Mathematicians have known for centuries that there are plenty of irrational numbers that can’t be expressed as ratios of integers. Let’s see how they behave compared with the rational numbers.
Rational-number “density”
Suppose we assign rational numbers to points on a horizontal line so the distance of any point from the origin is directly proportional to its absolute value. If a point is on the left-hand side of the point representing 0, then that point corresponds to a negative number; if it’s on the right-hand side, it corresponds to a positive number.
If we take any two points a and b on the line that correspond to rational numbers, then the point midway between them corresponds to the rational number (a+b)/2. (Do you recognize this as the formula for the average, or arithmetic mean, of two numbers?) We can keep cutting an inter-val in half, and if the end points are both rational numbers, then the midpoint is another rational number. Figure 9-1 shows an example of this, starting with the interval between 1 and 2.
It is tempting to suppose that the points on a rational-number line are “infinitely dense.”
The point midway between any two rational-number points always corresponds to another rational number. But do the rational numbers account for all of the points along a true geo-metric line? The answer is “No. They don’t even come close!”
Irrational numbers
As we have seen, an irrational number can’t be expressed as the ratio of two integers. Examples of irrational numbers include:
Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use.
• The length of the diagonal of a square that measures 1 unit on each edge.
• The length of the diagonal of a cube that measures 1 unit on each edge.
• The ratio of a circle’s circumference to its diameter.
• The decimal number 0.01001000100001000001...
Whenever we try to express an irrational a number in decimal form, the result is an endless nonrepeating decimal. (The last item in the above list has a pattern of sorts, but it is not a repeating pattern like the decimal expansion of a rational.) No matter how many digits we write down to the right of the decimal point, the expression is an approximation of the actual value. A pattern can never be found that allows us to convert the expression to a ratio of integers.
The set of irrationals can be denoted S. This set is disjoint from the set Q of rationals. No irrational number is rational, and no rational number is irrational. In set notation,
S∩Q= ∅ Real numbers
The set of real numbers, denoted by R,is the union of the set Q of all rationals and the set S of all irrationals:
R=Q∪S 1–1/2
1 2
1–1/2
1–3/4 2
1–1/2 1–3/4
1–5/8
1–1/2 1–5/8
1–9/16
1–9/16 1–5/8
1–19/32
Figure 9-1 An interval on the rational-number line can be cut in half over and over, and you can always find infinitely many numbers in it.
The Number Hierarchy 125
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!
The Number Hierarchy 127
S Q
R= Everything inside the large rectangle
Z N
S
N Q
Z
Figure 9-2 The relationship among the sets of real numbers (R), irrationals (S), rationals (Q), integers (Z), and natural numbers (N).
You can define a point between 1 and 2 that doesn’t correspond to any rational number. Take a square measuring exactly 1 unit on each edge, known as a unit square, and place it on the number line so one corner is at the point for 0 and the other corner is on the line between the points for 1 and 2, as shown in Fig. 9-3. The opposite corner of the square falls exactly on the point for the square root of 2, because the diagonal of a unit square is precisely 21/2 units long. (That fact comes from basic geometry.) The number 21/2 is irrational.
If you build a line using only the points corresponding to rational numbers, that line will be full of “holes.”
Are you still confused?
The rational numbers, when depicted as points along a line, are “dense,” but not as dense as the points on a line can get. No matter how close together two rational-number points on a line might be, there is always another rational-number point between them. But there are points on a continuous geometric number line that don’t correspond to any rational quantity. The set of reals is more “dense” than the set of rational numbers.
“All right,” you say. “This game is strange, but I’ll play along. How many times more dense is the real-number line than the rational line? Twice? A dozen times? A hundred times?” The answer might astonish you. The set of real numbers, when assigned to points on a line, is infinitely more “dense” than the set of rational numbers.
Here’s a crude analogy. Suppose you’re exploring the “planet Maths” and you stumble across a rational-number line and a real-rational-number line lying in a field. Both lines resemble straight, infinitely long, thin, rigid
128 Irrational and Real Numbers
wires. The rational-number line is gray. When you pick it up, you find that it’s weightless. The real-number line is black. When you lift it, you discover that it’s heavy. The real-number line contains more “stuff.”
Here’s a challenge!
Let’s try a little exercise that involves some plain-language logic, some set theory, and some Venn dia-gram reading skill. Which of the following statements are true, based on our knowledge of the number hierarchy?
• All rational numbers are real.
• All integers are real.
• Some integers are irrational.
• Some irrational numbers are real.
• All irrational numbers are real.
Solution
We can figure all of these out by looking at Fig. 9-2. The answers, bullet-by-bullet, are as follows.
• Set Q is entirely contained in set R. Therefore, all rationals are real.
• Set Z is entirely contained in set R. Therefore, all integers are real.
• Set Z is completely separate from set S. Therefore, no integers are irrational.
• Set S has elements in common with set R. Therefore, some irrationals are real.
• Set S is entirely contained in set R. Therefore, all irrationals are real.
0 1 2 Positive-number line
This point corresponds to the square root of 2
1 unit
1 unit
Figure 9-3 If you take a unit square and place it diagonally along the positive-number line with one corner at the point for 0, the opposite corner is at the point for 21/2, which is irrational.