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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide whole numbers. By the end of the module students should be able to understand the process of division, understand division of a nonzero number into zero, understand why division by zero is undefined, and use a calculator to divide one whole number by another.

Section overview

  • Division
  • Division into Zero (Zero As a Dividend: 0 a size 12{ { {0} over {a} } } {} , a 0 size 12{a<>0} {} )
  • Division by Zero (Zero As a Divisor: 0 a size 12{ { {0} over {a} } } {} , a 0 size 12{a<>0} {} )
  • Division by and into Zero (Zero As a Dividend and Divisor: 0 0 size 12{ { {0} over {0} } } {} )
  • Calculators

Division

Division is a description of repeated subtraction.

In the process of division, the concern is how many times one number is contained in another number. For example, we might be interested in how many 5's are contained in 15. The word times is significant because it implies a relationship between division and multiplication.

There are several notations used to indicate division. Suppose Q records the number of times 5 is contained in 15. We can indicate this by writing

Q 5 15 5 into 15 15 5 = Q 15 divided by 5

15 / 5 = Q 15 divided by 5 15 ÷ 5 = Q 15 divided by 5

Each of these division notations describes the same number, represented here by the symbol Q size 12{Q} {} . Each notation also converts to the same multiplication form. It is 15 = 5 × Q size 12{"15"=5 times Q} {}

In division,

Dividend

the number being divided into is called the dividend .

Divisor

the number dividing into the dividend is the divisor .

Quotient

the result of the division is called the quotient .

quotient divisor dividend

dividend divisor = quotient

dividend / divisor = quotient dividend ÷ divisor = quotient

Sample set a

Find the following quotients using multiplication facts.

18 ÷ 6 size 12{"18" div 6} {}

Since 6 × 3 = 18 size 12{6 times 3="18"} {} ,

18 ÷ 6 = 3 size 12{"18" div 6=3} {}

Notice also that

18 - 6 ̲ 12 - 6 ̲ 6 - 6 ̲ 0 Repeated subtraction

Thus, 6 is contained in 18 three times.

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24 3 size 12{ { {"24"} over {3} } } {}

Since 3 × 8 = 24 size 12{3 times 8="24"} {} ,

24 3 = 8 size 12{ { {"24"} over {3} } =8} {}

Notice also that 3 could be subtracted exactly 8 times from 24. This implies that 3 is contained in 24 eight times.

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36 6 size 12{ { {"36"} over {6} } } {}

Since 6 × 6 = 36 size 12{6 times 6="36"} {} ,

36 6 = 6 alignl { stack { size 12{ { {"36"} over {6} } =6} {} #{} } } {}

Thus, there are 6 sixes in 36.

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9 72

Since 9 × 8 = 72 size 12{9 times 8="72"} {} ,

8 9 72

Thus, there are 8 nines in 72.

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Practice set a

Use multiplication facts to determine the following quotients.

32 ÷ 8 size 12{"32" div 8} {}

4

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18 ÷ 9 size 12{"18" div 9} {}

2

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25 5 size 12{ { {"25"} over {5} } } {}

5

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48 8 size 12{ { {"48"} over {8} } } {}

6

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28 7 size 12{ { {"28"} over {7} } } {}

4

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Division into zero (zero as a dividend: 0 a , a 0 )

Let's look at what happens when the dividend (the number being divided into) is zero, and the divisor (the number doing the dividing) is any whole number except zero. The question is

What number, if any, is 0 any nonzero whole number size 12{ { {0} over {"any nonzero whole number"} } } {} ?

Let's represent this unknown quotient by Q size 12{Q} {} . Then,

0 any nonzero whole number = Q size 12{ { {0} over {"any nonzero whole number"} } =Q} {}

Converting this division problem to its corresponding multiplication problem, we get

0 = Q × ( any nonzero whole number ) size 12{0=Q times \( "any nonzero whole number" \) } {}

From our knowledge of multiplication, we can understand that if the product of two whole numbers is zero, then one or both of the whole numbers must be zero. Since any nonzero whole number is certainly not zero, Q size 12{Q} {} must represent zero. Then,

0 any nonzero whole number = 0 size 12{ { {0} over {"any nonzero whole number"} } =0} {}

Zero divided by any nonzero whole number is zero

Zero divided any nonzero whole number is zero.

Division by zero (zero as a divisor: a 0 , a 0 )

Now we ask,

What number, if any, is any nonzero whole number 0 size 12{ { {"any nonzero whole number"} over {0} } } {} ?

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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