ARITHMETIC OF FRACTIONS AND REAL NUMBERS

Objectives: To recall how fractions are added or multiplied to other fractions or real numbers.

Alternative Reading:  Section 1.2 of Beginning Algebra, by K. Elayn Martin-Gay.  Available on reserve in the library.

Recall 1: Basic definitions

A natural number is a number from the set

1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

where the three dots . . .  on the right mean that these numbers continue forever.

A whole number is a natural number or 0.

An integer is a number from the set

. . .  -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6,. . .

A fractional number  is a number of the form m/n , where both m and n are whole numbers, and n is different from  0.

So, 1/2, 3/5, 2/13, 4/1 and 0/1 are examples of frational numbers, but 1/0 and (-1)/2 are not.

A rational number is a number of the form m/n, where both m and n are integers, and n is different from 0.

So, 9/(-5), (-1)/2, 2/13 and 0/1 are examples of rational numbers, but 1/0 is not.

You may have forgotten the basic rules for multiplying and adding fractions. Here are some general rules.

1.1 , if a, b, c, d are integers,  with b and d different form 0.

1.2 ,  if a, b, c, d are integers,  with b and d different form 0.

It is easy to convince yourself that the two rules above are valid. Take a few sample numbers -- let a/b=1/2 and c/d=3/4. The question is whether or not the equalities

and

hold. The validity of these equations does not prove the general equations 1.1 and 1.2, but it does make 1.1 and 1.2 more believable. How can we test the validity of these equalities?

Notice that 3/4 may be represented by the picture

If the whole rectangle is divided into 4 equal pieces, then three of those (lightly shaded) pieces represent 3/4 of the whole rectangle. Multiplying 3/4 by 1/2 is the same thing as dividing by 2. Pictorially, this is represented by dividing the lightly shaded pieces in two. The darkly shaded areas therefore represent 1/2 of 3/4.

This picture has 8 parts. Six of them represent 3/4 of the whole; three of them represent 1/2 of 3/4 of the whole. This means that 1/2 of 3/4 equals 3/8.

Now we can see that the equality 1.1 is valid.

A similar argument would convince you of the validity of 1.2.

Recall 2: If a/b is a fractional or rational number, then b (the lower number) is called the denominator  and a (the upper number) is called the numerator.

If we are adding two fractional or rational numbers such as a/b and c/b, where the denominators are equal, then the result is

a/b+c/b =(a+c)/b .

In other words -- in this case -- we add the numerators and keep the denominator the same.

Warning

Keep in mind that adding the numerators is valid only in the special case where the denominators are equal.

Recall 3:

(a) Do you recall that adding two numbers, a and b is the same as adding b and a? In other words, a + b = b + a. It is the same with multiplying a and b. In other words axb = bxa. The laws that permit you to flip a and b are called the commutative laws.

(b) Do you recall that adding two numbers, a and b and then a third number c is the same as adding a to the sum of b and c? In other words, (a + b) + c = a + (b + c). The laws that permit you to move the parentheses around are called the associative laws.

(c) Do you recall that multiplying a number a by the sum of two other numbers, b and c is the same as adding axb and axc? In other words, ax(b + c) = axb + axc. The law that permits you to combine the multiplication and addition in this way is called the distributive law

Recall 4 When the numerator and denominator of a rational number are equal, then that rational number is 1. For example, the following rational numbers are all equivalent to the number 1.
1/1, (-1)/(-1), 17/17, x/x (x different from 0).

Recall 5 Multiplying any number by 1 does not change the number. If a is any number, rational or otherwise, then 1*a = a.

Recall 6 An expression is a symbol, or sequence of symbols, that represents a quantity. For example, the following are expressions:

Terminology

An expression of the form of a single fraction is called a rational expression. For example,

x^2/y^2, (x+1)y/2y and (x+1)/2

are rational expressions.

Recall 7 Do you recall how to divide two fractional expressions? Suppose that you are faced with the expression

Do you know how to evaluate such an expression? The general rule is that if you are faced with a rational expression of the form

you should multiply the numerator by the flip or inverse of the denominator; that is (a/b)*(d/c) . Therefore

Note

Certain ambiguities may arise through incorrect use of notation, as in the following case:

Does this rational number stand for 1/4 of 2/3, or 2 divided by 3/4?  It is not at all clear. The ambiguity is the fault of the notation. If it stands for 1/4 of 2/3, then it should be written as

Notice the difference between the long and short lines.

Note

The rational numbers 9/18, 3/6, 1/2 are all equivalent representations of 1/2. We say that 1/2 is the reduced form of the rational numbers of the form (1*a)/(2*a) , where a stands for any integer. In Recall 7 we found that

equals 18/28. The reduced form of 18/28 is 9/14.

Examples

Example 1. Simplify (x/y)(x/y)

Solution: From Rule 1 we know that the expression may be written as (xx)/(yy). We often denote xx as x^2; so this last expression may be written as x^2/y^2.

Example 2. Simplify  [(x+1)/y](y/2).

Solution: From Rule 1, we know that
[(x+1)/y](y/2)=[(x+1)y]/(y2)

If we now use the commutative law (Recall 3 a), we see that the expression on the right may be written as
y(x+1)/(y2)

Using Rule 1, once again, we may write this last expression as (y/y)(x+1)/2. By Recall 4, we know that (y/y)=1. So we have 1(x+1)/2. By Recall 5, we may write this last expression as
(x+1)/2

Example 3. Write 3/y+x/4 as a rational expression.

Solution: The rational number 3/y is equivalent to (4*3)/(4*y) and the rational number x/4 is equivalent to (x*y)/(4*y).

Both (4*3)/(4*y) and (x*y)/(4*y) have the same denominator; so we may add the numerators and keep the denominator 4*y. We have
3/y+x/4=(4*3+x*y)/4y=(12+xy)/4y.

Example 4. Write (x/y)[3/y+x/4] as a rational expression.

Solution: From Example 3, we know that 3/y+x/4 may be represented in its equivalent form as (12+xy)/4y. Therefore, we may start with (x/y)(12+xy)/4y. Using Rule 1, together with Recall 4, we may write this last expression as (12x+x^2*y)/(4*y^2), which is the desired rational expression.

Note

1) 0/1= 0 is not considered to be a rational number.

EXERCISES

Write each expression as a fractional expression

1. (3/4)(5/2)

2. (3/4)(2/2)

3. 5*(3/4)

4. (3/4)/(5/2)

5 (a) (2/3)/4    (b)2/(3/4)

6. 3/5+5/3

7. (4/3)(3/5+5/3)

8. -[(y+3)/x](x/5)

9. (x/y)(x/5+x/3)

10. 1/(2/x+2/y)

11. [(x+1)/y](y/2)

12. 3x/(2/x+4y)

13. (y+z)/[(2/x)(y/3)]

14. [1/(y+z)]/[(2/x)(y/3)]

15. [1/yz)/(3/x+y/3)

16. (2xy+1)[2x/(2/x+4y)]