2.4 - Division of Numbers

There are several notations for division:

They all mean to divide the number a by the number b. The number a is called the numerator (or sometimes dividend), the number b is called the denominator (or sometimes divisor), and the ratio a/b is called the quotient. The notation is exactly the same as the notation for fractions because when a and b are whole numbers then the quotient that results from dividing a by b is just the fraction a/b. Click here to review fractions. In this section we study division when a and b are any real numbers in general.

Division defined. The result of dividing a real number a by a real number b is that real number c such that a = b · c. In other words, division is defined in terms of multiplication, which we studied in the previous section.

For example:

Division (of a non-zero number) by zero is undefined. The reason is this. Suppose that it was possible that a / 0 (where a is non-zero) could equal some number c. Then by the definition of division this would mean that a = 0 · c. But this is impossible because zero times anything is zero.

Zero divided by zero is indeterminate. (Indeterminate means that 0 / 0 could equal any number c.) The reason is that by the definition of division this means that 0 = 0 · c. But this is true for any number c whatsoever.

Note that this does not mean that 0 / n is undefined or indeterminate. On the contrary, 0 / n = 0 for any non-zero n.

Reciprocals. Two numbers whose product is 1 are called reciprocals.

For example:
4/5 and 5/4 are reciprocals because  

8 and 1/8 are reciprocals because  

Just as subtraction can be replaced by the addition of a negative so division can be replaced by multiplication by a reciprocal:

The division of a by b is equivalent to the multiplication of a by the reciprocal of b :

Here are some examples:
replace this division by this multiplication
by the reciprocal

Division with negative numbers

Because division is defined in terms of multiplication, the rule for finding the sign for division is the same as the rule for finding the sign for multiplication: Count the number of negative factors. (It doesn't matter whether the factors are in the numerator or the denominator.) If the number is even then the result is positive. If the number is odd then the result is negative. The minus sign is usually put in front of the quotient. In this example there are 5 negative factors so the result is negative:

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