Concept of Rational Numbers

The Rational Numbers

Let us now turn our attention to the set of rational numbers. We have already considered the four operations of arithmetic with the natural numbers, quotient numbers, and integers. Since these sets are all subsets of F, we will want the definitions and rules we develop for the rational numbers to be consistent with those already discussed for N, Q, and I.
  Every quotient number can be represented by a fraction whose numerator and denominator are both natural number symbols. But many different fractions can represent the same quotient number. For example, 2/3, 4/6 and 10/15 all represent the same number, so we call them equal fractions. Notice that 2/3=4/6 and 2∙6=3∙4. More generally we say that if a/b and c/d represent quotient numbers, then a/b = c/d if and only if ad = bc. Every rational number can be represented by a fraction whose numerator and denominator are integer symbols. (But the denominator may not be 0.) Let us make an analogous assumption about the rational numbers. 

Example 1 Decide whether or not these pairs of numerals are equal.

(a) -8/(-3) and 8/3    (b) 4/(-7) and (-7)/4    (c) (-3)/4 and 6/(-8)

Solution

(a) -8/(-3)=8/3 Since (-8)(3) = (-3)(8)

(b) 4/(-7)!=(-7)/4 since (4)(4)!=(-7)(-7)

(c) (-3)/4=6/(-8) Since (-3)(-8) = (4)(6)

Canceling is a useful technique when working with fractions which represent quotient numbers. If the numerator and denominator of a fraction have exactly the same factor, this factor may be canceled. For example,

This same canceling property holds in the set F, as the following theorem proves.

Theorem 1

Proof We will show that the cross products are equal.

(ax)b = b(ax)      [Commutative property of multiplication with integers]
       = b(xa)     [Same as above]
         = (bx)a     [Associative property of multiplication with integers]

Example 2  Use the canceling theorem to simplify these.

(a) -24/(-30)    (b) (-64)/24    (c) 18/(-45)

Solution

Let us adopt the same rules for adding and multiplying rational numbers that we use for quotient numbers.

Example 3 Add or multiply as indicated.

(a) (-4)/5+(-3)/(-7)    (b) (-2)/(-5)∙7/(-9)

Solution

(a) -4/5+(-3)/(-7)=((-4)(-7)+(5)(-3))/((5)(-7))=(28+(-15))/(-35)=13/(-35)
(b) (-2)/(-5)∙7/(-9)=((-2)(7))/((-5)(-9))=(-14)/(45)

  There is an easier method for adding quotient numbers whose denominators are the same. For example,

          4/19+12/19=(4+12)/19=16/19

This method can also be used with rational numbers, as the next theorem illustrates.

Theorem 2

 
Proof  a/b+c/b=(ab+bc)/(b∙b)=(ba+bc)/(b∙b)=(b(a+c))/(b∙b)=(a+c)/b

Example 4  Add:

(a) 4/(-7)+36/(-7)     (b) (-5)/12+5/6

Solution

(a) 4/(-7)+36/(-7)=(4+36)/(-7)=40/(-7)
(b) (-5)/12+5/6=(-5)/12+10/12=(-5+10)/12=5/12

  We shall represent the additive inverse of a rational number in a manner analogous to that we used for the integers. The additive inverse of 5/7 is represented by − 5/7,  and the additive inverse of  -5/7  is − -5/7. This notation leads to some interesting results if we consider the fact that 5/7=(-5)/(-7)  [ since (5)(-7) = (7)(-5) ]. Then − 5/7 = − (-5)/(-7) that is, the additive inverse of 5/7 equals the additive inverse of (-5)/(-7). Also (-5)/7=5/(-7) [ since (-5)(-7)=7∙5 ]. Hence − (-5)/7 = − 5/(-7); the additive inverse of (-5)/7 equals the additive inverse of 5/(-7).

Example 5 Show that (-5)/7 = − 5/7 

Solution Since 5/7+(-5)/7 = 0, we see that (-5)/7 is the additive inverse of 5/7 or in symbols (-5)/7 = − 5/7.

  Of course the example also shows that − (-5)/7=5/7 (The additive inverse of (-5)/7 is 5/7). When we combine the results of the example and the notation discussed in the preceding paragraph, we see that (-5)/7, 5/(-7), − (-5)/(-7) and − 5/7 all represent the same rational number. Also, − (-5)/7 = − 5/(-7) = (-5)/(-7) = 5/7.

Example 6 Determine which of these are equal:

(-3)/(-4), 3/(-4), − 3/4, − (-3)/(-4), − 3/(-4), − (-3)/4, (-3)/4, 3/4.

Solution

(-3)/(-4) = − 3/(-4) = − (-3)/4 = 3/4 [Note that in each there is an even number of negative signs]

3/(-4)= − 3/4= −(-3)/(-4)= (-3)/4  [Note that each has an odd number of negative signs]

  Since there are so many ways to represent any rational number, we shall want to define the simplest form of each. If the rational number is an integer, we have already agreed to represent it with one of the symbols, ..., -2, -1, 0, 1, 2, ... . Positive rational numbers are also quotient numbers, and we have agreed to represent those which are not integers with fractions whose numerators and denominators are the simplest forms of natural numbers and are relatively prime. That leaves only the negative rational numbers which are not integers; for them let us agree that the simplest form is a fraction whose numerator is the simplest form of a negative integer and whose denominator is the simplest form of a positive integer. Of course, we will want the numerator and denominator to be relatively prime (that is, have no common factors which will cancel).

Example 7  Find the simplest form.

(a) 12/(-4)  (b) (-3)/(-8)  (c) (-8)/28  (d) − 6/(-48)  (e) − (-5)/(-15)

Solution  (a) -3   (b) 3/8   (c) (-2)/7  (d) 1/8   (e) (-1)/3

  Although we shall usually want to express a rational number in its simplest form, there are times when its decimal form will be helpful. We know that 3/2= 1.5. Then since (-3)/2 = − 3/2, we see that (-3)/2=-1.5. When the decimal form is not obvious, we may always find it by division as the next example illustrates.

Example 8  Find the decimal form.

(a) (-5)/16    (b) 8/3    (c) (-3)/11

Solution

We see that the division never ends and the decimal form has infinitely many digits which repeat in a set pattern. Let us call such a decimal a repeating decimal. We shall write 8/3 = 2.6... . Note the use of dots to indicate ‘‘and so forth” and the bar over the 6 to indicate that the 6, but not the 2 is repeating.

Note the bar over both the 2 and 7 to indicate both these digits repeat.   It can be shown that every rational number has a decimal form which either terminates, such as .3125 or is repeating, such as .27... . It is also true that every decimal which either terminates or is repeating represents a rational number.

  We are now ready to define subtraction for rational numbers. Example 5 suggests that the additive inverse of every rational number is also a rational number. In general if a/b {is-in} F, then − a/b = (-a)/b and so − a/b {is-in} F 
This fact and Theorem 2 on subtraction of integers lead us to make the following definition.

Example 9  Find the simplest form.

(a) -7/12 − 4/12    (b) (-8)/9 − 3/(-5)

Solution

(a) -7/12 − 4/12 = -7/12 + (-4)/12 = (-11)/12

(b) (-8)/9 − 3/(-5) = (-8)/9 + 3/5 = (-40+27)/45 = (-13)/45 

  Since -7/2∙2/-7=(-14)/(-14)=1, we see that -7/2 and 2/-7 are multiplicative inverses. Similarly, every rational number (except zero) has a multiplicative inverse. In general the multiplicative inverse of a/b is b/a, if a!= 0 and b!= 0.

Example 10 Find in simplest form the multiplicative inverse of each of these.

(a) 4/-9    (b) (-11)/(-13)    (c) -3    (d) 0

Solution

(a) -9/4     (b) 13/11     (c) -1/3     (d) There is none

  We shall now define division with rational numbers in a manner analogous to that for quotient numbers.

Example 11  Find the simplest form.

(a) -3/2 ÷ 2/5    (b) (-5)/(-8) ÷ 25/-6

Solution

In Example 9(a) we see that -7/12 − 4/12 = -11/12. It is worth noting that -11/12 + 4/12 = -7/12. The sum of the difference and the subtrahend equals the minuend. Also in Example 11(a) we have -3/2 ÷ 2/5 = -15/4. And -15/4∙2/5=-3/2. The product of the quotient and the divisor equals the dividend. It is interesting to observe that the definitions we have chosen, for subtraction and division with rational numbers are consistent with the definitions we have chosen for these operations with the integers.