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Math 370 Learning Objectives
This entire section is provided for you to review on your own, if needed.
Sets of Numbers
While the authors would like nothing more than to delve quickly and deeply into the sheer excitement that is Precalculus, experiencehas taught us that a brief refresher on some basic notions is welcome, if not completely necessary, at this stage. To that end, we present a brief summary of Set Theoryand some of the associated vocabulary and notations we use in the text. Like all good Math books, we begin with a definition.
Definition: Set
A set is a well-defined collection of objects which are called the elementsof the set.
Here, "well-defined"means that it is possible to determine if something belongs to the collection or not, without prejudice. For example, the collection of letters that make up the word "smolko" is well-defined and is a set, but the collection of the worst math teachers in the world is not well-defined, and so is not a set.In general, there are three ways to describe sets.
Ways to Describe Sets
- The Verbal Method: Use a sentence to define a set.
- The Roster Method: Begin with a left brace, \( \{ \), list each element of the set only once and then end with a right brace,\( \} \).
- The Set-Builder Method: A combination of the verbal and roster methods using a dummy variablesuch as \(x\).
For example, let \(S\) be the set described verbally as the set of letters that make up the word "smolko". A description of \(S\) usingroster notation would be \(\left\{ s, m, o, l, k \right\}\). Note that we listed "o"only once, even though it appears twice in "smolko". Also, the order of the elements doesn’t matter, so \(\left\{ k, l, m, o, s \right\}\) is also a roster description of \(S\). A description of \(S\) using set-builder notation is:
\[\{ x \mid x \text{ is a letter in the word "smolko".} \}. \nonumber \]
The way to read this is: "The set of elements \(x\) such that \(x\) is a letter in the word 'smolko'."In each of the above cases, we may use the familiar equals sign, \( = \),and write \(S = \left\{ s, m, o, l, k \right\}\) or \(\ S=\{x \mid x \text { is a letter in the word "smolko". }\}\). Clearly \(m\) is in \(S\) and \(q\) is not in \(S\). We express these sentiments mathematically by writing \(m \in S\) and \(q \notin S\). The symbol\( \in \)is read as "belongs to," and the symbol \( \notin \) is read as "does not belong to." Throughout your mathematical upbringing, you have encountered several famous sets of numbers. They are listed below.
Definition: Sets of Numbers
- The Empty Set: \(\emptyset=\{\}=\{x \mid x \neq x\}\).
This is the set with no elements and it plays a vital role in mathematics.1 - The Natural Numbers: \( \mathbb{N} = \{ 1, 2, 3, \ldots\}\).
The periods of ellipsis here indicate that the natural numbers contain \(1\), \(2\), \(3\), and so forth. - The Whole Numbers: \( \mathbb{W} = \{ 0, 1, 2, \ldots \}\)
- The Integers: \( \mathbb{Z} =\{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}\)
- The Rational Numbers: \( \mathbb{Q} = \left\{\frac{a}{b} \mid a \in \mathbb{Z} \text{ and } b \in \mathbb{Z} \right\}\).
Rational numbers are the ratios of integers (provided the denominator is not zero). It turns out that another way to describe the rational numbersis:
\[\mathbb{Q} = \{ x \mid x \text{ possesses a repeating or terminating decimal representation.}\} \nonumber \] - The Irrational Numbers: \(\mathbb{P} = \{x \mid x \text { does not have a repeating or terminating decimal representation, and } x \text{ does not have an imaginary part}\}\).2
- The Real Numbers: \(\mathbb{R} = \mathbb{Q} \cup \mathbb{P} \).
The symbol \( \cup \) is theunionof both sets. That is, the set of real numbers is the set comprised of joining the set of rational numbers with the set of irrational numbers. - The Complex Numbers: \(\mathbb{C} = \{ a + b i \mid a, b \in \mathbb{R} \text { and } i = \sqrt{-1}\}\).
\( a \) is called the real part and \( b \) is called theimaginary partof the complex number. Despite their importance, the complex numbers play only a minor role in the text.3
Get used to set notations like \( \mathbb{Z} \) and \( \mathbb{R} \) as we will use them extensively throughout this text. It is important to note that every natural number is a whole number, which, in turn, is an integer. Each integer is a rational number (take \(b =1\) in the above definition for \(\mathbb Q\)) and the rational numbers are all real numbers, since they possess decimal representations. If we take \(b=0\) in the above definition of \(\mathbb C\), we see that every real number is a complex number. In this sense, the sets \(\mathbb N\), \(\mathbb W\), \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\), and \(\mathbb C\) are nestedlike Matryoshka dolls(see Figure \( \PageIndex{1} \).
Figure \( \PageIndex{1} \)
In Figure \( \PageIndex{1} \), the grey-shaded area of \( \mathbb{R} \) represents the irrational numbers \( \mathbb{P} \) and the orange-shaded area in \( \mathbb{C} \) represents complex numbers with nonzero imaginary parts.
Interval Notation
For the most part, this textbook focuses on sets whose elements come from the real numbers, \(\mathbb R\). Recall that we may visualize \(\mathbb R\) as a line. Segments of this line are called intervals of numbers. Below is a summary of interval notations associated with given sets of numbers. For intervals with finite endpoints, we list the left endpoint, then the right endpoint.We use square brackets, \([\)or \(]\), if the endpoint is included in the interval and use a closeddot to indicate membership in the interval. Otherwise, we use parentheses, \((\)or \()\),and an opencircle to indicate that the endpoint is not part of the set. If the interval does not have finite endpoints, we use the symbols \(-\infty\) to indicate that the interval extends indefinitely to the left and \(\infty\) to indicate that the interval extends indefinitely to the right. Since infinity is a concept,4 and not a number, we always use parentheses when using these symbols in interval notation, and use an appropriate arrow to indicate that the interval extends indefinitely in one (or both) directions. Finally, the notations \( a \lt b \), \( a \leq b \), \( a \gt b \), and \( a \geq b \) are calledinequality notation.
Definition: Interval Notation
Let \(a\) and \(b\) be real numbers with \(a<b\).
| Set of Real Numbers | Inequality Notation | Interval Notation | Region on the Real Number Line |
|---|---|---|---|
| \(\ \{x \mid a<x<b\}\) | \( a \lt x \lt b \) | \(\ (a, b)\) |  |
| \(\ \{x \mid a \leq x<b\}\) | \( a \leq x \lt b \) | \(\ [a, b)\) |  |
| \(\ \{x \mid a<x \leq b\}\) | \( a \lt x \leq b \) | \(\ (a, b]\) |  |
| \(\ \{x \mid a \leq x \leq b\}\) | \( a \leq x \leq b \) | \(\ [a, b]\) |  |
| \(\ \{x \mid x<b\}\) | \( x \lt b \) | \(\ (-\infty, b)\) |  |
| \(\ \{x \mid x \leq b\}\) | \( x \leq b \) | \(\ (-\infty, b]\) |  |
| \(\ \{x \mid x>a\}\) | \( x \gt a \) | \(\ (a, \infty)\) |  |
| \(\ \{x \mid x \geq a\}\) | \( x \geq a \) | \(\ [a, \infty)\) |  |
| \(\ \mathbb{R}\) | \(\ (-\infty, \infty)\) |  |
Caution
One of the most common mistakes made by students when it comes to interval notation is writing the interval in the wrong order. Remember, an interval written in interval notation is always listed from lower number to higher number.
For an example, consider the sets of real numbers described below.
| Set of Real Numbers | Inequality Notation | Interval Notation | Region on the Real Number Line |
|---|---|---|---|
| \(\ \{x \mid 1 \leq x<3\}\) | \( 1 \leq x \lt 3 \) | \(\ [1, 3)\) |  |
| \(\ \{x \mid-1 \leq x \leq 4\}\) | \( -1 \leq x \leq 4 \) | \(\ [−1, 4]\) |  |
| \(\ \{x \mid x \leq 5\}\) | \( x \leq 5 \) | \(\ (-\infty, 5]\) |  |
| \(\ \{x \mid x>-2\}\) | \( x \gt -2\) | \(\ (-2, \infty)\) |  |
Subsection Footnotes
4While we will talk about infinity in this course, the true study of infinity and the infinitesimal occurs in Calculus. For now, it is important to understand that infinity does not play well with our general understanding of Arithmetic. For example, most people would agree that\[ 1 + 5 = 6,\nonumber \]and so, subtracting \( 5\) from both sides, we get\[ 1= 1. \nonumber \]However, while it is true that \[ 1 + \infty = \infty,\nonumber \]if we try to abuse arithmetic and subtract \( \infty \) from both sides, we get the contradiction\[ 1 = 0. \nonumber \]
Intersections and Unions of Sets
We will often have occasion to combine sets. There are two basic ways to combine sets: intersection and union. We define both of these concepts below.
Definition: Intersectionand Union
Suppose \(A\) and \(B\) are two sets.
- The intersection of \(A\) and \(B\): \(A \cap B = \{ x \mid x \in A \text{ and } x \in B \}\)
- The union of \(A\) and \(B\): \(A \cup B = \{ x \mid x \in A \text{ or } x\in B \text{ (or both)} \}\)
Said differently, the intersection of two sets is the overlap of the two sets – the elements which the sets have in common. The union of two sets consists of the totality of the elements in each of the sets, collected together. See Figure \( \PageIndex{2} \) below.
Figure \( \PageIndex{2} \): A Venn Diagram5 of the sets \( A \) (the leftmost ellipse) and \( B\) (the rightmost ellipse). The intersection, \( A \cap B \) is the white region. The union, \( A \cup B \), is the combination of the yellow, white, and blue regions.
For example, if \(A = \{ 1,2,3 \}\) and \(B = \{2,4,6 \}\), then \(A \cap B = \{2\}\) and \(A \cup B = \{1,2,3,4,6\}\). If \(A = [-5,3)\) and \(B = (1, \infty)\), then we can find \(A \cap B\) and \(A\cup B\) graphically. To find \(A\cap B\), we shade the overlap of the two and obtain \(A \cap B = (1,3)\). To find \(A \cup B\), we shade each of \(A\) and \(B\) and describe the resulting shaded region to find \(A \cup B = [-5,\infty)\).
While both intersection and union are important, we have more occasion to use union in this text than intersection, simply because most of the sets of real numbers we will be working with are either intervals or are unions of intervals, as the following example illustrates.
Example \( \PageIndex{1} \)
Express the following sets of numbers using interval notation.
- \(\ \{x \mid x \leq-2 \text { or } x \geq 2\}\)
- \(\ \{x \mid x \neq 3\}\)
- \(\ \{x \mid x \neq \pm 3\}\)
- \(\ \{x \mid-1<x \leq 3 \text { or } x=5\}\)
Solution
- The best way to proceed here is to graph the set of numbers on the number line and glean the answer from it. The inequality \(x \leq -2\) corresponds to the interval \((-\infty, -2]\) and the inequality \(x \geq 2\) corresponds to the interval \([2, \infty)\). Since we are looking to describe the real numbers \(x\) in one of these or the other, we have \(\{ x \, \mid \, x \leq -2 \, \, \text{or} \, \, x \geq 2 \} = (-\infty, -2] \cup [2, \infty)\).
- For the set \(\{ x \, \mid \, x \neq 3 \}\), we shade the entire real number line except \(x=3\), where we leave an open circle. This divides the real number line into two intervals, \((-\infty, 3)\) and \((3,\infty)\). Since the values of \(x\) could be in either one of these intervals or the other, we have that \(\{ x \, \mid \, x \neq 3 \} = (-\infty, 3) \cup (3,\infty)\).
- For the set \(\{ x \, \mid \, x \neq \pm 3 \}\), we proceed as before and exclude both \(x=3\) and \(x=-3\) from our set. This breaks the number line into three intervals, \((-\infty, -3)\), \((-3,3)\) and \((3, \infty)\). Since the set describes real numbers which come from the first, second or third interval, we have \(\{ x \, \mid \, x \neq \pm 3 \} = (-\infty, -3) \cup (-3,3) \cup (3, \infty)\).
- Graphing the set \(\{ x \, \mid \, -1 < x \leq 3 \,\, \text{or} \,\, x = 5\}\), we get one interval, \((-1,3]\) along with a single number, or point, \(\{ 5\}\). While we could express the latter as \([5,5]\), we choose to write our answer as \(\{ x \, \mid \, -1 < x \leq 3 \,\, \text{or} \,\, x = 5\} = (-1,3] \cup \{ 5\}\).
Subsection Footnotes
5The reader is encouraged to research Venn Diagrams.
The Cartesian Coordinate Plane
In order to visualize the pure excitement that is Precalculus, we need to unite Algebra and Geometry. Simply put, we must find a way to draw algebraic things. Let’s start with possibly the greatest mathematical achievement of all time: the Cartesian Coordinate Plane(named in honor of René Descartes).Imagine two real number lines crossing at a right angle at \(0\) as drawn below.
The horizontal number line is usually called the \(x\)-axis while the vertical number line is usually called the \(y\)-axis(these labels can vary depending on the context of application).As with the usual number line, we imagine these axes extending off indefinitely in both directions. Having two number lines allows us to locate the positions of points off of the number lines as well as points on the lines themselves.
For example, consider the point \(P\) in Figure \( \PageIndex{3}\)A. To use the numbers on the axes to label this point, we imagine dropping a vertical line from the \(x\)-axis to \(P\) and extending a horizontal line from the \(y\)-axis to \(P\). This process is sometimes called "projecting" the point \(P\) to the \(x\)- and \(y\)-axis, respectively. We then describe the point \(P\) using the ordered pair \((2,-4)\) (Figure \( \PageIndex{3}\)B). The first number in the ordered pair is called the abscissa or \(x\)-coordinate and the second is called the ordinate or \(y\)-coordinate. Taken together, the ordered pair \((2,-4)\) comprise the Cartesian coordinates, or rectangular coordinates,6 of the point \(P\).
Figures \( \PageIndex{3}\)A (left) and\( \PageIndex{3}\)B (right)
In practice, the distinction between a point and its coordinates is blurred; for example, we often speak of "the point" \((2,-4)\).We can think of \((2,-4)\) as instructions on how to reach \(P\) from \((0, 0)\) (also known as the origin) by moving \(2\) units to the right and \(4\) units downwards. Notice that the order in the pair is important - if we wish to plot the point \((-4,2)\), we would move to the left \(4\) units from the origin and then move upwards \(2\) units (refer back toFigure \( \PageIndex{1}\)B).
When we speak of the Cartesian Coordinate Plane, we mean the set of all possible ordered pairs \((x,y)\) as \(x\) and \(y\) take values from the real numbers. Below is a summary of important facts about Cartesian coordinates.
Important Facts about the Cartesian Coordinate Plane
- \((a,b)\) and \((c,d)\) represent the same point in the plane if and only if \(a = c\) and \(b = d\).
- \((x,y)\) lies on the \(x\)-axis if and only if \(y = 0\).
- \((x,y)\) lies on the \(y\)-axis if and only if \(x=0\).
- The origin is the point \((0,0)\). It is the only point common to both axes.
Example \( \PageIndex{2} \)
Plot the following points: \(A(5,8)\), \(B\left(-\frac{5}{2}, 3\right)\), \(C(-5.8, -3)\), \(D(4.5, -1)\), \(E(5,0)\), \(F(0,5)\), \(G(-7,0)\), \(H(0, -9)\), \(O(0,0)\).
Note
The letter \(O\) is almost always reserved for the origin.
Solution
To plot these points, we start at the origin and move to the right if the \(x\)-coordinate is positive; to the left if it is negative. Next, we move up if the \(y\)-coordinate is positive or down if it is negative. If the \(x\)-coordinate is \(0\), we start at the origin and move along the \(y\)-axis only. If the \(y\)-coordinate is \(0\) we move along the \(x\)-axis only.
The axes divide the plane into four regions called quadrants. They are labeled with Roman numerals and proceed counterclockwise around the plane
For example, \((1,2)\) lies in Quadrant I, \((-1,2)\) in Quadrant II, \((-1,-2)\) in Quadrant III and \((1,-2)\) in Quadrant IV. If a point other than the origin happens to lie on the axes, we typically refer to that point as lying on the positive or negative \(x\)-axis (if \(y = 0\)) or on the positive or negative \(y\)-axis (if \(x = 0\)). For example, \((0,4)\) lies on the positive \(y\)-axis whereas \((-117,0)\) lies on the negative \(x\)-axis. Such points do not belong to any of the four quadrants. We will be referencing the four quadrants extensively in Chapters 10 and 11.
Subsection Footnotes
6We will learn of a different way to reference a point on a plane in Section 11.4.
Symmetry
One of the most important concepts in all of mathematics is symmetry.There are many types of symmetry in mathematics, but three of them can be discussed easily using Cartesian Coordinates.
Definition: Symmetry
Two points \((a,b)\) and \((c,d)\) in the plane are said to be
- symmetric about the \(x\)-axis if \(a = c\) and \(b = -d\)
- symmetric about the \(y\)-axis if \(a = -c\) and \(b = d\)
- symmetric about the origin if \(a = -c\) and \(b = -d\)
Schematically,
Figure \( \PageIndex{4} \)
In Figure \( \PageIndex{4} \), \(P\) and \(S\) are symmetric about the \(x\)-axis, as are \(Q\) and \(R\); \(P\) and \(Q\) are symmetric about the \(y\)-axis, as are \(R\) and \(S\); and \(P\) and \(R\) are symmetric about the origin, as are \(Q\) and \(S\).
Example \( \PageIndex{3} \)
Let \(P\) be the point \((-2,3)\). Find the points which are symmetric to \(P\) about the:
- \(x\)-axis
- \(y\)-axis
- origin
Check your answer by plotting the points.
Solution
Figure \( \PageIndex{4} \) gives us a good way to think about finding symmetric points in terms of taking the opposites of the \(x\)- and/or \(y\)-coordinates of \(P(-2,3)\).
- To find the point symmetric about the \(x\)-axis, we replace the \(y\)-coordinate with its opposite to get \((-2,-3)\).
- To find the point symmetric about the \(y\)-axis, we replace the \(x\)-coordinate with its opposite to get \((2,3)\).
- To find the point symmetric about the origin, we replace the \(x\)- and \(y\)-coordinates with their opposites to get \((2,-3)\).
One way to visualize the processes in the previous example is with the concept of a reflection. If we start with our point \((-2,3)\) and pretend that the \(x\)-axis is a mirror, then the reflection of \((-2,3)\) across the \(x\)-axis would lie at \((-2,-3)\). If we pretend that the \(y\)-axis is a mirror, the reflection of \((-2,3)\) across that axis would be \((2,3)\). If we reflect across the \(x\)-axis and then the \(y\)-axis, we would go from \((-2,3)\) to \((-2,-3)\) then to \((2,-3)\), and so we would end up at the point symmetric to \((-2,3)\) about the origin. We summarize and generalize this process below.
Reflections
To reflect a point \((x,y)\) about the:
- \(x\)-axis, replace \(y\) with \(-y\).
- \(y\)-axis, replace \(x\) with \(-x\).
- origin, replace \(x\) with \(-x\) and \(y\) with \(-y\).
Distance in the Plane
Another important concept in Geometry is the notion of length. If we are going to unite Algebra and Geometry using the Cartesian Plane, then we need to develop an algebraic understanding of what distance in the plane means. Suppose we have two points, \(P\left(x_0, y_0\right)\) and \(Q\left(x_{1}, y_{1}\right),\) in the plane. By the distance \(d\) between \(P\) and \(Q\), we mean the length of the line segment joining \(P\) with \(Q\).7 Our goal now is to create an algebraic formula to compute the distance between these two points. Consider Figure \( \PageIndex{5}\)A.
Figures \( \PageIndex{5}\)A (left) and\( \PageIndex{5}\)B (right)
With a little more imagination, we can envision a right triangle whose hypotenuse has length \(d\) as drawn in Figure\( \PageIndex{5}\)B. We see that the lengths of the legs of the triangle are \(\left|x_{1} - x_0\right|\) and \(\left|y_{1} - y_0\right|\) so the Pythagorean Theorem gives us
\[\left|x_1 - x_0\right|^2 + \left|y_1 - y_0\right|^2 = d^2 \nonumber \] \[\left(x_1 - x_0\right)^2 + \left(y_1 - y_0\right)^2 = d^2 \nonumber \]
Using Extraction of Roots on the second equation and the fact that distance is never negative, we get the Distance Formula.
Theorem:Distance Formula
The distance \(d\) between the points \(P\left(x_0, y_0\right)\) and \(Q\left(x_{1}, y_{1}\right)\) is
\[d = \sqrt{ \left(x_1 - x_0\right)^2 + \left(y_1 - y_0\right)^2}. \nonumber \]
It is not always the case that the points \(P\) and \(Q\) lend themselves to constructing such a triangle. If the points \(P\) and \(Q\) are arranged vertically or horizontally, or describe the exact same point, we cannot use the above geometric argument to derive the Distance Formula. It is left to the reader to verify the Distance Formula for these cases in the homework.
Example \( \PageIndex{4} \)
Find and simplify the distance between \(P(-2,3)\) and \(Q(1,-3)\).
Solution
\[\begin{array}{rclr}
d & = & \sqrt{\left(x_1 - x_0 \right)^2 + \left(y_1 - y_0 \right)^2} & \left( \text{Distance Formula}\right) \\
& = & \sqrt{ (1-(-2))^2 + (-3-3)^2} & \\
& = & \sqrt{ (3)^2 + (-6)^2} &\\
& = & \sqrt{ 9 + 36} & \\
& = & \sqrt{45} &\\
& = & 3 \sqrt{5} \\
\end{array} \nonumber \]
Therefore, the distance between \( P \) and \( Q \) is \(3 \sqrt{5}\).
Example \( \PageIndex{5} \)
Find all the points with \(x\)-coordinate \(1\) which are \(4\) units from the point \((3,2)\).
Solution
We shall soon see that the points we wish to find are on the line \(x=1\), but for now we’ll just view them as points of the form \((1,y)\). Visually,
We require that the distance from \((3,2)\) to \((1,y)\) be \(4\). The Distance Formulayields
\[\begin{array}{crclr}
& d & = & \sqrt{\left(x_1-x_0\right)^2+\left(y_1-y_0\right)^2} & \left( \text{Distance Formula} \right) \\
\implies & 4 & = & \sqrt{(1-3)^2+(y-2)^2} & \\
\implies & 4 & = & \sqrt{(-2)^2+(y-2)^2} & \\
\implies & 4 & = & \sqrt{4+(y-2)^2} & \\
\implies & 4^2 & = & \left(\sqrt{4+(y-2)^2}\right)^2 & \\
\implies & 16& = & 4+(y-2)^2& \\
\implies & 12& = & (y-2)^2& \\
\implies & \pm \sqrt{12} & = & y - 2& \\
\implies & \pm 2 \sqrt{3} & = & y - 2&\\
\implies & 2 \pm 2 \sqrt{3} & = & y& \\
\end{array} \nonumber \]
We obtain two answers: \((1, 2 + 2 \sqrt{3})\) and \((1, 2-2 \sqrt{3}).\) The reader is encouraged to think about why there are two answers.
Related to finding the distance between two points is the problem of finding the midpoint of the line segment connecting two points. Given two points, \(P\left(x_0, y_0\right)\) and \(Q\left(x_{1}, y_{1}\right)\), the midpoint \(M\) of \(P\) and \(Q\) is defined to be the point on the line segment connecting \(P\) and \(Q\) whose distance from \(P\) is equal to its distance from \(Q\).
If we think of reaching \(M\) by going "halfway over" and "halfway up" we get the following formula.
Theorem: Midpoint Formula
The midpoint \(M\) of the line segment connecting \(P\left(x_0, y_0\right)\) and \(Q\left(x_{1}, y_{1}\right)\) is
\[M = \left( \dfrac{x_0 + x_1}{2} , \dfrac{y_0 + y_1}{2} \right). \nonumber \]
If we let \(d\) denote the distance between \(P\) and \(Q\), we leave it as a homework exercise to show that the distance between \(P\) and \(M\) is \(\frac{d}{2}\) which is the same as the distance between \(M\) and \(Q\). This suffices to show that the Midpoint Formulagives the coordinates of the midpoint.
Example \( \PageIndex{6} \)
Find the midpoint of the line segment connecting \(P(-2,3)\) and \(Q(1,-3)\).
Solution
\[\begin{array}{rclr}
M & = & \left( \dfrac{x_0+x_1}{2}, \dfrac{y_0+y_1}{2} \right) & \left( \text{Midpoint Formula} \right) \\
& = & \left( \dfrac{(-2)+1}{2}, \dfrac{3+(-3)}{2} \right) & \\
& = & \left( \dfrac{-1}{2}, \dfrac{0}{2} \right) & \\
& = & \left( -\dfrac{1}{2}, 0\right) & \\
\end{array} \nonumber \]
The midpoint is \(\left(- \frac{1}{2}, 0 \right)\).
We close with a more abstract application of the Midpoint Formula. We will revisit the following example in the homework for Section 2.1.
Example \( \PageIndex{7} \)
If \(a \neq b\), prove that the line \(y = x\) equally divides the line segment with endpoints \((a,b)\) and \((b,a)\).
Solution
As with most claims we want to prove, the first step is to understand what is being stated. As written, this claim is far too confusing. However, if we draw an example, it might help us understand what is being stated. If we let \( a=-4 \) and \( b=2 \), we get
This single visualization helps understand the claim immensely. We want to prove the following statement:
If\( a \neq b \), then themidpoint of the line segment connecting \( (a ,b) \) and \( (b,a) \) must be on the line \( y = x \).
Proof
To prove the claim, we use the Midpoint Formula.
\[\begin{array}{rclr}
M & = & \left( \dfrac{a+b}{2}, \dfrac{b+a}{2} \right) & \left( \text{Midpoint Formula} \right) \\
& = & \left( \dfrac{a+b}{2}, \dfrac{a+b}{2} \right) & \\
\end{array} \nonumber \]
Since the \(x\) and \(y\) coordinates of this point are the same, we find that the midpoint lies on the line \(y=x\), as required.
Subsection Footnotes
7Remember, given any two distinct points in the plane, there is a unique line containing both points.
