
- Multiply and Divide Whole Numbers
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- Multiplication as Repeated Addition
- Single Digit Multiplication
- Multiplication by 10, 100, and 1000
- Multiplication Without Carry
- Multiplication With Carry
- Multiplication With Trailing Zeros: Problem Type 1
- Multiplication With Trailing Zeros: Problem Type 2
- Multiplication of 2-digit Numbers With 2-digit Numbers
- Multiplication of a Single Digit Number With Large Numbers
- Multiplication of Large Numbers
- Multiples Problem Type 1
- Multiples Problem Type 2
- Division Facts
- Fact Families for Multiplication and Division
- Multiplication or Division of Whole Numbers (Word problems)
- Multiplication and Addition or Subtraction of Whole numbers (Word problems)
- Unit Rates and Ratios of Whole Numbers (Word problems)
- Division Without Carry
- Division With Carry
- Division Involving Zero
- Whole Number Division: 2-digit by 2-digit, No Remainder
- Whole Number Division: 3-digit by 2-digit, No Remainder
- Division With Trailing Zeros: Problem Type 1
- Division With Trailing Zeros: Problem Type 2
- Quotient and Remainder: Problem type 1
- Quotient and Remainder: Problem type 2
- Quotient and Remainder (Word problems)
Division Involving Zero
- For any real number a, $\frac{a}{0}$ is undefined
- For any non-zero real number a, $\frac{0}{a}$ = 0
- $\frac{0}{0}$ is indeterminate
For any real number a, $\frac{a}{0}$ is undefined
Dividing any real number by zero is undefined and sometimes taken as infinity. Division is splitting into equal parts or groups.
Let us consider an example: Suppose there are 12 ice cream cups and 4 friends want to share them. How do they divide the ice cream cups?
$\frac{12}{4}$ = 3; So they get 3 each: Now, let us try dividing the 12 ice cream cups among zero people. How much does each person get?
Does that question make sense? No, it doesn't.
We can't share among zero people, and we can't divide by 0.
Suppose we could get some number k by dividing any real number a by zero
Let us assume $\frac{a}{0}$ = k. Then k × 0 = a. There is no such number k which when multiplied by zero will give a. So k does not exist and therefore $\frac{a}{0}$ is said to be undefined.
For any non-zero real number a, $\frac{0}{a}$ = 0
If zero is divided by any non-zero real number a, we get 0 as the result. If zero items are divided among a number of people, share got by each person will be zero only
$\frac{0}{0}$ is indeterminate
Division of zero by zero is a quantity that cannot be found and is called indeterminate.
Find the value of $\frac{0}{5}$
Solution
Step 1:
For example, $\frac{3}{4}$ = 3 × $\frac{1}{4}$ .
Step 2:
Similarly, $\frac{0}{5}$ = 0 × $\frac{1}{5}$ = 0
as the product of zero and any number is zero.
Evaluate $\frac{7}{0}$
Solution
Step 1:
By definition, division of any number by zero is not defined.
Step 2:
So, $\frac{7}{0}$ is not defined.
Evaluate $\frac{0}{13}$
Solution
Step 1:
Zero divided by any number is zero.
Step 2:
So, $\frac{0}{13}$ = 0