Do Not Divide by Zero

Division of whole numbers can be done in two ways. (Long division combines the two.)
keep subtracting the divisor
Keep subtracting the divisor until what is left is smaller than the divisor. What is left over is the remainder. The quotient is the number of times you subtracted.
For example, to find divided by , do , , , . Stop now because is less than . There are four subtractions so the quotient is . The remainder is .
Another example: to divide by , do , , , . Stop because is less than . There are four subtractions so the quotient is . The remainder is .
Now for division by zero. When you subtract zero from a whole number larger than zero, you get the same whole number back. So you can subtract again and again without making any progress.
For example, to find divided by , do , , , …, forever. The result is never smaller than no matter how many times you subtract.
subtract the largest possible multiple of the divisor
To divide two whole numbers both greater than zero, find the largest multiple of the divisor that is not larger than the number being divided. That multiple is the quotient. The remainder is the difference between the number being divided and the product of the quotient and divisor.
For example, divide by . The multiples of are , , , , , , , …. The largest multiple less than is . The quotient is 4 because . The remainder is . So divided by is the quotient with remainder .
Another example: to divide by , do , so the quotient is and the remainder is zero.
When dividing by zero, all the multiples of zero are zero, so there is no largest one less than the number to be divided.
For example, to divide by , all of these multiples are zero and are less than : , , , , …,, …, , …. The quotient cannot be the largest of , , , , …, , …, , …, because there is no largest.
So do not divide by zero!



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Because , ,, and so on, it seems reasonable that should be , as well.
But in that case, . The number is not special; is any number you want.
Another possibility is that is not . Suppose it is instead. Then . So is after all.
Whatever we assume to be, we can use fractions to show it would be anything else. So we cannot give any value.
So do not divide by zero!
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Related Curriculum Standards

US Common Core State Standards, Mathematics

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