5.5
Rational Numbers
In this section we will show how the integers are extended to the rational num- bers.
Exercise 5.5.1. Give two ways in which the integers and the rational numbers are different. That is, describe two things that you can do with rational numbers that you cannot do with integers.
Notice that in the list of axioms in the previous section, there is a “missing” axiom: an M-5 axiom. This axiom would say that every integer has a mul- tiplicative inverse, a number such that when multiplied by the given number, gives 1.
But we know this doesn’t happen: for example, 2 does not have such an inverse. Rational numbers were invented to deal with this “defect.”
We may view the integers as an attempt to solve certain linear equations of the form ax + b = 0. However, when we are confronted with an equation like 3x + 1 = 0, we cannot find an integer solution. The reason is the missing multiplicative inverse.
The rational numbers have this missing axiom. They satisfy all the axioms of the integers in the previous section, plus one more:
M-5 (Inverse). Every rational number except 0 has a special “twin” which, when multiplied by it, yields 1. That is, for each rational a 6= 0, there is another rational, called a−1, such that a · a−1 = 1.
*Exercise 5.5.2. In Exercise 5.4.5 you saw that polynomials with integer co- efficients satisfied the same addition, multiplication and distributive axioms as the integers. What “extension” of the polynomials will also satisfy the M-5 Axiom?
Another name for a−1 is 1a. We now define, for integers a and b, a 6= 0,
b a = b ·
1
a. This definition, together with the axioms stated earlier, give us the
usual arithmetic on fractions. The b is called the numerator, the a is called the denominator. The set of all numbers of the form a/b where a and b are integers is called rational numbers. A common notation for the rational numbers is Q. Exercise 5.5.3. Show that between every pair of distinct rational numbers there is a third rational number. Is the same thing true for integers? That is, is it true that for every pair of distinct integers there is a third, different, integer which lies between the first two? Explain.
We need to clarify two points about this fractional notation for rational numbers. The first point is that it is not unique. If the numerator and denom- inator have a common factor, that factor can be “canceled” from both parts. Thus, the fraction 6
15 and the fraction 2
5 both represent the same rational num-
ber. Usually, we try to reduce as much as possible, so that the numerator and denominator have no common factor.
Divide Into Get Remainder 13 40 3 1 13 10 0 10 13 100 7 9 13 90 6 12 13 120 9 3 13 30 2 4 13 40 3 1
Table 5.2: Long division
Exercise 5.5.4. If the numerator and denominator have no common factor, what is their GCD?
The second point is that if n and d are positive integers with n ≥ d, then n
d = q + r
d, (5.3)
where q is a positive integer and r is an integer between 0 and d − 1. For instance, 194 = 4 +34.
Exercise 5.5.5. Explain why Equation (5.3) is just a restatement of Equa- tion (5.1).
Fractional values can be effectively described by our positional base ten system described earlier. We represent a number by a sequence of digits, a decimal point, and a second sequence of digits. The number represented has as its integer portion the number represented by the first sequence of digits, using Equation (5.2). The fractional portion is represented by the sequence of digits to the right of the decimal point. Notice that this sequence may be infinitely long. If this sequence is f1f2f3· · · , the fraction it represents is
f1· 1 10+ f2· 1 102+ f3· 1 103 + · · · . (5.4)
In this format, rational numbers have a particularly simple representation. Rational numbers can always be represented as repeating or terminating deci- mals. Let’s see how this works.
If we want to write 1/3 as a decimal, we would divide 3 into 1. Performing the long division gives 0.333 · · · . At each stage of the division, we get a remainder of 1. When we “bring down” the 0, we are always dividing 10 by 3.
A more complicated example might by 4/13. The steps are in Table 5.2. After the last step in Table 5.2, the process will repeat, and we will get 0.307692307692 · · · .
A third example is 15/22. When the long division is performed, we get 15/22 = 0.61818 · · · . Notice that the first digit does not repeat.
5.5. RATIONAL NUMBERS 123
To save space, we will describe these repeating decimals with a bar over the repeating section. Thus 0.333 · · · = 0.3, 0.307692307692 · · · = 0.307692 and 0.61818 · · · = 0.618.
Calculators are a poor tool to use when working with repeating decimals. Some repeating decimals don’t look “repeating” on the calculator.
Exercise 5.5.6. Is 7/17 a rational number? Use your calculator to compute 7/17. Does it “look like” a rational number on your calculator? Explain. Exercise 5.5.7. Convert to decimal: 7/17.
And some non-repeating decimals appear repeating. Exercise 5.5.8. Use your calculator to compute
√ 2 − 1
13861− 1 297195528.
Do you think this number is rational? (Remember that√2 is irrational.)
We have been discussing one kind of rational number: repeating decimals. Another kind of rational number “divides evenly.” For example, 1/2 = 0.5. These are called terminating decimals.
Exercise 5.5.9. Which of these rationals are repeating decimals and which are terminating: 2/5, 5/6, 4/15, 3/40? In general, which rationals are terminating and which are repeating?
Exercise 5.5.10. Why will every rational number be a repeating decimal or a terminating decimal? How long can the repeating sequence be for the fraction b/a, where b and a 6= 0 are integers?
The converse of the statement in Exercise 5.5.10 is also true: every repeating decimal or terminating decimal is a rational number. Here is an easy way to convert a repeating decimal to a rational. Let’s demonstrate with 0.42
Let x = 0.42. Then 100x = 42.42. Now subtract these two equations: 99x = 42 or x = 14/33.
Exercise 5.5.11. Why did we multiply by 100 in this process? Exercise 5.5.12. Convert to rationals: 0.4453, 13.221, 1.112.
One way of seeing that every repeating decimal corresponds to a rational number is to use Equation (5.4). Again, using 0.42 to demonstrate, we know that 0.4242 · · · stands for
4 · 1 10+ 2 · 1 100 + 4 · 1 1000+ 2 · 1 10000+ · · · .
This can be seen to be the sum of a geometric sequence, something we learned about in Chapter 1. The common ratio of the geometric sequence is less than one, so by the results of that earlier chapter, this “infinite” sum can be evaluated as a fraction.
Uniqueness is a thorny issue when dealing with representations of rational numbers. We have already seen that the fractional form p/q is not unique (remember 1/2 = 2/4). The representation as repeating or terminating deci- mals also has its problems. For instance, is 2.4 terminating? Or is 2.4 = 2.40 repeating? Even more problematic is the next exercise.
Exercise 5.5.13. What rational number does 0.9 represent?
Exercise 5.5.14. Find another decimal representation of 0.9. Find another decimal representation of 3.2159. Find another decimal representation of 190. Exercise 5.5.15. Restate the following sentence so that it is true: every ra- tional has a unique representation as a repeating or terminating decimal. Hint: Use Exercise 5.5.14 to explain what exception must be ruled out.
Exercise 5.5.16. Make a list of situations where the fraction form for rational numbers is “better,” and make a list of situations where the repeating or ter- minating decimal form is “better.” Explain what definition of “better” you are using.
We summarize our results regarding the representation of rational numbers as decimals as follows.
Theorem 11. Every rational number has a unique representation as a terminating decimal or as a repeating decimal where the repeating section is not 9.
We can write our rational numbers in other number bases, just as we wrote the integers in other bases. Instead of “decimal fractions,” we have “basal fractions.” Instead of a “decimal point,” we use a “basal point.” And just as with decimal fractions, rationals are repeating or terminating basal fractions.
For example, in binary, 0.1two is one-half.
*Exercise 5.5.17. Convert to base two, base five and base eight: 0.5, 0.2, 1/3. *Exercise 5.5.18. Convert to decimal fractions: 0.5eight, 0.2three.
*Exercise 5.5.19. True or false (and give reasons): For each rational number, there is a number base such that the given rational number can be expressed as a terminating basal fraction in that base.
In a later chapter, we will discuss the next extension of our number system, the set of real numbers which includes irrational numbers. Irrational numbers are those whose decimal representations neither terminate nor repeat. An ex- ample is 1.6060060006 · · · . Another is π = 3.14159 · · · .