2.2: Concepts of Division of Whole Numbers

Learning Objectives

• understand the process of division
• understand division of a nonzero number into zero
• understand why division by zero is undefined
• be able to use a calculator to divide one whole number by another

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

\(\begin{matrix} \underbrace{\begin{array} {r} {Q} \\ {5 \overline{)15}} \end{array}} \\ {\text{5 into 15}} \end{matrix}\) \(\begin{matrix} \underbrace{\dfrac{15}{5} = Q} \\ {\text{15 divided by 5}} \end{matrix}\)

\(\begin{matrix} \underbrace{15/5 = Q} \\ {\text{15 divided by 5}} \end{matrix}\) \(\begin{matrix} \underbrace{15 \div 5 = Q} \\ {\text{15 divided by 5}} \end{matrix}\)

Each of these division notations describes the same number, represented here by the symbol \(Q\). Each notation also converts to the same multiplication form. It is \(15 = 5 \times Q\)

Definition: Dividend

In division, the number being divided into is called the dividend.

Definition: Divisor

In division, the number dividing into the dividend is the divisor.

Definition: Quotient

In division, the result of the division is called the quotient.

\[\begin{align*} \begin{array} {r} {\text{quotient}} \\ {\text{divisor} \overline{)\text{dividend}}} &\end{array} \\[4pt] \dfrac{\text{dividend}}{\text{divisor}} &= \text{quotient} \\[4pt] \text{dividend/divisor} &= \text{quotient} \\[4pt] \text{dividend} \div \text{divisor} &= \text{quotient} \end{align*}\]

Sample Set A

Find the following quotients using multiplication facts.

\(18 \div 6\)

Solution

Since \(6 \times 3 = 18\),

\(18 \div 6 = 3\)

Notice also that

\(\left \{ \begin{array} {r} {18} \\ {\underline{-6}} \\ {12} \\ {\underline{-6}} \\ {6} \\ {\underline{-6}} \\ {0} \end{array} \right \} \text{ Repeated subtraction}\)

Thus, 6 is contained in 18 three times.

Sample Set A

\(\dfrac{24}{3}\)

Solution

Since \(3 \times 8 = 24\),

\(\dfrac{24}{3} = 8\)

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

Sample Set A

\(\dfrac{36}{6}\)

Solution

Since \(6 \times 6 = 36\),

\(\dfrac{36}{6} = 6\)

Thus, there are 6 sixes in 36.

Sample Set A

\(9\overline{)72}\)

Solution

Since \(9 \times 8 = 72\),

\(\begin{array} {r} {8} \\ {9\overline{)72}} \end{array}\)

Thus, there are 8 nines in 72.

Practice Set A

Use multiplication facts to determine the following quotients.

\(32 \div 8\)

Answer

4

Practice Set A

\(18 \div 9\)

Answer

2

Practice Set A

\(\dfrac{25}{5}\)

Answer

5

Practice Set A

\(\dfrac{48}{8}\)

Answer

6

Practice Set A

\(\dfrac{28}{7}\)

Answer

4

Practice Set A

\(4\overline{)36}\)

Answer

4

Division into Zero (Zero as a Dividend: \(\dfrac{0}{a}\), \(a \ne 0\))

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

\[\dfrac{0}{\text{any nonzero whole number}}? \nonumber\]

Let's represent this unknown quotient by \(Q\). Then,

\[\dfrac{0}{\text{any nonzero whole number}} = Q\nonumber\]

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

\[0 = Q \times \text{(any nonzero whole number)}\nonumber\]

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, \(Q\) must represent zero. Then,

\[\dfrac{0}{\text{any nonzero whole number}} = 0\nonumber\]

Therefore zero divided any nonzero whole number is zero.

Division by Zero (Zero as a Divisor: \(\dfrac{a}{0}, a \ne 0\))

Now we ask: What number, if any, is

\[\dfrac{\text{any nonzero whole number}}{0}? \nonumber\]

Letting \(Q\) represent a possible quotient, we get

Converting to the corresponding multiplication form, we have

\[\text{(any nonzero whole number)} = Q \times 0 \nonumber\]

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

\[\dfrac{\text{any nonzero whole number}}{0} \nonumber\]

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: \(\frac{0}{0}\))

We are now curious about zero divided by zero \((\dfrac{0}{0})\). If we let \(Q\) represent a potential quotient, we get

\[\dfrac{0}{0} = Q\]

Converting to the multiplication form,

\[0 = Q \times 0\]

This results in

\[0 = 0\]

This is a statement that is true regardless of the number used in place of \(Q\). For example,

\[\dfrac{0}{0} = 5\]

since \(0 = 5 \times 0\).

\[\dfrac{0}{0} = 31\]

since \(0 = 31 \times 0\).

\[\dfrac{0}{0} = 286\]

since \(0 = 286 \times 0\).

A unique quotient cannot be determined.

Definition: Indeterminant

Since the result of the divisions above are inconclusive, we say that \(\dfrac{0}{0}\) is indeterminant.

Sample Set B

Perform, if possible, each division.

\(\dfrac{19}{0}\). Since division by 0 does not name a whole number, no quotient exists, and we state \(\dfrac{19}{0}\) is undefined

Sample Set B

\(0\overline{)14}\). Since division by 0 does not name a defined number, no quotient exists, and we state \(0\overline{)14}\) is undefined

Sample Set B

\(9\overline{)0}\). Since division into 0 by any nonzero whole number results in 0, we have \(\begin{array} {r} {0} \\ {9\overline{)0}} \end{array}\)

Sample Set B

\(\dfrac{0}{7}\). Since division into 0 by any nonzero whole number results in 0, we have \(\dfrac{0}{7} = 0\)

Practice Set B

Perform, if possible, the following divisions.

\(\dfrac{5}{0}\)

Answer

undefined

Practice Set B

\(\dfrac{0}{4}\)

Answer

0

Practice Set B

\(0\overline{)0}\)

Answer

indeterminant

Practice Set B

\(0\overline{)9}\)

Answer

undefined

Practice Set B

\(\dfrac{9}{0}\)

Answer

undefined

Practice Set B

\(\dfrac{0}{1}\)

Answer

0

Calculators

Divisions can also be performed using a calculator.

Sample Set C

Divide 24 by 3.

Solution

Display Reads
Type	24	24
Press	\(\div\)	24
Type	3	3
Press	=	8

The display now reads 8, and we conclude that \(24 \div 3 = 8\).

Sample Set C

Divide 0 by 7.

Solution

Display Reads
Type	0	0
Press	\(\div\)	0
Type	7	7
Press	=	8

The display now reads 0, and we conclude that \(0 \div 7 = 0\).

Sample Set C

Divide 7 by 0.

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

Solution

Display Reads
Type	7	7
Press	\(\div\)	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.

\(35 \div 7\)

Answer

5

Practice Set C

\(56 \div 8\)

Answer

7

Practice Set C

\(0 \div 6\)

Answer

0

Practice Set C

\(3 \div 0\)

Answer

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

Practice Set C

\(0 \div 0\)

Answer

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

Exercises

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

Exercise \(\PageIndex{1}\)

\(4\overline{)32}\)

Answer

8

Exercise \(\PageIndex{2}\)

\(7\overline{)42}\)

Exercise \(\PageIndex{3}\)

\(6\overline{)18}\)

Answer

3

Exercise \(\PageIndex{4}\)

\(2\overline{)14}\)

Exercise \(\PageIndex{5}\)

\(3\overline{)27}\)

Answer

9

Exercise \(\PageIndex{6}\)

\(1\overline{)6}\)

Exercise \(\PageIndex{7}\)

\(4\overline{)28}\)

Answer

7

Exercise \(\PageIndex{8}\)

\(\dfrac{30}{5}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{16}{4}\)

Answer

4

Exercise \(\PageIndex{10}\)

\(24 \div 8\)

Exercise \(\PageIndex{11}\)

\(10 \div 2\)

Answer

5

Exercise \(\PageIndex{12}\)

\(21 \div 7\)

Exercise \(\PageIndex{13}\)

\(21 \div 3\)

Answer

7

Exercise \(\PageIndex{14}\)

\(0 \div 6\)

Exercise \(\PageIndex{15}\)

\(8 \div 0\)

Answer

not defined

Exercise \(\PageIndex{16}\)

\(12 \div 4\)

Exercise \(\PageIndex{17}\)

\(3\overline{)9}\)

Answer

3

Exercise \(\PageIndex{18}\)

\(0\overline{)0}\)

Exercise \(\PageIndex{19}\)

\(7\overline{)0}\)

Answer

0

Exercise \(\PageIndex{20}\)

\(6\overline{)48}\)

Exercise \(\PageIndex{21}\)

\(\dfrac{15}{3}\)

Answer

5

Exercise \(\PageIndex{22}\)

\(\dfrac{35}{0}\)

Exercise \(\PageIndex{23}\)

\(56 \div 7\)

Answer

8

Exercise \(\PageIndex{24}\)

\(\dfrac{0}{9}\)

Exercise \(\PageIndex{25}\)

\(72 \div 8\)

Answer

9

Exercise \(\PageIndex{26}\)

Write \(\dfrac{16}{2} = 8\) using three different notations.

Exercise \(\PageIndex{27}\)

Write \(\dfrac{27}{9} = 3\) using three different notations.

Answer

\(27 \div 9 = 3\); \(9\overline{)27} = 3\); \(\dfrac{27}{9} = 3\)

Exercise \(\PageIndex{28}\)

In the statement \(\begin{array} {r} {4} \\ {6 \overline{)24}} \end{array}\)

6 is called the.

24 is called the.

4 is called the.

Exercise \(\PageIndex{29}\)

In the statement \(56 \div 8 = 7\).

7 is called the.

8 is called the.

56 is called the.

Answer

7 is quotient; 8 is divisor; 56 is dividend

Exercises for Review

Exercise \(\PageIndex{28}\)

What is the largest digit?

Exercise \(\PageIndex{29}\)

Find the sum. \(\begin{array} {r} {8,006} \\ {\underline{+ 4,118}} \end{array}\)

Answer

12,124

Exercise \(\PageIndex{28}\)

Find the difference. \(\begin{array} {r} {631} \\ {\underline{-589}} \end{array}\)

Exercise \(\PageIndex{29}\)

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

Answer

\(\begin{array} {c} {(2 + 3) + 7 = 2 + (3 + 7) = 12} \\ {5 + 7 = 2 + 10 = 12} \end{array}\)

Exercise \(\PageIndex{28}\)

Find the product. \(\begin{array} {r} {86} \\ {\underline{\times 12}} \end{array}\)