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Concepts of Division of Whole Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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: 0a0a size 12{ { {0} over {a} } } {}, a≠0a≠0 size 12{a <> 0} {})
  • Division by Zero (Zero As a Divisor: 0a0a size 12{ { {0} over {a} } } {}, a≠0a≠0 size 12{a <> 0} {})
  • Division by and into Zero (Zero As a Dividend and Divisor: 0000 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 QQ records the number of times 5 is contained in 15. We can indicate this by writing

Q 5 15 ︸ 5 into 15 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 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 QQ size 12{Q} {}. Each notation also converts to the same multiplication form. It is 15=5×Q15=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 quotient divisor dividend

dividend divisor = quotient dividend divisor =quotient

dividend / divisor = quotient dividend ÷ divisor = quotient dividend/divisor=quotientdividend÷divisor=quotient

Sample Set A

Find the following quotients using multiplication facts.

Example 1

18 ÷ 6 18 ÷ 6 size 12{"18" div 6} {}

Since 6×3=186×3=18 size 12{6 times 3="18"} {},

18 ÷ 6 = 3 18 ÷ 6 = 3 size 12{"18" div 6=3} {}

Notice also that

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

Thus, 6 is contained in 18 three times.

Example 2

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

Since 3×8=243×8=24 size 12{3 times 8="24"} {},

24 3 = 8 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.

Example 3

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

Since 6×6=366×6=36 size 12{6 times 6="36"} {},

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

Thus, there are 6 sixes in 36.

Example 4

9 72 9 72

Since 9×8=729×8=72 size 12{9 times 8="72"} {},

8 9 72 8 9 72

Thus, there are 8 nines in 72.

Practice Set A

Use multiplication facts to determine the following quotients.

Exercise 1

32÷832÷8 size 12{"32" div 8} {}

Solution

4

Exercise 2

18÷918÷9 size 12{"18" div 9} {}

Solution

2

Exercise 3

255255 size 12{ { {"25"} over {5} } } {}

Solution

5

Exercise 4

488488 size 12{ { {"48"} over {8} } } {}

Solution

6

Exercise 5

287287 size 12{ { {"28"} over {7} } } {}

Solution

4

Exercise 6

4 36 4 36

Solution

9

Division into Zero (Zero as a Dividend: 0a0a , a≠0a≠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 0any nonzero whole number0any nonzero whole number size 12{ { {0} over {"any nonzero whole number"} } } {}?

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

0 any nonzero whole number = Q 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 ) 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, QQ size 12{Q} {} must represent zero. Then,

0 any nonzero whole number = 0 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:a0a0 , a≠0a≠0)

Now we ask,

What number, if any, is any nonzero whole number0any nonzero whole number0 size 12{ { {"any nonzero whole number"} over {0} } } {} ?

Letting QQ size 12{Q} {} represent a possible quotient, we get

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

Converting to the corresponding multiplication form, we have

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

Since Q×0=0Q×0=0 size 12{Q times 0=0} {}, (any nonzero whole number)=0(any nonzero whole number)=0 size 12{ \( "any nonzero whole number" \) =0} {}. But this is absurd. This would mean that 6=06=0 size 12{6=0} {}, or 37=037=0 size 12{"37"=0} {}. A nonzero whole number cannot equal 0! Thus,

any nonzero whole number0any nonzero whole number0 size 12{ { {"any nonzero whole number"} over {0} } } {}does not name a number

Division by Zero is Undefined

Division by zero does not name a number. It is, therefore, undefined.

Division by and Into Zero (Zero as a Dividend and Divisor:0000)

We are now curious about zero divided by zero 0000 size 12{ left ( { {0} over {0} } right )} {}. If we let QQ size 12{Q} {} represent a potential quotient, we get

0 0 = Q 0 0 = Q size 12{ { {0} over {0} } =Q} {}

Converting to the multiplication form,

0 = Q × 0 0 = Q × 0 size 12{0=Q times 0} {}

This results in

0 = 0 0 = 0 size 12{0=0} {}

This is a statement that is true regardless of the number used in place of QQ size 12{Q} {}. For example,

00=500=5 size 12{ { {0} over {0} } =5} {}, since 0=5×00=5×0 size 12{0=5 times 0} {}.

00=3100=31 size 12{ { {0} over {0} } ="31"} {}, since 0=31×00=31×0 size 12{0="31" times 0} {}.

00=28600=286 size 12{ { {0} over {0} } ="286"} {}, since 0=286×00=286×0 size 12{0="286" times 0} {}.

A unique quotient cannot be determined.

Indeterminant

Since the result of the division is inconclusive, we say that 0000 size 12{ { {0} over {0} } } {} is indeterminant.

0000 size 12{ { {0} over {0} } } {} is Indeterminant

The division 0000 size 12{ { {0} over {0} } } {} is indeterminant.

Sample Set B

Perform, if possible, each division.

Example 5

190190 size 12{ { {"19"} over {0} } } {}. Since division by 0 does not name a whole number, no quotient exists, and we state 190190 size 12{ { {"19"} over {0} } } {} is undefined

Example 6

0 14 0 14 . Since division by 0 does not name a defined number, no quotient exists, and we state 0 14 0 14 is undefined

Example 7

9 0 9 0 . Since division into 0 by any nonzero whole number results in 0, we have 0 9 0 0 9 0

Example 8

0707 size 12{ { {0} over {7} } } {}. Since division into 0 by any nonzero whole number results in 0, we have 07=007=0 size 12{ { {0} over {7} } =0} {}

Practice Set B

Perform, if possible, the following divisions.

Exercise 7

5050 size 12{ { {5} over {0} } } {}

Solution

undefined

Exercise 8

0404 size 12{ { {0} over {4} } } {}

Solution

0

Exercise 9

0 0 0 0

Solution

indeterminant

Exercise 10

0 8 0 8

Solution

undefined

Exercise 11

9090 size 12{ { {9} over {0} } } {}

Solution

undefined

Exercise 12

0101 size 12{ { {0} over {1} } } {}

Solution

0

Calculators

Divisions can also be performed using a calculator.

Sample Set C

Example 9

Divide 24 by 3.

Table 1
Display Reads
Type 24 24
Press ÷ 24
Type 3 3
Press = 8

The display now reads 8, and we conclude that 24÷3=824÷3=8 size 12{"24" div 3=8} {}.

Example 10

Divide 0 by 7.

Table 2
Display Reads
Type 0 0
Press ÷ 0
Type 7 7
Press = 0

The display now reads 0, and we conclude that 0÷7=00÷7=0 size 12{0 div 7=0} {}.

Example 11

Divide 7 by 0.

Since division by zero is undefined, the calculator should register some kind of error message.

Table 3
Display Reads
Type 7 7
Press ÷ 7
Type 0 0
Press = Error

The error message indicates an undefined operation was attempted, in this case, division by zero.

Practice Set C

Use a calculator to perform each division.

Exercise 13

35÷735÷7 size 12{"35" div 7} {}

Solution

5

Exercise 14

56÷856÷8 size 12{"56" div 8} {}

Solution

7

Exercise 15

0÷60÷6 size 12{0 div 6} {}

Solution

0

Exercise 16

3÷03÷0 size 12{3 div 0} {}

Solution

An error message tells us that this operation is undefined. The particular message depends on the calculator.

Exercise 17

0÷00÷0 size 12{0 div 0} {}

Solution

An error message tells us that this operation cannot be performed. Some calculators actually set 0 ÷ 0 0÷0 equal to 1. We know better! 0 ÷ 0 0÷0 is indeterminant.

Exercises

For the following problems, determine the quotients (if possi­ble). You may use a calculator to check the result.

Exercise 18

4 32 4 32

Solution

8

Exercise 19

7 42 7 42

Exercise 20

6 18 6 18

Solution

3

Exercise 21

2 14 2 14

Exercise 22

3 27 3 27

Solution

9

Exercise 23

1 6 1 6

Exercise 24

4 28 4 28

Solution

7

Exercise 25

30 5 30 5 size 12{ { {"30"} over {5} } } {}

Exercise 26

164164 size 12{ { {"16"} over {4} } } {}

Solution

4

Exercise 27

24 ÷ 8 24 ÷ 8 size 12{"24" div 8} {}

Exercise 28

10÷210÷2 size 12{"10" div 2} {}

Solution

5

Exercise 29

21 ÷ 7 21 ÷ 7 size 12{"21" div 7} {}

Exercise 30

21÷321÷3 size 12{"21" div 3} {}

Solution

7

Exercise 31

0 ÷ 6 0 ÷ 6 size 12{0 div 6} {}

Exercise 32

8÷08÷0 size 12{8 div 0} {}

Solution

not defined

Exercise 33

12 ÷ 4 12 ÷ 4 size 12{"12" div 4} {}

Exercise 34

3 9 3 9

Solution

3

Exercise 35

0 0 0 0

Exercise 36

7 0 7 0

Solution

0

Exercise 37

6 48 6 48

Exercise 38

153153 size 12{ { {"15"} over {3} } } {}

Solution

5

Exercise 39

35 0 35 0 size 12{ { {"35"} over {0} } } {}

Exercise 40

56÷756÷7 size 12{"56" div 7} {}

Solution

8

Exercise 41

0 9 0 9 size 12{ { {0} over {9} } } {}

Exercise 42

72÷872÷8 size 12{"72" div 8} {}

Solution

9

Exercise 43

Write 162=8162=8 size 12{ { {"16"} over {2} } =8} {} using three different notations.

Exercise 44

Write 279=3279=3 size 12{ { {"27"} over {9} } =3} {} using three different notations.

Solution

27÷9=327÷9=3 size 12{"27" div 9=3} {}; 9 27 = 3 9 27 =3 ; 279=3279=3 size 12{ { {"27"} over {9} } =3} {}

Exercise 45

In the statement 4 6 24 4 6 24

6 is called the
.

24 is called the
.

4 is called the
.

Exercise 46

In the statement 56÷8=756÷8=7 size 12{"56" div 8=7} {},

7 is called the
.

8 is called the
.

56 is called the
.

Solution

7 is quotient; 8 is divisor; 56 is dividend

Exercises for Review

Exercise 47

((Reference)) What is the largest digit?

Exercise 48

((Reference)) Find the sum. 8,006 + 4,118 ̲ 8,006 + 4,118 ̲

Solution

12,124

Exercise 49

((Reference)) Find the difference. 631 - 589 ̲ 631 - 589 ̲

Exercise 50

((Reference)) Use the numbers 2, 3, and 7 to illustrate the associative property of addition.

Solution

(2+3)+7=2+(3+7)=125+7=2+10=12(2+3)+7=2+(3+7)=125+7=2+10=12alignl { stack { size 12{ \( 2+3 \) +7=2+ \( 3+7 \) ="12"} {} # size 12{5+7=2+"10"="12"} {} } } {}

Exercise 51

((Reference)) Find the product. 86 × 12 ̲ 86 × 12 ̲

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