Numbers

The most basic and familiar kinds of numbers are natural numbers, the elements of the set

They arise naturally in counting objects of all kinds. What does it mean to count a set of objects, say a pile of dollar bills'? When we count dollar bills, we lake each element in the set of dollar bills and pair it wilh an element of Z+, starting with 1 and moving successively through the sel. When we have exhausted the elements of the first set (of dollar bills), the element of Z, that we have reached in this process gives us the number of dollar hills. This number is called the cardinality of the set of dollar bills; the cardinality of a set is the number of objects it contains. As we saw in section 2.1, Z+ has an infinite number of elements. Given any positive integer, we can always find a larger one. We now define some properties of natural numbers and show that other types of numbers arise from the operations of addition, multiplication, subtraction, and division.

Consider the addition or multiplication of any two elements, a and b. of Zr. These two operations are closely related, since multiplying a and b is simply adding a to itself b times or b to itself« limes. The first thing we note is thai the result of this is itself always an element of Z+:

This is formally expressed by saying that the set Z. is closed under the operations of addition and multiplication.

Is Z+ closed under the operations of subtraction and division'7 Clearly not. If u < b. we have u - b < 0 £ Z+

so Zt is not closed under subtraction, li is easy to find cases where a/h is not an integer.

The facl that Z. is not closed under subtraction leads, naturally to the definition of the set of integers

where the three dots indicate that we go out lo infinity in each direction. Z is a somewhat more absiract set than Z„. since it is hard to imagine observing a set containing, say —3 objects. Nevertheless, ii is impossible in general to consider solving the simplest kind of equation x + b = a

Figure 2.4 The set Z

unless we have negative numbers available. Intuitively we can think of a negative number as "something owing"—a kind of debt. It is also useful to represent Z as in figure 2.4. Along a horizontal line we mark off intervals of equal length, choose a central point to represent zero, and measure the positive integers in ascending value to the right and negative integers in descending value to the left (so that -a < -b if a > b • 0). It is also clear that Z+ C Z.

The set of integers is closed under addition, subtraction, and multiplication. However, neither Z nor Z^ is closed under division, since, for example, a/b Z if a = -2 and b = 3, though both a and b are in Z. This means, for example, that we could not in all cases find x t Z such that bx = a, a.b € Z This leads to the definition of the set Q of rational numbers

Note that Z c Q, since we could clearly choose a = kh for k e Z. Note also that we rule out division by zero. We say that any expression involving zero in the denominator is undefined. The reason for this can be seen from the equation bx = a. I fb - 0, then no.v exists such that bx — a for any u ^ 0, and so ruling out division by zero recognizes this fact. (The term "rational number" comes from the fact these numbers are ratios of integers.) Figure 2.5 shows the set of rational numbers and indicates that these include points between the integers. Take any two points on the line that give distinct rational numbers. There is an infinity of other rational numbers between those points. To see this, consider the two rational numbers 1 and 2, and note that 1 + (2- l)/cw\thc € Z+, must be a rational number between 1 and 2. Each value of e gives a different rational number, and since Z+ has an infinite number of elements, there must be an infinite number of rational numbers between I and 2. Replacing 1 by a and 2 by b > a, where a and b are any elements in Q, shows that this must be true in general.

9 5 3 I 7 9 4 "4 4 2 4 4 ■ I I-1-M—I-1-1-1-1—»—I-1-►

Figure 2.5 The set Q

An important observation is that the segment between the numbers 1 and 2 (or between any two rational numbers) is not entirely composed of rational numbers. Other numbers exist. In figure 2.6 we have constructed a right-angle triangle with sides and B of length I and hypotenuse of length L lying along the line. From the Pythagorean theorem we know that the square of the distance L along the line is given by

so L = J2. The following theorem, the proof of which was well known to the ancient Greeks, shows that Jl is not a rational number. This tells us that Q is not closed under the operation of taking square roots. Alternatively, it tells us thai there must be numbers other than rational numbers. These are irrational numbers, which cannot be expressed as the ratio of two integers.

Theorem 2.1 The number Jl is not a rational number, that is. J2 f. Q

Proof

The proof is by contradiction. Suppose that -¿2 6 Q. Then there exist integers /> and q such that p/q ~ >/2, where we choose the smallest such p and q. Now p2/q2 = 2, or p2 — 2q2, so p2 must be an even number. Since the square of an odd number is always odd. p is even, and we may write p — 2r. where r is also an integer. We then have

and so q2 = 2r2, Clearly, q2, and hence q, must be even. This result contradicts the assumption that p and q were the smallest numbers to give p/q — *J2. Thus the statement s/2 e Q must be false. ■

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