Session 1, 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. Note 2

Use the number line in the Interactive Activity that follows to complete Problems C1-C6. Note 3

This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. For a non-interactive version of this activity, draw your own number line on a large sheet of paper, and then follow the instructions below.

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.

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. Close Tip

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. Close Tip

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? Close Tip

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?

If you are using a VCR, 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 x^{2} + 1 = 0. Close Tip

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. Close Tip

Video Segment 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.

If you are using a VCR, 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 i^{2} = -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.)