
- List the rules for ray tracking for thin lenses.
- Illustrate the formation of images using the technique of ray tracking.
- Determine power of a lens given the focal length.
Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom lens. In this section, we will use the law of refraction to explore the properties of lenses and how they form images.
The word lens derives from the Latin word for a lentil bean, the shape of which is similar to the convex lens in Figure \(\PageIndex{1}\). The convex lens shown has been shaped so that all light rays that enter it parallel to its axis cross one another at a single point on the opposite side of the lens. (The axis is defined to be a line normal to the lens at its center, as shown in Figure \(\PageIndex{1}\).) Such a lens is called a converging (or convex) lens for the converging effect it has on light rays. An expanded view of the path of one ray through the lens is shown, to illustrate how the ray changes direction both as it enters and as it leaves the lens. Since the index of refraction of the lens is greater than that of air, the ray moves towards the perpendicular as it enters and away from the perpendicular as it leaves. (This is in accordance with the law of refraction.) Due to the lens’s shape, light is thus bent toward the axis at both surfaces. The point at which the rays cross is defined to be the focal point F of the lens. The distance from the center of the lens to its focal point is defined to be the focal length \(f\) of the lens. Figure \(\PageIndex{2}\) shows how a converging lens, such as that in a magnifying glass, can converge the nearly parallel light rays from the sun to a small spot.
Definition: CONVERGING OR CONVEX LENS
The lens in which light rays that enter it parallel to its axis cross one another at a single point on the opposite side with a converging effect is called converging lens.
Definition: FOCAL POINT F
The point at which the light rays cross is called the focal point F of the lens.
Definition: FOCAL LENGTH \(f\)
The distance from the center of the lens to its focal point is called focal length \(f\).
The greater effect a lens has on light rays, the more powerful it is said to be. For example, a powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length than a weak lens. The light will also focus into a smaller and more intense spot for a more powerful lens. The power \(P\) of a lens is defined to be the inverse of its focal length. In equation form, this is
\[P=\frac{1}{f}. \nonumber \]
Definition: POWER \(P\)
The power \(P\) of a lens is defined to be the inverse of its focal length. In equation form, this is
\[P=\frac{1}{f}. \nonumber\]
where \(f\) is the focal length of the lens, which must be given in meters (and not cm or mm). The power of a lens P has the unit diopters (D), provided that the focal length is given in meters. That is, \(1 ~\mathrm{D}=1 / \mathrm{m}\), or \(1 \mathrm{~m}^{-1}\). (Note that this power (optical power, actually) is not the same as power in watts. It is a concept related to the effect of optical devices on light.) Optometrists prescribe common spectacles and contact lenses in units of diopters.
Example \(\PageIndex{1}\): What is the Power of a Common Magnifying Glass?
Suppose you take a magnifying glass out on a sunny day and you find that it concentrates sunlight to a small spot 8.00 cm away from the lens. What are the focal length and power of the lens?
Strategy
The situation here is the same as those shown in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\). The Sun is so far away that the Sun’s rays are nearly parallel when they reach Earth. The magnifying glass is a convex (or converging) lens, focusing the nearly parallel rays of sunlight. Thus the focal length of the lens is the distance from the lens to the spot, and its power is the inverse of this distance (in m).
Solution
The focal length of the lens is the distance from the center of the lens to the spot, given to be 8.00 cm. Thus,
\[f=8.00 \mathrm{~cm}. \nonumber\]
To find the power of the lens, we must first convert the focal length to meters; then, we substitute this value into the equation for power. This gives
\[P=\frac{1}{f}=\frac{1}{0.0800 \mathrm{~m}}=12.5 ~\mathrm{D}. \nonumber\]
Discussion
This is a relatively powerful lens. The power of a lens in diopters should not be confused with the familiar concept of power in watts. It is an unfortunate fact that the word “power” is used for two completely different concepts. If you examine a prescription for eyeglasses, you will note lens powers given in diopters. If you examine the label on a motor, you will note energy consumption rate given as a power in watts.
Figure \(\PageIndex{3}\) shows a concave lens and the effect it has on rays of light that enter it parallel to its axis (the path taken by ray 2 in the figure is the axis of the lens). The concave lens is a diverging lens , because it causes the light rays to bend away (diverge) from its axis. In this case, the lens has been shaped so that all light rays entering it parallel to its axis appear to originate from the same point, \(F\), defined to be the focal point of a diverging lens. The distance from the center of the lens to the focal point is again called the focal length \(f\) of the lens. Note that the focal length and power of a diverging lens are defined to be negative. For example, if the distance to \(F\) in Figure \(\PageIndex{3}\) is 5.00 cm, then the focal length is \(f=-5.00 \mathrm{~cm}\) and the power of the lens is \(P=-20 ~\mathrm{D}\). An expanded view of the path of one ray through the lens is shown in the figure to illustrate how the shape of the lens, together with the law of refraction, causes the ray to follow its particular path and be diverged.
Definition: DIVERGING LENS
A lens that causes the light rays to bend away from its axis is called a diverging lens.
As noted in the initial discussion of the law of refraction, the paths of light rays are exactly reversible. This means that the direction of the arrows could be reversed for all of the rays in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{3}\). For example, if a point light source is placed at the focal point of a convex lens, as shown in Figure \(\PageIndex{4}\), parallel light rays emerge from the other side.
Ray Tracing and Thin Lenses
Ray tracing is the technique of determining or following (tracing) the paths that light rays take. For rays passing through matter, the law of refraction is used to trace the paths. Here we use ray tracing to help us understand the action of lenses in situations ranging from forming images on film to magnifying small print to correcting nearsightedness. While ray tracing for complicated lenses, such as those found in sophisticated cameras, may require computer techniques, there is a set of simple rules for tracing rays through thin lenses. A thin lens is defined to be one whose thickness allows rays to refract, as illustrated in Figure \(\PageIndex{1}\), but does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces but the lens is thin enough to assume that light rays bend only once. A thin symmetrical lens has two focal points, one on either side and both at the same distance from the lens. (See Figure \(\PageIndex{5}\).) Another important characteristic of a thin lens is that light rays through its center are deflected by a negligible amount, as seen in Figure \(\PageIndex{6}\).
Definition: THIN LENS
A thin lens is defined to be one whose thickness allows rays to refract but does not allow properties such as dispersion and aberrations.
TAKE-HOME EXPERIMENT: A VISIT TO THE OPTICIAN
Look through your eyeglasses (or those of a friend) backward and forward and comment on whether they act like thin lenses.
Using paper, pencil, and a straight edge, ray tracing can accurately describe the operation of a lens. The rules for ray tracing for thin lenses are based on the illustrations already discussed:
- Principal Ray 1 : a ray entering a converging lens parallel to its axis passes through the focal point F of the lens on the other side (see rays 1 and 3 in Figure \(\PageIndex{1}\)); a ray entering a diverging lens parallel to its axis seems to come from the focal point F (see rays 1 and 3 in Figure \(\PageIndex{3}\))
- Principal Ray 2 : a ray passing through the center of either a converging or a diverging lens does not change direction (see Figure \(\PageIndex{6}\), and see ray 2 in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{3}\))
- Principal Ray 3 : a ray entering a converging lens through its focal point exits parallel to its axis (the reverse of rays 1 and 3 in Figure \(\PageIndex{1}\)); a ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis (the reverse of rays 1 and 3 in Figure \(\PageIndex{3}\)). The third principal ray is optional and may be used to verify the accuracy of image location.
Image Formation by Thin Lenses
In some circumstances, a lens forms an obvious image, such as when a movie projector casts an image onto a screen. In other cases, the image is less obvious. Where, for example, is the image formed by eyeglasses? We will use ray tracing for thin lenses to illustrate how they form images, and we will develop equations to describe the image formation quantitatively.
Consider an object some distance away from a converging lens, as shown in Figure \(\PageIndex{7}\). To find the location and size of the image formed, we trace the paths of principal rays originating from one point on the object, in this case the top of the person’s head. The figure shows the three principal rays from the top of the object as described above. (Note that there are many light rays leaving this point going in many directions, but we concentrate on the three principal rays which can be traced by simple rules.) The first ray is one that enters the lens parallel to its axis and passes through the focal point on the other side (Principal Ray 1). The second ray passes through the center of the lens without changing direction (Principal Ray 2). The third ray passes through the nearer focal point on its way into the lens and leaves the lens parallel to its axis (Principal Ray 3). If correctly drawn, all three rays cross at the same point on the other side of the lens. The image of the top of the person’s head is located at this point. All rays, including those that are not principal rays, that come from the same point on the top of the person’s head are refracted in such a way as to cross at the point shown; the principal rays are what we use to find this point. Rays from another point on the object, such as her belt buckle, will also cross at another common point, forming a complete image, as shown. Although three rays are traced in Figure \(\PageIndex{7}\), only two are necessary to locate the image. Before applying ray tracing to other situations, let us consider the example shown in Figure \(\PageIndex{7}\) in more detail.
The image formed in Figure \(\PageIndex{7}\) is a real image , meaning that it can be projected. That is, light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye, for example. Figure \(\PageIndex{8}\) shows how such an image would be projected onto film by a camera lens. This figure also shows how a real image is projected onto the retina by the lens of an eye. Note that the image is there whether it is projected onto a screen or not.
Definition: REAL IMAGE
The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image.
Several important distances appear in Figure \(\PageIndex{7}\). We define \(d_{\mathrm{o}}\) to be the object distance, the distance of an object from the center of a lens. Image distance \(d_{\mathrm{i}}\) is defined to be the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols \(h_{\mathrm{o}}\) and \(h_{\mathrm{i}}\), respectively. Images that appear upright relative to the object have heights that are positive and those that are inverted have negative heights. Using the rules of ray tracing and making a scale drawing with paper and pencil, like that in Figure \(\PageIndex{8}\), we can accurately describe the location and size of an image. But the real benefit of ray tracing is in visualizing how images are formed in a variety of situations. To obtain numerical information, we use a pair of equations that can be derived from a geometric analysis of ray tracing for thin lenses. The thin lens equations are
\[\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f} \nonumber \]
and
\[\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m. \nonumber \]
We define the ratio of image height to object height (\(h_{\mathrm{i}} / h_{\mathrm{o}}\)) to be the magnification \(m\). (The minus sign in the equation above will be discussed shortly.) The thin lens equations are broadly applicable to all situations involving thin lenses (and “thin” mirrors, as we will see later). We will explore many features of image formation in the following worked examples.
Definition: IMAGE DISTANCE
The distance of the image from the center of the lens is called image distance.
THIN LENS EQUATIONS AND MAGNIFICATION
\[\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f} \nonumber\]
\[\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m \nonumber\]
Example \(\PageIndex{2}\): Finding the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure \(\PageIndex{9}\). Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.
Strategy and Concept
Since the object is placed farther away from a converging lens than the focal length of the lens, this situation is analogous to those illustrated in Figure \(\PageIndex{7}\) and Figure \(\PageIndex{8}\). Ray tracing to scale should produce similar results for di. Numerical solutions for \(d_{\mathrm{i}}\) and \(m\) can be obtained using the thin lens equations, noting that \(d_{\mathrm{o}}=0.750 \mathrm{~m} \text { and } f=0.500 \mathrm{~m}\).
Solutions (Ray tracing)
The ray tracing to scale in Figure \(\PageIndex{9}\) shows two rays from a point on the bulb’s filament crossing about 1.50 m on the far side of the lens. Thus the image distance \(d_{\mathrm{i}}\) is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus \(m\) is about –2. The minus sign indicates that the image is inverted.
The thin lens equations can be used to find \(d_{\mathrm{i}}\) from the given information:
\[\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}. \nonumber\]
Rearranging to isolate \(d_{\mathrm{i}}\) gives
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}-\frac{1}{d_{\mathrm{o}}}. \nonumber\]
Entering known quantities gives a value for \(1 / d_{\mathrm{i}}\):
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{0.500 \mathrm{~m}}-\frac{1}{0.750 \mathrm{~m}}=\frac{0.667}{\mathrm{~m}}. \nonumber\]
This must be inverted to find \(d_{\mathrm{i}}\):
\[d_{\mathrm{i}}=\frac{\mathrm{m}}{0.667}=1.50 \mathrm{~m}. \nonumber\]
Note that another way to find \(d_{\mathrm{i}}\) is to rearrange the equation:
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}-\frac{1}{d_{\mathrm{o}}}. \nonumber\]
This yields the equation for the image distance as:
\[d_{\mathrm{i}}=\frac{f d_{\mathrm{o}}}{d_{\mathrm{o}}-f}. \nonumber\]
Note that there is no inverting here.
The thin lens equations can be used to find the magnification \(m\), since both \(d_{\mathrm{i}}\) and \(d_{\mathrm{o}}\) are known. Entering their values gives
\[m=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=-\frac{1.50 \mathrm{~m}}{0.750 \mathrm{~m}}=-2.00. \nonumber\]
Discussion
Note that the minus sign causes the magnification to be negative when the image is inverted. Ray tracing and the use of the thin lens equations produce consistent results. The thin lens equations give the most precise results, being limited only by the accuracy of the given information. Ray tracing is limited by the accuracy with which you can draw, but it is highly useful both conceptually and visually.
Real images, such as the one considered in the previous example, are formed by converging lenses whenever an object is farther from the lens than its focal length. This is true for movie projectors, cameras, and the eye. We shall refer to these as case 1 images. A case 1 image is formed when \(d_{0}>f\) and \(f\) is positive, as in Figure \(\PageIndex{10}\)(a). (A summary of the three cases or types of image formation appears at the end of this section.)
A different type of image is formed when an object, such as a person's face, is held close to a convex lens. The image is upright and larger than the object, as seen in Figure \(\PageIndex{10}\)(b), and so the lens is called a magnifier. If you slowly pull the magnifier away from the face, you will see that the magnification steadily increases until the image begins to blur. Pulling the magnifier even farther away produces an inverted image as seen in Figure \(\PageIndex{10}\)(a). The distance at which the image blurs, and beyond which it inverts, is the focal length of the lens. To use a convex lens as a magnifier, the object must be closer to the converging lens than its focal length. This is called a case 2 image. A case 2 image is formed when \(d_{\mathrm{o}}<f\) and \(f\) is positive.
Figure \(\PageIndex{11}\) uses ray tracing to show how an image is formed when an object is held closer to a converging lens than its focal length. Rays coming from a common point on the object continue to diverge after passing through the lens, but all appear to originate from a point at the location of the image. The image is on the same side of the lens as the object and is farther away from the lens than the object. This image, like all case 2 images, cannot be projected and, hence, is called a virtual image . Light rays only appear to originate at a virtual image; they do not actually pass through that location in space. A screen placed at the location of a virtual image will receive only diffuse light from the object, not focused rays from the lens. Additionally, a screen placed on the opposite side of the lens will receive rays that are still diverging, and so no image will be projected on it. We can see the magnified image with our eyes, because the lens of the eye converges the rays into a real image projected on our retina. Finally, we note that a virtual image is upright and larger than the object, meaning that the magnification is positive and greater than 1.
Definition: VIRTUAL IMAGE
An image that is on the same side of the lens as the object and cannot be projected on a screen is called a virtual image.
Example \(\PageIndex{3}\): Image Produced by a Magnifying Glass
Suppose the book page in Figure \(\PageIndex{11}\)(a) is held 7.50 cm from a convex lens of focal length 10.0 cm, such as a typical magnifying glass might have. What magnification is produced?
Strategy and Concept
We are given that \(d_{\mathrm{o}}=7.50 \mathrm{~cm}\) and \(f=10.0 \mathrm{~cm}\), so we have a situation where the object is placed closer to the lens than its focal length. We therefore expect to get a case 2 virtual image with a positive magnification that is greater than 1. Ray tracing produces an image like that shown in Figure \(\PageIndex{11}\), but we will use the thin lens equations to get numerical solutions in this example.
Solution
To find the magnification m, we try to use magnification equation, \(m=-d_{\mathrm{i}} / d_{\mathrm{o}}\). We do not have a value for \(d_{\mathrm{i}}\), so that we must first find the location of the image using lens equation. (The procedure is the same as followed in the preceding example, where \(d_{\mathrm{o}}\) and \(f\) were known.) Rearranging the magnification equation to isolate \(d_{\mathrm{i}}\) gives
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}-\frac{1}{d_{\mathrm{o}}}. \nonumber\]
Entering known values, we obtain a value for \(1 / d_{\mathrm{i}}\):
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{10.0 \mathrm{~cm}}-\frac{1}{7.50 \mathrm{~cm}}=\frac{-0.0333}{\mathrm{~cm}}. \nonumber\]
This must be inverted to find \(d_{\mathrm{i}}\):
\[d_{\mathrm{i}}=-\frac{\mathrm{cm}}{0.0333}=-30.0 \mathrm{~cm}. \nonumber\]
Now the thin lens equation can be used to find the magnification \(m\), since both \(d_{\mathrm{i}}\) and \(d_{\mathrm{o}}\) are known. Entering their values gives
\[m=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=-\frac{-30.0 \mathrm{~cm}}{7.5 \mathrm{~cm}}=4.00. \nonumber\]
Discussion
A number of results in this example are true of all case 2 images, as well as being consistent with Figure \(\PageIndex{11}\). Magnification is indeed positive (as predicted), meaning the image is upright. The magnification is also greater than 1, meaning that the image is larger than the object—in this case, by a factor of 3. Note that the image distance is negative. This means the image is on the same side of the lens as the object. Thus the image cannot be projected and is virtual. (Negative values of \(d_{\mathrm{i}}\) occur for virtual images.) The image is farther from the lens than the object, since the image distance is greater in magnitude than the object distance. The location of the image is not obvious when you look through a magnifier. In fact, since the image is bigger than the object, you may think the image is closer than the object. But the image is farther away, a fact that is useful in correcting farsightedness, as we shall see in a later section.
A third type of image is formed by a diverging or concave lens. Try looking through eyeglasses meant to correct nearsightedness. (See Figure \(\PageIndex{12}\).) You will see an image that is upright but smaller than the object. This means that the magnification is positive but less than 1. The ray diagram in Figure \(\PageIndex{13}\) shows that the image is on the same side of the lens as the object and, hence, cannot be projected—it is a virtual image. Note that the image is closer to the lens than the object. This is a case 3 image, formed for any object by a negative focal length or diverging lens.
Example \(\PageIndex{4}\): Image Produced by a Concave Lens
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?
Strategy and Concept
This example is identical to the preceding one, except that the focal length is negative for a concave or diverging lens. The method of solution is thus the same, but the results are different in important ways.
Solution
To find the magnification \(m\), we must first find the image distance \(d_{\mathrm{i}}\) using thin lens equation
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}-\frac{1}{d_{\mathrm{o}}}, \nonumber\]
or its alternative rearrangement
\[d_{i}=\frac{f d_{\mathrm{o}}}{d_{\mathrm{o}}-f}. \nonumber\]
We are given that \(f=-10.0 \mathrm{~cm}\) and \(d_{\mathrm{o}}=7.50 \mathrm{~cm}\). Entering these yields a value for \(1 / d_{\mathrm{i}}\):
\[\frac{1}{d_{\mathrm{i}}}=\frac{1}{-10.0 \mathrm{~cm}}-\frac{1}{7.50 \mathrm{~cm}}=\frac{-0.2333}{\mathrm{~cm}}. \nonumber\]
This must be inverted to find \(d_{\mathrm{i}}\):
\[d_{\mathrm{i}}=-\frac{\mathrm{cm}}{0.2333}=-4.29 \mathrm{~cm}. \nonumber\]
Or
\[d_{\mathrm{i}}=\frac{(7.5)(-10)}{(7.5-(-10))}=-75 / 17.5=-4.29 \mathrm{~cm}. \nonumber\]
Now the magnification equation can be used to find the magnification \(m\), since both \(d_{\mathrm{i}}\) and \(d_{\mathrm{o}}\) are known. Entering their values gives
\[m=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=-\frac{-4.29 \mathrm{~cm}}{7.50 \mathrm{~cm}}=0.571. \nonumber\]
Discussion
A number of results in this example are true of all case 3 images, as well as being consistent with Figure \(\PageIndex{13}\). Magnification is positive (as predicted), meaning the image is upright. The magnification is also less than 1, meaning the image is smaller than the object—in this case, a little over half its size. The image distance is negative, meaning the image is on the same side of the lens as the object. (The image is virtual.) The image is closer to the lens than the object, since the image distance is smaller in magnitude than the object distance. The location of the image is not obvious when you look through a concave lens. In fact, since the image is smaller than the object, you may think it is farther away. But the image is closer than the object, a fact that is useful in correcting nearsightedness, as we shall see in a later section.
Table \(\PageIndex{1}\) summarizes the three types of images formed by single thin lenses. These are referred to as case 1, 2, and 3 images. Convex (converging) lenses can form either real or virtual images (cases 1 and 2, respectively), whereas concave (diverging) lenses can form only virtual images (always case 3). Real images are always inverted, but they can be either larger or smaller than the object. For example, a slide projector forms an image larger than the slide, whereas a camera makes an image smaller than the object being photographed. Virtual images are always upright and cannot be projected. Virtual images are larger than the object only in case 2, where a convex lens is used. The virtual image produced by a concave lens is always smaller than the object—a case 3 image. We can see and photograph virtual images only by using an additional lens to form a real image.
TAKE-HOME EXPERIMENT: CONCENTRATING SUNLIGHT
Find several lenses and determine whether they are converging or diverging. In general those that are thicker near the edges are diverging and those that are thicker near the center are converging. On a bright sunny day take the converging lenses outside and try focusing the sunlight onto a piece of paper. Determine the focal lengths of the lenses. Be careful because the paper may start to burn, depending on the type of lens you have selected.
Section Summary
- Light rays entering a converging lens parallel to its axis cross one another at a single point on the opposite side.
- For a converging lens, the focal point is the point at which converging light rays cross; for a diverging lens, the focal point is the point from which diverging light rays appear to originate.
- The distance from the center of the lens to its focal point is called the focal length \(f\).
- Power \(P\) of a lens is defined to be the inverse of its focal length, \(P=\frac{1}{f}\).
- A lens that causes the light rays to bend away from its axis is called a diverging lens.
- Ray tracing is the technique of graphically determining the paths that light rays take.
- The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image.
- Thin lens equations are \(\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}\) and \(\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m\) (magnification).
- The distance of the image from the center of the lens is called image distance.
- An image that is on the same side of the lens as the object and cannot be projected on a screen is called a virtual image.
Glossary
- converging lens
- a convex lens in which light rays that enter it parallel to its axis converge at a single point on the opposite side
- diverging lens
- a concave lens in which light rays that enter it parallel to its axis bend away (diverge) from its axis
- focal point
- for a converging lens or mirror, the point at which converging light rays cross; for a diverging lens or mirror, the point from which diverging light rays appear to originate
- focal length
- distance from the center of a lens or curved mirror to its focal point
- magnification
- ratio of image height to object height
- power
- inverse of focal length
- real image
- image that can be projected
- virtual image
- image that cannot be projected


11.7: References
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Merck & Co. (2023). Merck manual: Professional version . https://www.merckmanuals.com/professional
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Patient Safety Authority. (2006). Improving the safety of telephone or verbal orders. PA-PSRS Patient Safety Advisory , 3 (2), 1, 3–7. patientsafety.pa.gov/ADVISORI...0606_01b.aspx#
Tariq, R. A., Vashisht, R., Sinha, A., & Scherbak, Y. (2023). Medication dispensing errors and prevention. In StatPearls [Internet]. StatPearls Publishing. www.ncbi.nlm.nih.gov/books/NBK519065/
The Joint Commission. (2022). Standards FAQs. https://www.jointcommission.org/stan...psg/000001545/
U.S. Food and Drug Administration. (2022). Finding and learning about side effects (adverse reactions) . https://www.fda.gov/drugs/informatio...erse-reactions


21.3: The Adaptive Immune Response - T Lymphocytes and Their Functional Types
Learning Objectives
- Explain the advantages of the adaptive immune response over the innate immune response
- List the various characteristics of an antigen
- Describe the types of T cell antigen receptors
- Outline the steps of T cell development
- Describe the major T cell types and their functions
Innate immune responses (and early induced responses) are in many cases ineffective at completely controlling pathogen growth. However, they slow pathogen growth and allow time for the adaptive immune response to strengthen and either control or eliminate the pathogen. The innate immune system also sends signals to the cells of the adaptive immune system, guiding them in how to attack the pathogen. Thus, these are the two important arms of the immune response.
The Benefits of the Adaptive Immune Response
The specificity of the adaptive immune response—its ability to specifically recognize and make a response against a wide variety of pathogens—is its great strength. Antigens, the small chemical groups often associated with pathogens, are recognized by receptors on the surface of B and T lymphocytes. The adaptive immune response to these antigens is so versatile that it can respond to nearly any pathogen. This increase in specificity comes because the adaptive immune response has a unique way to develop as many as 10 11 , or 100 trillion, different receptors to recognize nearly every conceivable pathogen. How could so many different types of antibodies be encoded? And what about the many specificities of T cells? There is not nearly enough DNA in a cell to have a separate gene for each specificity. The mechanism was finally worked out in the 1970s and 1980s using the new tools of molecular genetics
Primary Disease and Immunological Memory
The immune system’s first exposure to a pathogen is called a primary adaptive response . Symptoms of a first infection, called primary disease, are always relatively severe because it takes time for an initial adaptive immune response to a pathogen to become effective.
Upon re-exposure to the same pathogen, a secondary adaptive immune response is generated, which is stronger and faster that the primary response. The secondary adaptive response often eliminates a pathogen before it can cause significant tissue damage or any symptoms. Without symptoms, there is no disease, and the individual is not even aware of the infection. This secondary response is the basis of immunological memory , which protects us from getting diseases repeatedly from the same pathogen. By this mechanism, an individual’s exposure to pathogens early in life spares the person from these diseases later in life.
Self Recognition
A third important feature of the adaptive immune response is its ability to distinguish between self-antigens, those that are normally present in the body, and foreign antigens, those that might be on a potential pathogen. As T and B cells mature, there are mechanisms in place that prevent them from recognizing self-antigen, preventing a damaging immune response against the body. These mechanisms are not 100 percent effective, however, and their breakdown leads to autoimmune diseases, which will be discussed later in this chapter.
T Cell-Mediated Immune Responses
The primary cells that control the adaptive immune response are the lymphocytes, the T and B cells. T cells are particularly important, as they not only control a multitude of immune responses directly, but also control B cell immune responses in many cases as well. Thus, many of the decisions about how to attack a pathogen are made at the T cell level, and knowledge of their functional types is crucial to understanding the functioning and regulation of adaptive immune responses as a whole.
T lymphocytes recognize antigens based on a two-chain protein receptor. The most common and important of these are the alpha-beta T cell receptors (Figure \(\PageIndex{1}\)).
There are two chains in the T cell receptor, and each chain consists of two domains. The variable region domain is furthest away from the T cell membrane and is so named because its amino acid sequence varies between receptors. In contrast, the constant region domain has less variation. The differences in the amino acid sequences of the variable domains are the molecular basis of the diversity of antigens the receptor can recognize. Thus, the antigen-binding site of the receptor consists of the terminal ends of both receptor chains, and the amino acid sequences of those two areas combine to determine its antigenic specificity. Each T cell produces only one type of receptor and thus is specific for a single particular antigen.
Antigens
Antigens on pathogens are usually large and complex, and consist of many antigenic determinants. An antigenic determinant (epitope) is one of the small regions within an antigen to which a receptor can bind, and antigenic determinants are limited by the size of the receptor itself. They usually consist of six or fewer amino acid residues in a protein, or one or two sugar moieties in a carbohydrate antigen. Antigenic determinants on a carbohydrate antigen are usually less diverse than on a protein antigen. Carbohydrate antigens are found on bacterial cell walls and on red blood cells (the ABO blood group antigens). Protein antigens are complex because of the variety of three-dimensional shapes that proteins can assume, and are especially important for the immune responses to viruses and worm parasites. It is the interaction of the shape of the antigen and the complementary shape of the amino acids of the antigen-binding site that accounts for the chemical basis of specificity (Figure \(\PageIndex{2}\)).
Antigen Processing and Presentation
Although Figure \(\PageIndex{2}\) shows T cell receptors interacting with antigenic determinants directly, the mechanism that T cells use to recognize antigens is, in reality, much more complex. T cells do not recognize free-floating or cell-bound antigens as they appear on the surface of the pathogen. They only recognize antigen on the surface of specialized cells called antigen-presenting cells. Antigens are internalized by these cells. Antigen processing is a mechanism that enzymatically cleaves the antigen into smaller pieces. The antigen fragments are then brought to the cell’s surface and associated with a specialized type of antigen-presenting protein known as a major histocompatibility complex (MHC) molecule. The MHC is the cluster of genes that encode these antigen-presenting molecules. The association of the antigen fragments with an MHC molecule on the surface of a cell is known as antigen presentation and results in the recognition of antigen by a T cell. This association of antigen and MHC occurs inside the cell, and it is the complex of the two that is brought to the surface. The peptide-binding cleft is a small indentation at the end of the MHC molecule that is furthest away from the cell membrane; it is here that the processed fragment of antigen sits. MHC molecules are capable of presenting a variety of antigens, depending on the amino acid sequence, in their peptide-binding clefts. It is the combination of the MHC molecule and the fragment of the original peptide or carbohydrate that is actually physically recognized by the T cell receptor (Figure \(\PageIndex{3}\)).
Two distinct types of MHC molecules, MHC class I and MHC class II , play roles in antigen presentation. Although produced from different genes, they both have similar functions. They bring processed antigen to the surface of the cell via a transport vesicle and present the antigen to the T cell and its receptor. Antigens from different classes of pathogens, however, use different MHC classes and take different routes through the cell to get to the surface for presentation. The basic mechanism, though, is the same. Antigens are processed by digestion, are brought into the endomembrane system of the cell, and then are expressed on the surface of the antigen-presenting cell for antigen recognition by a T cell. Intracellular antigens are typical of viruses, which replicate inside the cell, and certain other intracellular parasites and bacteria. These antigens are processed in the cytosol by an enzyme complex known as the proteasome and are then brought into the endoplasmic reticulum by the transporter associated with antigen processing (TAP) system, where they interact with class I MHC molecules and are eventually transported to the cell surface by a transport vesicle.
Extracellular antigens, characteristic of many bacteria, parasites, and fungi that do not replicate inside the cell’s cytoplasm, are brought into the endomembrane system of the cell by receptor-mediated endocytosis. The resulting vesicle fuses with vesicles from the Golgi complex, which contain pre-formed MHC class II molecules. After fusion of these two vesicles and the association of antigen and MHC, the new vesicle makes its way to the cell surface.
Professional Antigen-presenting Cells
Many cell types express class I molecules for the presentation of intracellular antigens. These MHC molecules may then stimulate a cytotoxic T cell immune response, eventually destroying the cell and the pathogen within. This is especially important when it comes to the most common class of intracellular pathogens, the virus. Viruses infect nearly every tissue of the body, so all these tissues must necessarily be able to express class I MHC or no T cell response can be made.
On the other hand, class II MHC molecules are expressed only on the cells of the immune system, specifically cells that affect other arms of the immune response. Thus, these cells are called “professional” antigen-presenting cells to distinguish them from those that bear class I MHC. The three types of professional antigen presenters are macrophages, dendritic cells, and B cells (Table).
Macrophages stimulate T cells to release cytokines that enhance phagocytosis. Dendritic cells also kill pathogens by phagocytosis (see Figure \(\PageIndex{3}\)), but their major function is to bring antigens to regional draining lymph nodes. The lymph nodes are the locations in which most T cell responses against pathogens of the interstitial tissues are mounted. Macrophages are found in the skin and in the lining of mucosal surfaces, such as the nasopharynx, stomach, lungs, and intestines. B cells may also present antigens to T cells, which are necessary for certain types of antibody responses, to be covered later in this chapter.
| Classes of Antigen-presenting Cells | |||
|---|---|---|---|
| MHC | Cell type | Phagocytic? | Function |
| Class I | Many | No | Stimulates cytotoxic T cell immune response |
| Class II | Macrophage | Yes | Stimulates phagocytosis and presentation at primary infection site |
| Class II | Dendritic | Yes, in tissues | Brings antigens to regional lymph nodes |
| Class II | B cell | Yes, internalizes surface Ig and antigen | Stimulates antibody secretion by B cells |
T Cell Development and Differentiation
The process of eliminating T cells that might attack the cells of one’s own body is referred to as T cell tolerance . While thymocytes are in the cortex of the thymus, they are referred to as “double negatives,” meaning that they do not bear the CD4 or CD8 molecules that you can use to follow their pathways of differentiation (Figure \(\PageIndex{4}\)). In the cortex of the thymus, they are exposed to cortical epithelial cells. In a process known as positive selection , double-negative thymocytes bind to the MHC molecules they observe on the thymic epithelia, and the MHC molecules of “self” are selected. This mechanism kills many thymocytes during T cell differentiation. In fact, only two percent of the thymocytes that enter the thymus leave it as mature, functional T cells.
Later, the cells become double positives that express both CD4 and CD8 markers and move from the cortex to the junction between the cortex and medulla. It is here that negative selection takes place. In negative selection , self-antigens are brought into the thymus from other parts of the body by professional antigen-presenting cells. The T cells that bind to these self-antigens are selected for negatively and are killed by apoptosis. In summary, the only T cells left are those that can bind to MHC molecules of the body with foreign antigens presented on their binding clefts, preventing an attack on one’s own body tissues, at least under normal circumstances. Tolerance can be broken, however, by the development of an autoimmune response, to be discussed later in this chapter.
The cells that leave the thymus become single positives, expressing either CD4 or CD8, but not both (see Figure \(\PageIndex{4}\)). The CD4 + T cells will bind to class II MHC and the CD8 + cells will bind to class I MHC. The discussion that follows explains the functions of these molecules and how they can be used to differentiate between the different T cell functional types.
Mechanisms of T Cell-mediated Immune Responses
Mature T cells become activated by recognizing processed foreign antigen in association with a self-MHC molecule and begin dividing rapidly by mitosis. This proliferation of T cells is called clonal expansion and is necessary to make the immune response strong enough to effectively control a pathogen. How does the body select only those T cells that are needed against a specific pathogen? Again, the specificity of a T cell is based on the amino acid sequence and the three-dimensional shape of the antigen-binding site formed by the variable regions of the two chains of the T cell receptor (Figure \(\PageIndex{5}\)). Clonal selection is the process of antigen binding only to those T cells that have receptors specific to that antigen. Each T cell that is activated has a specific receptor “hard-wired” into its DNA, and all of its progeny will have identical DNA and T cell receptors, forming clones of the original T cell.
Clonal Selection and Expansion
The clonal selection theory was proposed by Frank Burnet in the 1950s. However, the term clonal selection is not a complete description of the theory, as clonal expansion goes hand in glove with the selection process. The main tenet of the theory is that a typical individual has a multitude (10 11 ) of different types of T cell clones based on their receptors. In this use, a clone is a group of lymphocytes that share the same antigen receptor . Each clone is necessarily present in the body in low numbers. Otherwise, the body would not have room for lymphocytes with so many specificities.
Only those clones of lymphocytes whose receptors are activated by the antigen are stimulated to proliferate. Keep in mind that most antigens have multiple antigenic determinants, so a T cell response to a typical antigen involves a polyclonal response. A polyclonal response is the stimulation of multiple T cell clones. Once activated, the selected clones increase in number and make many copies of each cell type, each clone with its unique receptor. By the time this process is complete, the body will have large numbers of specific lymphocytes available to fight the infection (see Figure \(\PageIndex{5}\)).
The Cellular Basis of Immunological Memory
As already discussed, one of the major features of an adaptive immune response is the development of immunological memory.
During a primary adaptive immune response, both memory T cells and effector T cells are generated. Memory T cells are long-lived and can even persist for a lifetime. Memory cells are primed to act rapidly. Thus, any subsequent exposure to the pathogen will elicit a very rapid T cell response. This rapid, secondary adaptive response generates large numbers of effector T cells so fast that the pathogen is often overwhelmed before it can cause any symptoms of disease. This is what is meant by immunity to a disease. The same pattern of primary and secondary immune responses occurs in B cells and the antibody response, as will be discussed later in the chapter.
T Cell Types and their Functions
In the discussion of T cell development, you saw that mature T cells express either the CD4 marker or the CD8 marker, but not both. These markers are cell adhesion molecules that keep the T cell in close contact with the antigen-presenting cell by directly binding to the MHC molecule (to a different part of the molecule than does the antigen). Thus, T cells and antigen-presenting cells are held together in two ways: by CD4 or CD8 attaching to MHC and by the T cell receptor binding to antigen (Figure \(\PageIndex{6}\)).
Although the correlation is not 100 percent, CD4-bearing T cells are associated with helper functions and CD8-bearing T cells are associated with cytotoxicity. These functional distinctions based on CD4 and CD8 markers are useful in defining the function of each type.
Helper T Cells and their Cytokines
Helper T cells (Th) , bearing the CD4 molecule, function by secreting cytokines that act to enhance other immune responses. There are two classes of Th cells, and they act on different components of the immune response. These cells are not distinguished by their surface molecules but by the characteristic set of cytokines they secrete (Table).
Th1 cells are a type of helper T cell that secretes cytokines that regulate the immunological activity and development of a variety of cells, including macrophages and other types of T cells.
Th2 cells , on the other hand, are cytokine-secreting cells that act on B cells to drive their differentiation into plasma cells that make antibody. In fact, T cell help is required for antibody responses to most protein antigens, and these are called T cell-dependent antigens.
Cytotoxic T cells
Cytotoxic T cells (Tc) are T cells that kill target cells by inducing apoptosis using the same mechanism as NK cells. They either express Fas ligand, which binds to the fas molecule on the target cell, or act by using perforins and granzymes contained in their cytoplasmic granules. As was discussed earlier with NK cells, killing a virally infected cell before the virus can complete its replication cycle results in the production of no infectious particles. As more Tc cells are developed during an immune response, they overwhelm the ability of the virus to cause disease. In addition, each Tc cell can kill more than one target cell, making them especially effective. Tc cells are so important in the antiviral immune response that some speculate that this was the main reason the adaptive immune response evolved in the first place.
Regulatory T Cells
Regulatory T cells (Treg) , or suppressor T cells, are the most recently discovered of the types listed here, so less is understood about them. In addition to CD4, they bear the molecules CD25 and FOXP3. Exactly how they function is still under investigation, but it is known that they suppress other T cell immune responses. This is an important feature of the immune response, because if clonal expansion during immune responses were allowed to continue uncontrolled, these responses could lead to autoimmune diseases and other medical issues.
Not only do T cells directly destroy pathogens, but they regulate nearly all other types of the adaptive immune response as well, as evidenced by the functions of the T cell types, their surface markers, the cells they work on, and the types of pathogens they work against (see Table).
| Functions of T Cell Types and Their Cytokines | ||||||
|---|---|---|---|---|---|---|
| T cell | Main target | Function | Pathogen | Surface marker | MHC | Cytokines or mediators |
| Tc | Infected cells | Cytotoxicity | Intracellular | CD8 | Class I | Perforins, granzymes, and fas ligand |
| Th1 | Macrophage | Helper inducer | Extracellular | CD4 | Class II | Interferon-γ and TGF-β |
| Th2 | B cell | Helper inducer | Extracellular | CD4 | Class II | IL-4, IL-6, IL-10, and others |
| Treg | Th cell | Suppressor | None | CD4, CD25 | ? | TGF-β and IL-10 |
Chapter Review
T cells recognize antigens with their antigen receptor, a complex of two protein chains on their surface. They do not recognize self-antigens, however, but only processed antigen presented on their surfaces in a binding groove of a major histocompatibility complex molecule. T cells develop in the thymus, where they learn to use self-MHC molecules to recognize only foreign antigens, thus making them tolerant to self-antigens. There are several functional types of T lymphocytes, the major ones being helper, regulatory, and cytotoxic T cells.
Review Chapter
Q. T cells that secrete cytokines that help antibody responses are called ________.
A. Th1
B. Th2
C. regulatory T cells
D. thymocytes
Answer: B
Q. The taking in of antigen and digesting it for later presentation is called ________.
A. antigen presentation
B. antigen processing
C. endocytosis
D. exocytosis
Answer: B
Q. Why is clonal expansion so important?
A. to select for specific cells
B. to secrete cytokines
C. to kill target cells
D. to increase the numbers of specific cells
Answer: D
Q. The elimination of self-reactive thymocytes is called ________.
A. positive selection.
B. negative selection.
C. tolerance.
D. clonal selection.
Answer: B
Q. Which type of T cell is most effective against viruses?
A. Th1
B. Th2
C. cytotoxic T cells
D. regulatory T cells
Answer: C
Critical Thinking Questions
Q. Describe the processing and presentation of an intracellular antigen.
A. The antigen is digested by the proteasome, brought into the endoplasmic reticulum by the TAP transporter system, where it binds to class I MHC molecules. These are taken to the cell surface by transport vesicles.
Q. Describe clonal selection and expansion.
A. Antigen-specific clones are stimulated as their antigen receptor binds to antigen. They are then activated and proliferate, expanding their numbers. The result is a large number of antigen-specific lymphocytes.
Glossary
- antigenic determinant
- (also, epitope) one of the chemical groups recognized by a single type of lymphocyte antigen receptor
- antigen presentation
- binding of processed antigen to the protein-binding cleft of a major histocompatibility complex molecule
- antigen processing
- internalization and digestion of antigen in an antigen-presenting cell
- antigen receptor
- two-chain receptor by which lymphocytes recognize antigen
- clone
- group of lymphocytes sharing the same antigen receptor
- clonal expansion
- growth of a clone of selected lymphocytes
- clonal selection
- stimulating growth of lymphocytes that have specific receptors
- constant region domain
- part of a lymphocyte antigen receptor that does not vary much between different receptor types
- cytotoxic T cells (Tc)
- T lymphocytes with the ability to induce apoptosis in target cells
- effector T cells
- immune cells with a direct, adverse effect on a pathogen
- helper T cells (Th)
- T cells that secrete cytokines to enhance other immune responses, involved in activation of both B and T cell lymphocytes
- immunological memory
- ability of the adaptive immune response to mount a stronger and faster immune response upon re-exposure to a pathogen
- major histocompatibility complex (MHC)
- gene cluster whose proteins present antigens to T cells
- memory T cells
- long-lived immune cell reserved for future exposure to an pathogen
- MHC class I
- found on most cells of the body, it binds to the CD8 molecule on T cells
- MHC class II
- found on macrophages, dendritic cells, and B cells, it binds to CD4 molecules on T cells
- negative selection
- selection against thymocytes in the thymus that react with self-antigen
- polyclonal response
- response by multiple clones to a complex antigen with many determinants
- primary adaptive response
- immune system’s response to the first exposure to a pathogen
- positive selection
- selection of thymocytes within the thymus that interact with self, but not non-self, MHC molecules
- regulatory T cells (Treg)
- (also, suppressor T cells) class of CD4 T cells that regulates other T cell responses
- secondary adaptive response
- immune response observed upon re-exposure to a pathogen, which is stronger and faster than a primary response
- T cell tolerance
- process during T cell differentiation where most T cells that recognize antigens from one’s own body are destroyed
- Th1 cells
- cells that secrete cytokines that enhance the activity of macrophages and other cells
- Th2 cells
- cells that secrete cytokines that induce B cells to differentiate into antibody-secreting plasma cells
- variable region domain
- part of a lymphocyte antigen receptor that varies considerably between different receptor types


8.2: A Key Property of Fractions
We know that \(\frac{a}{b}\) is the answer to a division problem; for example, if \(a\) and \(b\) are both positive integers, then \(\frac{a}{b}\) represents the amount of pie an individual child receives when pies are shared equally by children.
What happens if we double the number of pie and double the number of kids? Nothing! The amount of pie per child is still the same. In other words, we have the following fact.
\(\frac{2a}{2b} = \frac{a}{b}\)
For example, as the picture shows, \(\frac{6}{3}\) and \(\frac{12}{6}\) both give two pies for each child.
And tripling the number of pies and the number of children also does not change the final amount of pies per child, nor does quadrupling each number, or one trillion-billion-tupling the numbers!
\(\frac{6}{3} = \frac{12}{6} = \frac{18}{9} = \ldots =\) two pies per child}
This leads us to want to believe what turns out to be a key property of fractions.
For any real numbers \(a\), \(b\), and \(x\) such that \(b \neq 0\) and \(x \neq 0\), \(\frac{xa}{xb} = \frac{a}{b}\).
Note that we have only thought about this for the special case where \(a\), \(b\), and \(x\) are positive integers; but it is actually true for many more possibilities.
We say that the fractions \(\frac{xa}{xb}\) and \(\frac{a}{b}\) are equivalent .
Write down at least 3 fractions equivalent to each of the following. Share your answers with another person and see if they agree that your answers are appropriate.
- \(\frac{1}{2}\)
- \(\frac{10}{3}\)
- 1
Note that in each case above, we started with smaller values in the numerator and denominator and then considered cases in which the values are larger. What if we went the other way.
\(\frac{20}{32}\) is the value equal to the amount an individual child gets when 32 kids share 20 pies.
This is the same result for when 8 kids share 5 pies.
\(\frac{20}{32} = \frac{5 \cdot 4}{8 \cdot 4} = \frac{5}{8}\)
Many people say we have cancelled the 4. Unfortunately, the word "cancelled" gets used for many situations in mathematics and can hide what is actually going on. It can also make it seem like we just get to take the 4 out because it's a weird trick or magic. Instead, the following phrases are better.
- \(\frac{20}{32} = \frac{5 \cdot 4}{8 \cdot 4} = \frac{5}{8}\)shows what happens when we divide out the common factor 4 from the numerator and denominator.
- \(\frac{20}{32} = \frac{5 \cdot 4}{8 \cdot 4} = \frac{5}{8}\) shows the process of reducing the fraction to lowest terms.
- \(\frac{20}{32} = \frac{5 \cdot 4}{8 \cdot 4} = \frac{5}{8}\) illustrates ; simplifying the fraction.
Each of these phrasings have advantages. It is a good idea to be comfortable with all of them and to try to use them instead of "cancelling".
As another example, \(\frac{280}{350}\) can certainly be simplified by noticing that there is a common factor of 10 in both the numerator and the denominator.
\(\frac{280}{350} = \frac{28 \cdot 10}{35 \cdot 10} = \frac{28}{35}\)
We can go further as 28 and 35 are both multiples of 7.
\(\frac{28}{35} = \frac{4 \cdot 7}{5 \cdot 7} = \frac{4}{5}\)
Thus, sharing 280 pies among 350 children gives the same result as sharing 4 pies among 5 children!
\(\frac{280}{350} = \frac{4}{5}\)
Since 4 and 5 share no common factors, this is as far as we can go with this example (while keeping the numerator and denominator equal to positive integers).
On Your Own
Mix and Match: Which simplified fraction goes with each non-simplified fraction fraction? (Notice that there are fewer simplified fractions than non-simplified frctions!)
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See if you can come up with a good explanation for why all of this works. Then share it with another person and see if they believe your explanation.


1.2.5: Overview of Data Collection Principles
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The first step in conducting research is to identify topics or questions that are to be investigated. A clearly laid out research question is helpful in identifying what subjects or cases should be studied and what variables are important. It is also important to consider how data are collected so that they are reliable and help achieve the research goals.
Populations and samples
Consider the following three research questions:
- What is the average mercury content in sword sh in the Atlantic Ocean?
- Over the last 5 years, what is the average time to degree for Duke undergraduate students?
- Does a new drug reduce the number of deaths in patients with severe heart disease?
Each research question refers to a target population. In the rst question, the target population is all sword sh in the Atlantic ocean, and each sh represents a case. Often times, it is too expensive to collect data for every case in a population. Instead, a sample is taken. A sample represents a subset of the cases and is often a small fraction of the population. For instance, 60 sword sh (or some other number) in the population might be selected, and this sample data may be used to provide an estimate of the population average and answer the research question.
Exercise
Exercise 1.7 For the second and third questions above, identify the target population and what represents an individual case. 10
Anecdotal Evidence
Consider the following possible responses to the three research questions:
- A man on the news got mercury poisoning from eating sword sh, so the average mercury concentration in sword sh must be dangerously high.
- I met two students who took more than 7 years to graduate from Duke, so it must take longer to graduate at Duke than at many other colleges.
- My friend's dad had a heart attack and died after they gave him a new heart disease drug, so the drug must not work.
Each of the conclusions are based on some data. However, there are two problems. First, the data only represent one or two cases. Second, and more importantly, it is unclear whether these cases are actually representative of the population. Data collected in this haphazard fashion are called anecdotal evidence .
10 (2) Notice that the rst question is only relevant to students who complete their degree; the average cannot be computed using a student who never nished her degree. Thus, only Duke undergraduate students who have graduated in the last ve years represent cases in the population under consideration. Each such student would represent an individual case. (3) A person with severe heart disease represents a case. The population includes all people with severe heart disease.
Anecdotal evidence
Be careful of data collected in a haphazard fashion. Such evidence may be true and veri able, but it may only represent extraordinary cases.
Anecdotal evidence typically is composed of unusual cases that we recall based on their striking characteristics. For instance, we are more likely to remember the two people we met who took 7 years to graduate than the six others who graduated in four years. Instead of looking at the most unusual cases, we should examine a sample of many cases that represent the population.
Sampling from a Population
We might try to estimate the time to graduation for Duke undergraduates in the last 5 years by collecting a sample of students. All graduates in the last 5 years represent the population, and graduates who are selected for review are collectively called the sample. In general, we always seek to randomly select a sample from a population. The most basic type of random selection is equivalent to how raffles are conducted. For example, in selecting graduates, we could write each graduate's name on a raffle ticket and draw 100 tickets. The selected names would represent a random sample of 100 graduates. Why pick a sample randomly? Why not just pick a sample by hand? Consider the following scenario.
Example
Example 1.8 Suppose we ask a student who happens to be majoring in nutrition to select several graduates for the study. What kind of students do you think she might collect? Do you think her sample would be representative of all graduates?
Perhaps she would pick a disproportionate number of graduates from health-related fields. Or perhaps her selection would be well-representative of the population. When selecting samples by hand, we run the risk of picking a biased sample, even if that bias is unintentional or difficult to discern.
If someone was permitted to pick and choose exactly which graduates were included in the sample, it is entirely possible that the sample could be skewed to that person's interests, which may be entirely unintentional. This introduces bias into a sample. Sampling randomly helps resolve this problem. The most basic random sample is called a simple random sample , and it is the equivalent of using a raffle to select cases. This means that each case in the population has an equal chance of being included and there is no implied connection between the cases in the sample.
The act of taking a simple random sample helps minimize bias, however, bias can crop up in other ways. Even when people are picked at random, e.g. for surveys, caution must be exercised if the non-response is high. For instance, if only 30% of the people randomly sampled for a survey actually respond, then it is unclear whether the results are representative of the entire population. This non-response bias can skew results.
Another common downfall is a convenience sample , where individuals who are easily accessible are more likely to be included in the sample. For instance, if a political survey is done by stopping people walking in the Bronx, this will not represent all of New York City. It is often diffcult to discern what sub-population a convenience sample represents.
Exercise
Exercise 1.9 We can easily access ratings for products, sellers, and companies through websites. These ratings are based only on those people who go out of their way to provide a rating. If 50% of online reviews for a product are negative, do you think this means that 50% of buyers are dissatisfied with the product? 11
11 Answers will vary. From our own anecdotal experiences, we believe people tend to rant more about products that fell below expectations than rave about those that perform as expected. For this reason, we suspect there is a negative bias in product ratings on sites like Amazon. However, since our experiences may not be representative, we also keep an open mind should data on the subject become available.
Explanatory and Response Variables
Consider the following question from page 7 for the county data set:
(1) Is federal spending, on average, higher or lower in counties with high rates of poverty?
If we suspect poverty might a ect spending in a county, then poverty is the explanatory variable and federal spending is the response variable in the relationship. 12 If there are many variables, it may be possible to consider a number of them as explanatory variables.
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TIP: Explanatory and response variables To identify the explanatory variable in a pair of variables, identify which of the two is suspected of a ecting the other and plan an appropriate analysis. \[\text{explanatory variable}\xrightarrow { \text {might affect}} \text {response variable}\] |
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Caution: association does not imply causation Labeling variables as explanatory and response does not guarantee the relationship between the two is actually causal, even if there is an association identi ed between the two variables. We use these labels only to keep track of which variable we suspect a ects the other. |
In some cases, there is no explanatory or response variable. Consider the following question from page 7:
(2) If homeownership is lower than the national average in one county, will the percent of multi-unit structures in that county likely be above or below the national average?
It is difficult to decide which of these variables should be considered the explanatory and response variable, i.e. the direction is ambiguous, so no explanatory or response labels are suggested here.
12 Sometimes the explanatory variable is called the independent variable and the response variable is called the dependent variable. However, this becomes confusing since a pair of variables might be independent or dependent, so we avoid this language.
Introducing observational studies and experiments
There are two primary types of data collection: observational studies and experiments.
Researchers perform an observational study when they collect data in a way that does not directly interfere with how the data arise. For instance, researchers may collect information via surveys, review medical or company records, or follow a cohort of many similar individuals to study why certain diseases might develop. In each of these situations, researchers merely observe the data that arise. In general, observational studies can provide evidence of a naturally occurring association between variables, but they cannot by themselves show a causal connection.
When researchers want to investigate the possibility of a causal connection, they conduct an experiment . Usually there will be both an explanatory and a response variable. For instance, we may suspect administering a drug will reduce mortality in heart attack patients over the following year. To check if there really is a causal connection between the explanatory variable and the response, researchers will collect a sample of individuals and split them into groups. The individuals in each group are assigned a treatment. When individuals are randomly assigned to a group, the experiment is called a randomized experiment . For example, each heart attack patient in the drug trial could be randomly assigned, perhaps by flipping a coin, into one of two groups: the first group receives a placebo (fake treatment) and the second group receives the drug. See the case study in Section 1.1 for another example of an experiment, though that study did not employ a placebo.
TIP: association \(\ne\) causation
In general, association does not imply causation, and causation can only be inferred from a randomized experiment.


5.6: Remote sensing of primary production and biomass
Understanding regional variability in carbon cycle processes requires a dramatically more spatially detailed analysis of global land surface processes than was available to climate scientists before 2000. In March 2000, the Moderate Resolution Imaging Spectroradiometer (MODIS), aboard NASA's Terra and Aqua satellites, began producing a regular global estimate of near-weekly gross photosynthesis (GPP) and annual net primary production (NPP) of the entire terrestrial Earth surface at up to 1 km spatial resolution. However, ecosystem respiration, and therefore the full terrestrial carbon balance, cannot be estimated from satellite measurements, but only from models or ground-based measurements.
The GPP and NPP products are designed to provide an accurate, regular measure of the production activity or growth of terrestrial vegetation. (Note: only NPP is available in the NEO site.) These products have both theoretical and practical utility. The theoretical use is primarily for defining the seasonally dynamic terrestrial surface carbon dioxide uptake for global carbon cycle studies such as answering the "missing sink question" of carbon. The spatial and seasonal dynamics of carbon dioxide flux are also of high interest in global climate modeling because carbon dioxide is an important greenhouse gas. Both GPP and RE, and therefore the terrestrial carbon balance, can be modeled. Currently, global carbon cycle models are being integrated with climate models with the goal of creating integrated Earth Systems Models that will represent the dynamic interactions between the atmosphere, biosphere, and oceans. The weekly and monthly MODIS GPP and NPP products are useful to constrain models that are used to answer carbon dioxide flux questions.
The practical utility of these GPP/ NPP products includes routine measurements of crop yield, range forage, and forest production, and other economically and socially significant products of vegetation growth. The value of an unbiased, regular source of crop, range, and forest production estimates for global political and economic decision-making is immense. Moreover, these products are freely available for all users worldwide. This daily computed GPP more correctly defines terrestrial carbon dioxide fluxes than simple Normalized Difference Vegetation Index (NDVI) correlations currently done to increase understanding on how the seasonal fluxes of net photosynthesis are related to seasonal variations of driving factors such as atmospheric carbon dioxide, temperature, light, and precipitation.
Source:
NASA Earth Observatory, Net Primary Production, Accessed December 2023, https://neo.gsfc.nasa.gov/view.php?datasetId=MOD17A3H_Y_NPP


10: Leadership Last updated Save as PDF Page ID 47569 Lumen Learning Lumen Learning 10.1: Introduction to Transformational and Transactional Theories of Leadership 10.2: Transformational and Transactional Theories of Leadership 10.3: Putting It Together- Leadership 10.4: Assignment- Leadership at Sun City Boards 10.5: Assignment- Evaluating Leadership 10.6: Discussion- Sun City Boards’ Self-Leadership Audit 10.7: Discussion- What Makes a Good Leader? 10.8: Why It Matters- Leadership 10.9: Introduction to Leadership 10.10: What Is Leadership? 10.11: Introduction to What Makes an Effective Leader 10.12: What Makes an Effective Leader? 10.13: Introduction to Effective vs. Poor Leadership 10.14: Effective vs. Poor Leadership 10.15: Introduction to Situational Theories of Leadership 10.16: Situational Theories of Leadership


18.6: Summary
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In this chapter I talked about several key programming concepts, things that you should know about if you want to start converting your simple scripts into full fledged programs:
- Writing and using scripts (Section 8.1).
- Using loops (Section 8.2) and implicit loops (Section 8.5).
- Making conditional statements (Section 8.3)
- Writing your own functions (Section 8.4)
As always, there are lots of things I’m ignoring in this chapter. It takes a lot of work to become a proper programmer, just as it takes a lot of work to be a proper psychologist or a proper statistician, and this book is certainly not going to provide you with all the tools you need to make that step. However, you’d be amazed at how much you can achieve using only the tools that I’ve covered up to this point. Loops, conditionals and functions are very powerful things, especially when combined with the various tools discussed in Chapters 3, 4 and 7. Believe it or not, you’re off to a pretty good start just by having made it to this point. If you want to keep going, there are (as always!) several other books you might want to look at. One that I’ve read and enjoyed is “A first course in statistical programming with R” Braun and Murdoch (2007), but quite a few people have suggested to me that “The art of programming with R” Matloff and Matloff (2011) is worth the effort too.
References
Braun, John, and Duncan J Murdoch. 2007. A First Course in Statistical Programming with R . Cambridge University Press Cambridge.
Matloff, Norman, and Norman S Matloff. 2011. The Art of R Programming: A Tour of Statistical Software Design . No Starch Press.
- The quote comes from Count Zero (1986)
- Okay, I lied. Sue me. One of the coolest features of Rstudio is the support for R Markdown , which lets you embed R code inside a Markdown document, and you can automatically publish your R Markdown to the web on Rstudio’s servers. If you’re the kind of nerd interested in this sort of thing, it’s really nice. And, yes, since I’m also that kind of nerd, of course I’m aware that iPython notebooks do the same thing and that R just nicked their idea. So what? It’s still cool. And anyway, this book isn’t called Learning Statistics with Python now, is it? Hm. Maybe I should write a Python version…
- As an aside: if there’s only a single command that you want to include inside your loop, then you don’t actually need to bother including the curly braces at all. However, until you’re comfortable programming in R I’d advise always using them, even when you don’t have to.
- Okay, fine. This example is still a bit ridiculous, in three respects. Firstly, the bank absolutely will not let the couple pay less than the amount required to terminate the loan in 30 years. Secondly, a constant interest rate of 30 years is hilarious. Thirdly, you can solve this much more efficiently than through brute force simulation. However, we’re not exactly in the business of being realistic or efficient here.
- Lexical scope.
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The
assign()
function. - Yes.
- Or a list of such variables.


1.92: Semidirect product
In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal.
Let G be a group, N a normal subgroup of G (i.e., N ◁ G ) and H a subgroup of G . G is a semidirect product of N and H if there exists a homomorphism G → H which is the identity on H and whose kernel is N . This is equivalent to say that:
- G = NH and N ∩ H = {1} (where "1" is identity element of G )
- G = HN and N ∩ H = {1}
- Every element of G can be written as a unique product of an element of N and an element of H
- Every element of G can be written as a unique product of an element of H and an element of N
One also says that " G splits over N ".


2.2: Introduction
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Native Americans long dominated the vastness of the American West. Linked culturally and geographically by trade, travel, and warfare, various indigenous groups controlled most of the continent west of the Mississippi River deep into the nineteenth century. Spanish, French, British, and later American traders had integrated themselves into many regional economies, and American emigrants pushed ever westward, but no imperial power had yet achieved anything approximating political or military control over the great bulk of the continent. But then the Civil War came and went and decoupled the West from the question of slavery just as the United States industrialized and laid down rails and pushed its ever-expanding population ever farther west.
Indigenous Americans had lived in North America for over ten millennia and, into the late nineteenth century, perhaps as many as 250,000 Natives still inhabited the American West. 1 But then unending waves of American settlers, the American military, and the unstoppable onrush of American capital conquered all. The United States removed Native groups to ever-shrinking reservations, incorporated the West first as territories and then as states, and, for the first time in its history, controlled the enormity of land between the two oceans.
The history of the late-nineteenth-century West is many-sided. Tragedy for some, triumph for others, the many intertwined histories of the American West marked a pivotal transformation in the history of the United States.


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cisgender
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individuals whose sex assigned at birth and gender identity are congruent
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gender dysphoria
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a deep sense of unease, anxiety, or discomfort that may occur in people whose gender does not align with their sex assigned at birth
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gender expression
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ways in which a person outwardly expresses their gender identity
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gender fluid
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a person who does not identify with a single fixed gender or has a changing or unfixed gender identity
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gender identity
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a person’s deeply held beliefs about who they are from a sexual perspective and with which, if any, gender they identify
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gender nonconforming
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a gender presentation that is outside the typical binary presentation; it can also define an individual who identifies outside the man–woman binary.
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gynecomastia
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a condition in which male breast tissue becomes enlarged and overdeveloped
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nonbinary
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an individual whose gender identity does not fit into the man–woman dichotomy
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transgender
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a broad term used by individuals whose sex assigned at birth does not match the gender they feel they are
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transsexual
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a term that sometimes may be used interchangeably with
transgender
to indicate an individual whose sex assigned at birth and gender identity do not match


7.4: Partnerships
What you’ll learn to do: discuss the advantages and disadvantages of partnerships
In this section we’ll discuss different kinds of partnerships and the pros and cons of each.
Learning Objectives
- Describe the difference between general and limited partnerships
- Discuss the advantages and disadvantages of partnerships
Partnerships
A partnership is a single business in which two or more people share ownership.
Each partner contributes to all aspects of the business, including money, property, labor, or skill. In return, each partner shares in the profits and losses of the business.
Because partnerships entail more than one person in the decision-making process, it’s important to discuss a wide variety of issues up front and develop a legal partnership agreement. This agreement should document how future business decisions will be made, including how the partners will divide profits, resolve disputes, change ownership (bring in new partners or buy out current partners), and how to dissolve the partnership. Although partnership agreements are not legally required, they are strongly recommended, and it’s considered extremely risky to operate without one.
Types of Partnerships
There are two general types of partnership arrangements:
- General Partnerships assume that profits, liability, and management duties are divided equally among partners. If you opt for an unequal distribution, the percentages assigned to each partner must be documented in the partnership agreement.
- Limited Partnerships (also known as a partnership with limited liability) are more complex than general partnerships. Limited partnerships allow partners to have limited liability as well as limited input with management decisions. These limits depend on the extent of each partner’s investment percentage. Limited partnerships are attractive to investors of short-term projects.
Forming a Partnership
To form a partnership, you must register your business with your state, a process generally handled through your Secretary of State’s office.
You’ll also need to establish your business name. For partnerships, your legal name is the name given in your partnership agreement. If you choose to operate under a name different from the officially registered name, you will most likely have to file a fictitious name (also known as an assumed name, trade name, or DBA name, short for “doing business as”).
Once your business is registered, you must obtain business licenses and permits. Regulations vary by industry, state, and locality.
Partnership Taxes
Most businesses will need to register with the IRS, register with state and local revenue agencies, and obtain a tax ID number or permit. An additional requirement for partnerships is that they must file an “annual information return” to report the income, deductions, gains and losses from the business’s operations, but the business itself does not pay income tax . Instead, the business “passes through” any profits or losses to its partners. Partners include their respective share of the partnership’s income or loss on their personal tax returns. Like sole proprietors, partners in the partnership are responsible for several additional taxes, including income tax, self-employment tax, and estimated tax. Since partnerships can be complex, having a professional to advise the partnership and partners on tax matters is crucial.
Advantages of a Partnership
- Easy and Inexpensive . Partnerships are generally an inexpensive and easily formed business structure. The majority of time spent starting a partnership often focuses on developing the partnership agreement.
- Shared Financial Commitment . In a partnership, each partner is equally invested in the success of the business. Partnerships have the advantage of pooling resources to obtain capital. This can be beneficial in terms of securing credit or by simply doubling the seed money available.
- Complementary Skills. A good partnership should capitalize on the benefits of being able to utilize the strengths, resources, and expertise of each partner.
- Partnership Incentives for Employees. Partnerships have an employment advantage over other entities if they offer employees the opportunity to become a partner. Partnership incentives often attract highly motivated and qualified employees.
Disadvantages of a Partnership
- Joint and Individual Liability. Similar to sole proprietorships, partnerships retain full, shared liability among the owners. Partners are not only liable for their own actions but also for the business debts and decisions made by other partners. In addition, the personal assets of all partners can be used to satisfy the partnership’s debt.
- Disagreements Among Partners . With multiple partners, there are bound to be disagreements. Partners should consult one another on all decisions, make compromises, and resolve disputes as amicably as possible.
- Shared Profits. Because partnerships are jointly owned, each partner must share the successes and profits of their business with the other partners. An unequal contribution of time, effort, or resources can cause discord among partners.
tw construction or t&t construction?
For several months Tom has been operating as a sole proprietorship and enjoying the control he maintains over his work and finances. Business is picking up and he has recently been contacted by a construction firm that wants to hire him to provide the trim carpentry for several large oceanfront homes they are building. He mentions this to his friend Todd, who seems very happy that Tom’s new business venture appears to be succeeding. A month later, Todd calls and asks Tom to meet him for dinner at Sandbar’s. During dinner, Todd proposes to Tom that the two of them form a general partnership: T&T Construction. Todd points out that taking on several large jobs as a sole proprietor is very risky—a partnership would mean shared risk and responsibility. He also offers to contribute some initial capital to the newly formed partnership, which would provide financial support for their day-to-day operations. Finally, Todd makes the case that, as a frame carpenter, he has skills that would complement Tom’s and potentially yield additional business opportunities.
Surprised by the proposal, Tom tells his friend that he needs some time to think it over before committing. During the next few days, he calls his accountant to find out how the partnership would impact his business. He learns that he would have to share control of the business and also share the profits. That doesn’t sound bad to Tom, especially if the business really grew—which it might, with the addition of Todd’s skills and labor. Tom is leaning toward accepting the offer. But when he finds out that he would be held responsible not only for the debts of the business but also the actions of his partner, he sours on the idea. He knows Todd has made some business decisions and deals that were a little on the sketchy side. Under the proposed General Partnership structure, if Todd made similar kinds of decisions or deals without Tom’s knowledge, Tom could still be held responsible and liable for the consequences. He’s realizes he’s not willing to accept that kind of risk. He decides to turn down Todd’s offer and keep running his business as a sole proprietor.


4.2.4: Formation of Ionic Compounds
Binary compounds are composed of just two different elements. For example, carbon dioxide (CO 2 ), is a binary compound because it contains just two elements: carbon and oxygen.
More specifically, binary ionic compounds are composed of just two elements, where one of the elements is a metal (which forms the cations) and the other is a nonmetal (which forms the anions). For example, NaCl is a binary ionic compound. We can think about the formation of such compounds in terms of the periodic properties of the elements. Many metallic elements have relatively low ionization potentials and lose electrons easily. These elements lie to the left in a period or near the bottom of a group on the periodic table. Nonmetal atoms have relatively high electron affinities and thus readily gain electrons lost by metal atoms, thereby filling their valence shells. Nonmetallic elements are found in the upper-right corner of the periodic table.
As all substances must be electrically neutral, the total number of positive charges on the cations of an ionic compound must equal the total number of negative charges on its anions. The formula of an ionic compound represents the simplest ratio of the numbers of ions necessary to give identical numbers of positive and negative charges. For example, the formula for aluminum oxide, Al 2 O 3 , indicates that this ionic compound contains two aluminum cations, Al 3 + , for every three oxide anions, O 2− [thus, (2 × +3) + (3 × –2) = 0].
It is important to note, however, that the formula for an ionic compound does not represent the physical arrangement of its ions. It is incorrect to refer to a sodium chloride (NaCl) “molecule” because there is not a single ionic bond, per se, between any specific pair of sodium and chloride ions. The attractive forces between ions are isotropic—the same in all directions—meaning that any particular ion is equally attracted to all of the nearby ions of opposite charge. This results in the ions arranging themselves into a tightly bound, three-dimensional lattice structure. Sodium chloride, for example, consists of a regular arrangement of equal numbers of Na + cations and Cl – anions (Figure Figure \(\PageIndex{2}\)).