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Private: Learning Math: Number and Operations
What Is a Number System? Part C: Building the Number Line (35 minutes)
After exploring a number system different from our own, we are now ready to begin examining the real number system. We will begin to classify and examine the different types of numbers we use and look at how the numbers and operations relate to one another. We will start with the counting numbers on a number line and then add more numbers to the line as they occur in our study of operations.
Note2: If you are working in a group, discuss first how the number line gets filled up and why we need more numbers than just the counting numbers. This question will be revisited in the next session.
Use the number line in the Interactive Activity that follows to complete Problems C1-C6.
Note 3: If you are working in a group, discuss first how the number line gets filled up and why we need more numbers than just the counting numbers. This question will be revisited in the next session.
Instructions for non-interactive activity:
Enter the number 1 near the center of the line, followed by the next several counting numbers (2, 3, 4, 5, …) to the right of the 1. Make sure that the distance between any two adjacent numbers is the same.
Problem C1
a. Suppose that the only numbers on this number line were the counting numbers. Assuming that your number line is infinitely long and your set of counting numbers is infinite, can you add any number in the set of counting numbers to any other number and stay within that set?
b. Can you multiply any two numbers in the set of counting numbers and stay within that set?
Problem C2
a. Moving on to subtraction, what other elements must you include on your number line to be able to subtract?
b. Enter the numbers you need for subtraction, again making sure that the distances are precise.
Problem C3
a. What elements must you include on your number line to be able to divide?
You may notice that by taking two numbers on the number line — 1 and 3, for example — and dividing the smaller by the larger, it will be necessary to add fractions between the integers already on the number line.
b. What do multiplicative inverses have to do with division?
The multiplicative inverse of a number is the number by which you must multiply the original number by to get the multiplicative identity element, or 1.
c. Will you ever be able to find a multiplicative inverse for 0? Why or why not?
You can see that your number line is filling up. Each of the various arithmetic operations — addition, subtraction, multiplication, and division — filled in more empty space.
Related to the number of elements in a given number set is the concept of density. If a set is dense, then no matter what two elements in the set you choose, you will be able to find another element of the same type between the two.
Problem C4
Of the sets you’ve included on your number line so far — counting numbers, integers, and rational numbers — which are dense?
For the counting numbers, think about whether you can find another counting number between 2 and 3. How about the rational numbers? Is there another number between 2.5 and 2.6? Is there one between 2.55 and 2.6?
Video Segment
In this video segment, Professor Findell explains the concept of density and why rational numbers, unlike the counting numbers or integers, are dense. Watch this segment after you’ve completed Problem C4.
Can you think of any other sets that are dense?
You can find this segment on the session video approximately 18 minutes and 38 seconds after the Annenberg Media logo.
You have accounted for the four main arithmetic operations by building a number line made up of counting numbers, then integers, then rational numbers.
Problem C5
a. Are there other kinds of operations, procedures, or algorithms that we use in mathematics that produce different number solutions?
b. What kinds of numbers do they produce?
Consider such situations as finding the length of a hypotenuse of a right triangle, finding the circumference of a circle, computing continuous compound interest, or solving an equation, such as x2 + 1 = 0.
Problem C6
a. Could you represent as a rational number? How do you know?
b. Determine the length of on your number line.
First, think about how to obtain 
using the Pythagorean theorem.
In this video segment, Vicky and Maria explore how they can calculate and then construct the value of 
as a physical distance on the number line. Note that the answer to the quadratic equation is
, but only positive values are used for measuring distances. Watch this segment after you’ve completed Problem C6.
Think about how you would use a similar method to construct other square root values.
You can find this segment on the session video approximately 20 minutes and 23 seconds after the Annenberg Media logo.
The roots and powers are now on the number line, but the line is still not complete. There are other types of numbers that can be represented as a length or a distance from 0.
A familiar value you use to calculate the circumference or area of a circle is 
. The value of is approximately, but not exactly, equal to 22/7, or 3.141593. In fact, you cannot express as the ratio of two integers, so it, like , is an irrational number.
Another irrational number is e, which is approximately equal to 2.7183; e appears in several mathematical computations, such as continuous compound interest; as the base of natural logarithms; and in calculus.
Problem C7
How could and e be represented on the number line? What are their distances from 0?
It’s now time to introduce another kind of number: complex numbers. Complex numbers are numbers formed by the addition of imaginary and real number elements. They are in the form a + bi, where a and b are real numbers, and i can be represented as i2 = -1 (a number such that when you square it, you get -1).
In order to represent complex numbers on a graph, draw a second line perpendicular to the original line and passing through the point (0,0). You can represent the value of a on the horizontal axis and the value of b on the vertical axis.
Problem C8
a. How could the real numbers be represented in this coordinate system?
b. How could the pure imaginary numbers (numbers in the form of bi) be represented? (Remember that imaginary numbers cannot be represented by lengths on the number line.)
Solutions
Problem C1
a. Yes, the set of counting numbers is closed for addition.
b. Yes, the set of counting numbers is closed for multiplication.
Problem C2
We must include 0 (to subtract things like 4 – 4) and negative integers (to subtract things like 23 – 831).
Problem C3
a. We must include all fractional numbers of the form p/q, where p and q are integers (positive, negative, or zero counting numbers), with the restriction that q cannot be 0 (dividing by 0 is not defined). For example, we will need numbers like 5/2 and 82/7 and -1/2.
b. Take a division problem like 5
3 = r. This is the equivalent to saying, “What number multiplied by 3 gives us 5?” The equation for this is 3 • r = 5. To solve this equation, we must isolate r on one side of it. Doing this requires dividing by 3 or multiplying 3 by its multiplicative inverse. The multiplicative inverse of 3 is usually written as 1/3. Multiplying both sides by 1/3 produces the following:
1/3 • (3 • r) = 1/3 • 5
(1/3 • 3) • r = 1/3 • 5
1 • r = 1/3 • 5
r = 5/3
c. No. If y is the multiplicative inverse of 0, then y • 0 = 1. But every real number multiplied by 0 equals 0, so y cannot be a real number — and there is no multiplicative inverse for 0. That’s why we can’t divide by 0.
Problem C4
Counting numbers are not dense. There is no counting number between 2 and 3. The integers are not dense either. However, we can always find a rational number between any two given rational numbers; for example, the average of any two fractions must always be a fraction between the two given fractions. Therefore, rational numbers are dense. One rational number between 2.5 and 2.6 is 2.55. One rational number between 2.55 and 2.6 is 2.555.
Problem C5
a. Some major examples include raising a number to a power (exponentiation) and its inverse function (taking roots, such as square or cube roots), working with circles (and the number , approximately 3.141593), and solving equations with exponents (such as 2x = 3).
b. Such operations produce irrational numbers, like 
, , or e (the base of natural logarithms; e is a mathematical constant approximately equal to 2.7183). Roots such as and 
are algebraic irrationals since they can be solutions to polynomial equations: numbers such as and e are called transcendental irrationals since they cannot be solutions to polynomial equations. Other equations, like x2 = -1, do not have a solution on the number line at all; this solution would be an imaginary number.
Problem C6
a. No. Proving this is actually pretty difficult, but for the to be rational, we would have to be able to write it as p/q in reduced form, where p and q are integers that are relatively prime. This would mean that p and q are solutions to the equation p2= 2q2, which cannot be solved if p and q can only be counting numbers.
b. The length of the is the hypotenuse of a right triangle whose legs are 1 and 1 (i.e., x2 = 12 + 12 or x2 = 2 or x = 
). So we know it has a “physical” distance and therefore can be located on the number line. This is about 1.414, but no decimal could ever express the exactly.
Problem C7
Each is on the number line some specific distance from 0 (since each number is a constant). As with the , the distance cannot be expressed as a terminating or repeating decimal. is approximately 3.141593, while e is approximately 2.7183.
Problem C8
a. The real numbers could be represented as the horizontal axis (similar to the number line). All real numbers, like 2, 1/2, -3, and e, would be on this line.
b. The pure imaginary numbers, like 2i, (1/2)i, -3i, and ei could be represented as the vertical axis. The coordinates of a real number are (x,0), where x is the real part. The coordinates of a pure imaginary number are (0,yi), where yi is the imaginary part.
