Properties of Division
We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know 12 ÷ 4 = 3 because 3 · 4 = 12. Knowing all the multiplication number facts is very important ...
Divide Whole Numbers
We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know \(12÷4=3\) because \(3·4=12.\) Knowing all the multiplication number facts is very important when doing division.
We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. From the example in the previous lesson, we know \(24÷8=3\) is correct because \(3·8=24.\)
Example
Divide. Then check by multiplying. (a) \(42÷6\) (b) \(\frac{72}{9}\) (c) \(7\overline{)63}\)
Solution
| (a) | |
| \(42÷6\) | |
| Divide 42 by 6. | \(7\) |
| Check by multiplying. \(7·6\) | |
| \(42✓\) |
| (b) | |
| \(\cfrac{72}{9}\) | |
| Divide 72 by 9. | \(8\) |
| Check by multiplying. \(8·9\) | |
| \(72✓\) |
| (c) | |
| \(7\overline{)63}\) | |
| Divide 63 by 7. | \(9\) |
| Check by multiplying. \(9·7\) |
|
| \(63✓\) |
Extra:
Divide. Then check by multiplying:
(a) \(54÷6\) (b) \(\frac{27}{9}\)
Answer
(a) 9 (b) 3
Extra:
Divide. Then check by multiplying:
(a) \(\frac{36}{9}\) (b) \(8\overline{)40}\)
Solution
(a) 4 (b) 5
What is the quotient when you divide a number by itself?
Dividing any number \(\mathrm{(except 0)}\) by itself produces a quotient of \(1.\) Also, any number divided by \(1\) produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Division Properties of One:
| Any number (except 0) divided by itself is one. | \(a÷a=1\) |
| Any number divided by one is the same number. | \(a÷1=a\) |
Example
Divide. Then check by multiplying:
- \(11÷11\)
- \(\frac{19}{1}\)
- \(1\overline{)7}\)
Solution
| a. | |
| \(11÷11\) | |
| A number divided by itself is 1. | \(1\) |
| Check by multiplying. \(1·11\) |
|
| \(11✓\) |
| b. | |
| \(\frac{19}{1}\) | |
| A number divided by 1 equals itself. | \(19\) |
| Check by multiplying. \(19·1\) |
|
| \(19✓\) |
| c. | |
| \(1\overline{)7}\) | |
| A number divided by 1 equals itself. | \(7\) |
| Check by multiplying. \(7·1\) | |
| \(7✓\) |
Extra:
Divide. Then check by multiplying:
(a) \(14÷14\) (b) \(\frac{27}{1}\)
Answer
(a) 1 (b) 27
Extra:
Divide. Then check by multiplying:
(a) \(\frac{16}{1}\) (b) \(1\overline{)4}\)
Solution
(a) 16 (b) 4
Suppose we have \(\mathrm{$0},\) and want to divide it among \(3\) people. How much would each person get? Each person would get \(\mathrm{$0}.\) Zero divided by any number is \(0.\)
Now suppose that we want to divide \(\mathrm{$1}0\) by \(0.\) That means we would want to find a number that we multiply by \(0\) to get \(10.\) This cannot happen because \(0\) times any number is \(0.\) Division by zero is said to be undefined.
These two ideas make up the Division Properties of Zero.
Division Properties of Zero:
| Zero divided by any number is 0. | \(0÷a=0\) |
| Dividing a number by zero is undefined. | \(a÷0\) undefined |
Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away \(0\) from \(10?\) Because subtracting \(0\) will never change the total, we will never get an answer. So we cannot divide a number by \(0.\)
Example
Divide. Check by multiplying: (a) \(0÷3\) (b) \(10/0.\)
Solution
| (a) | |
| \(0÷3\) | |
| Zero divided by any number is zero. | \(0\) |
| Check by multiplying.
|
|
| \(0✓\) |
| (b) | |
| \(10/0\) | |
| Division by zero is undefined. | undefined |
Extra:
Divide. Then check by multiplying:
(a) \(0÷2\) (b) \(17/0\)
Solution
(a) 0 (b) undefined
Extra:
Divide. Then check by multiplying:
(a) \(0÷6\) (b) \(13/0\)
Solution
(a) 0 (b) undefined
When the divisor or the dividend has more than one digit, it is usually easier to use the \(4\overline{)12}\) notation. This process is called long division. Let’s work through the process by dividing \(78\) by \(3.\)
| Divide the first digit of the dividend, 7, by the divisor, 3. | |
| The divisor 3 can go into 7 two times since \(2×3=6\). Write the 2 above the 7 in the quotient. | |
| Multiply the 2 in the quotient by 2 and write the product, 6, under the 7. | |
| Subtract that product from the first digit in the dividend. Subtract \(7-6\). Write the difference, 1, under the first digit in the dividend. | |
| Bring down the next digit of the dividend. Bring down the 8. | |
| Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times. | |
| Write 6 in the quotient above the 8. | |
| Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18. |
We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.
Check by multiplying the quotient times the divisor to get the dividend. Multiply \(26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}3\) to make sure that product equals the dividend, \(78.\)
It does, so our answer is correct.
Optional Video: Dividing Whole Numbers With a Remainder
This video below by Mathispower4u provides an example of dividing a 4 digit whole number by a 2 digit whole number with a remainder.
This lesson is part of:
Introducing Numbers