Algebra Know-It-ALL

126 Irrational and Real Numbers

We can envision the reals as corresponding to points on a continuous, straight, infinitely

long line, in the same way as we can imagine the rationals. But there are more points on a

real-number line than there are on a rational-number line. (Whether or not the real numbers

can be paired off one-to-one with the points on a true geometric line is a question that goes

far beyond the scope of this book!) The set of real numbers is related to the sets of rational

numbersQ, integers Z, and natural numbers N like this:

N⊂Z⊂Q⊂R

The operations of addition, subtraction, and multiplication can be defined over R. If # repre-

sents any of these operations and x and y are elements of R, then:

x # y∈R

This is a fancy way of saying that whenever you add, subtract, or multiply a real number by

another real number, you always get a real number. This is not generally true of division, expo-

nentiation (raising to a power), or taking a root. You can’t divide by 0, take 0 to the 0th power,

or take the 0th root of anything and get a real number. Also, you can’t take an even-integer

root of a negative number and get a real number.

Russian dolls

Now we can see the full hierarchy of number types. We started with the set of naturals, N.

Then we built the set of integers, Z, by introducing the notion of negative values. From there,

we generated the set of rationals, Q, by dividing integers by each other. Now, we have found

out about the set of irrationals, S, and the set of reals, R. The sets N, Z, Q, and R fit inside

each other like Russian dolls:

N⊂Z⊂Q⊂R

The set S is a proper subset of R, but it’s “standoffish” in the sense that it does not allow any of

the rationals, integers, or naturals into its “realm.” The Venn diagram of Fig. 9-2 shows how

all these sets are related.

Later on, we’ll learn about a set of numbers that’s even larger than the reals. Those are the

imaginary numbers and complex numbers. They result from taking the square roots of negative

reals and adding those quantities to other real numbers.

Are you confused?

Take the interval with end points on the number line corresponding to 1 and 2, as shown in Fig. 9-1. It

seems reasonable to think that there must be enough rational numbers (there are infinitely many, after all)

to account for every possible point in this interval. If that were true, then you could take any point P in the

interval and slice finer and finer intervals around it, with a rational number at the middle of each interval,

until finally you got an interval with P right in the middle. But you can’t always do this!