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

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

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

any nonzero whole number 0 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: 0 0 )

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

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

Converting to the multiplication form,

0 = Q × 0 size 12{0=Q times 0} {}

This results in

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

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

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

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

0 0 = 286 size 12{ { {0} over {0} } ="286"} {} , since 0 = 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 0 0 size 12{ { {0} over {0} } } {} is indeterminant .

0 0 size 12{ { {0} over {0} } } {} Is indeterminant

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

Sample set b

Perform, if possible, each division.

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

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

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

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

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Practice set b

Perform, if possible, the following divisions.

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

undefined

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

0

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

undefined

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

0

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Calculators

Divisions can also be performed using a calculator.

Sample set c

Divide 24 by 3.

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

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

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Divide 0 by 7.

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

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

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Divide 7 by 0.

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

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.

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Practice set c

Use a calculator to perform each division.

35 ÷ 7 size 12{"35" div 7} {}

5

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56 ÷ 8 size 12{"56" div 8} {}

7

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0 ÷ 6 size 12{0 div 6} {}

0

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3 ÷ 0 size 12{3 div 0} {}

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

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0 ÷ 0 size 12{0 div 0} {}

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

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Exercises

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

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

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

4

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24 ÷ 8 size 12{"24" div 8} {}

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10 ÷ 2 size 12{"10" div 2} {}

5

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21 ÷ 7 size 12{"21" div 7} {}

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21 ÷ 3 size 12{"21" div 3} {}

7

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0 ÷ 6 size 12{0 div 6} {}

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8 ÷ 0 size 12{8 div 0} {}

not defined

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12 ÷ 4 size 12{"12" div 4} {}

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

5

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

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56 ÷ 7 size 12{"56" div 7} {}

8

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

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72 ÷ 8 size 12{"72" div 8} {}

9

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

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

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

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In the statement 4 6 24

6 is called the .

24 is called the .

4 is called the .

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

7 is called the .

8 is called the .

56 is called the .

7 is quotient; 8 is divisor; 56 is dividend

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Exercises for review

( [link] ) What is the largest digit?

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( [link] ) Find the sum. 8,006 + 4,118 ̲

12,124

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( [link] ) Find the difference. 631 - 589 ̲

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

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

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( [link] ) Find the product. 86 × 12 ̲

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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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