Early in a beginning algebra course, students are taught how to add and subtract signed numbers. Addition of signed numbers is taught by giving students a set of rules that they can follow to get the right answer. Some number line diagrams are thrown in to illustrate addition of signed numbers. Subtraction of signed numbers is usually taught by the “add the opposite” rule. This rule sounds plausible, however, it does not give students any insight into what subtraction is all about from a geometric point of view. You might ask, “Isn’t learning how to apply a set of rules to get the answer sufficient?” In my view, it’s not. As I stated several times in previous posts, whenever possible, students should understand a math concept from both an algebraic and geometric point of view. The primary purpose of this post to show how I have used vector diagrams to illustrate addition and subtraction of real numbers and 2D vectors. I will also discuss geometric interpretations of expressions and relationships such as (a + b)/2, |x – 4| < 5, and |x – 5| > 7.
Based on my own experience, most kids quickly learn how to apply the rules for adding and subtracting positive and negative numbers. When simplifying polynomial expressions, rational polynomial expressions, and polynomial long division, most mistakes occur at the step where a negative number is subtracted. Students are generally pretty good at adding signed numbers, but a little weak when it comes to subtracting signed numbers. This is why my favorite high school math teacher, Vivian Jones, said, “When subtracting a number, just change the sign of the number and add”. This is what I tell my college algebra students when they do polynomial long division. As taken from a developmental algebra textbook, the rules for adding and subtracting signed numbers are as follows:
Adding Two Real Numbers a and b
- If a and b have the same sign, add their absolute values. Use the common sign of a and b as the sign of the answer.
- If a and b have different signs, subtract their absolute values. The sign of the answer is the sign of the number that has the largest absolute value.
Subtracting Two Real Numbers a and b
- a – b = a + (-b)
- In other words, a – b equals a plus the opposite of b
Diagrams B and C illustrate the subtraction rule for signed numbers. The problem with the subtraction rule and the corresponding vector diagram is that they don’t give us any insight into what subtraction is all about from a geometric point of view.
Diagrams D and E below illustrate the idea that subtracting two real numbers gives us the directed distance between two reals numbers. The concept of directed distance is fundamental in understanding subtraction from a geometric point of view. For real numbers a and b, the directed distance from b to a = a – b, and the directed distance from a to b = b – a. The positive distance or just the distance from a to b = |a – b| = |b – a|. The distance concept is related to many concepts in mathematics such as the amount of change in a variable, how much a data value deviates from some fixed constant, margin of error, etc. Notice that the subtraction problems in diagrams D and E are the same problems in diagrams B and C. After comparing diagrams B and C with diagrams D and E, it becomes clear that diagrams D and E give us a much better way to understand subtraction from a geometric point of view.
We will now take a look at 2D vectors which are essential in understanding a variety of concepts in math and physics. We can treat 2D vectors as line segments that have the properties of length and direction. Any two vectors are equivalent if and only if they have the same length and direction. In this post, 2D vectors will be denoted by bold face capital letters, and a pair of vertical absolute value bars will denote the length or magnitude of a vector. The arrow at one tip of a segment indicates the direction the vector. Geometric rays have infinite length, and therefore 2D vectors are not geometric rays. In passing, I will mention that 2D vectors are just a special case of an abstract mathematical object named vector. If you want to stretch your mind, take a course in infinite dimensional vector spaces. The student and teacher versions of my free handout, Introduction to Vectors, can be downloaded by going to the trigonometry section of our instructional content page.
The graph below shows how two 2D vectors are added. All vectors drawn in the same color are equal to each other because they have the same length and direction. All 2D vectors can be represented by a pair of real numbers of the form < x, y > where x and y equal the x-component and y-components of the vector. Knowing the x-y components of a vector, it’s easy, at least for trig students, to calculate the vector’s magnitude and direction. Note the following as you study the graph:
- A 2D vector can be expressed as the sum of its x-component and y-component. Example: Vector A = < -7, 0 > + < 0, 3 > = < -7, 3>.
- To find the x-y components of a vector, start at the tail of the vector and count the number of spaces left/right and the number of spaces up/down to the head of the vector.
- The x-y components of any two equivalent vectors are equal.
- The tail of the second vector in a vector sum is located at the tip of the first vector.
- Vector addition is commutative. In other words, it makes no difference in what order vectors are added.
- Any two equivalent vector pairs will always result in equivalent vector sums.
- The theorem of Pythagoras is used to calculate the length or magnitude of a vector.
- The inverse tangent function, Tan-1(x), is used to calculate the direction of a vector.
- Vectors A, B and S = A + B = B + A in the diagram have the following properties:
- A = < -7, 3 >, |A| = √(58) units, and direction of A ≈ 156.800
- B = < 3, 5 >, |B| = √(34) units, and direction of B ≈ 59.040
- S = < -4, 8 >, |S| = √(80) units, and direction of S ≈ 116.570
- |A| + |B| > |S|
The graph below shows how two 2D vectors are subtracted. Vector subtraction gives us a vector that represents a difference vector between the tips of the two vectors. Vector A – B has its tail at the tip of B and its head at the tip of A. Vector B – A has its tail at the tip of A and its head at the tip of B. All vectors drawn in the same color are equal to each other because they have the same length and direction. Note the following as you study the graph:
- The difference vector connects the tip of one vector to the tip of the other vector.
- Vector subtraction is not commutative. Vectors A – B and B – A have the same length, but they point in opposite directions; they are vector opposites.
- Vectors A, B, A – B, B – A have the following properties:
- A = < 6, 2 >, |A| = √(40) units, and the direction of A ≈ 156.800
- B = < 2, 9 >, |B| = √(85) units, and the direction of B ≈ 59.040
- A – B = < 4, -7 >, |A – B| = √(65), and the direction of A – B ≈ 299.750
- B – A = < -4, 7 >, |B – A| = √(65), and the direction of B – A ≈ 119.750
Once a student understands addition and subtraction from a geometric point of view, many math problems become much easier to solve. Consider the three routine math problems shown below.
Problem 1: Suppose the IQ score I of a person is in the normal range if the IQ score deviates from 100 by 10 points or less. What interval on a number line and inequality describes a normal IQ score?
Solution: The expression |I – 100| gives us the positive distance of the variable I from 100. Therefore the range of normal IQ scores is described by the inequality |I – 100| ≤10.
Problem 2: A part will fail inspection if its diameter d deviates from 2.5 cm by more than 0.001 cm. What interval on a number line and inequality describes the rejection region?
Solution: A part will fail inspection if the positive distance from 2.5 to d is more than 0.001 cm. Therefore the rejection region can be described by the inequality |d – 2.5| > 0.001. Of course, we tacitly assume that there are practical restrictions on values of d.
Problem 3: The graph of the closed interval [0.84, 2.68] is shown below. Find the following:
- Length of the interval
- Coordinate of the midpoint M
- Radius of the interval
- Write an inequality that describes the interval.
Solution: (Many of my elementary statistics students initially struggle with review problems like this.)
- Length of interval = 2.68 – 0.84 = 1.84
- Coordinate of midpoint M = (0.84 + 2.68)/2 = 1.76
- Radius of the interval = 1.84/2 = 2.68 – 1.76 = 1.76 – 0.84 = 0.92
- Inequality: |x – 1.76| ≤ 0.92
Miscellaneous facts I tell my students:
- If you subtract a smaller number from a bigger number, the answer is positive.
- If you subtract a bigger number from a smaller number, the answer is negative.
- If you subtract a positive number from n, the answer is smaller than n.
- If a positive influence is removed from your personal life, the quality of your personal life goes down.
- If you subtract a negative number from n, the answer is bigger than n.
- If you remove a negative influence from your person life, your personal life gets better.
- If you add a negative number to n, the answer is smaller than n.
- If a negative influence is introduced into your personal life, your personal life gets worse.
- To find out how far apart two numbers are, subtract the numbers.
- To find a number half way between two numbers, find the average of the numbers.
I will close this post with a true story about an epiphany I experienced early in my teaching career. The class was a regular high school geometry class. We were learning how to solve story problems involving complementary and supplementary angles. I could see that little Elmo (not his real name) was not getting the idea that if x is the measure of an acute angle, then 90 – x is the measure of the complement of the angle. So I asked Elmo a series of about 5 questions like: If an angle measures 200, what is the degree measure of the complement of the angle? Elmo got every one of my questions right. I then asked Elmo the following question: If an acute angle measures n degrees, what is the degree measure of the complement of the angle? All I got from Elmo was a blank stare. I’m thinking to myself, why doesn’t he get it? Then it hit me. When asked the degree measure of the complement of a 650, Elmo figured out how many degrees he needed to add to 650 to get 900. For me, this was an enlightening and humbling experience.