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# Division

In symbols, "p divided by q" can be written as  , or p ÷ q.

Remember the names for the different parts of a division problem:

• The dividend, p, is the thing being divided up. And sprinkled onto your salad, if you so choose.

• The divisor, q, is the thing that performs the dividing. It may also perform the conquering.

• The quotient is the answer.

"p divided by q" means the same thing as "q divided into p." When the numbers p and q are both whole numbers, division can be thought of as dividing up a bunch of things into smaller groups in either of two ways:

• If we split p into groups of size q will be how many groups (and perhaps fractions of a group) we get. By the way, "faction" is a synonym of "group," so you technically may be looking for the fraction of a faction. (We hope you're not mad at us for throwing that in there just to torture you. We wouldn't want there to be any friction.)

• If we split p into q groups,  will be the size of each group.

Sample: "20 divided by 4" could mean:

20 split into groups of size 5 or...

20 split into 4 groups

The idea is the same when p and q are real numbers instead of whole numbers, except that now the groups can have partial objects. Like if you're grouping chocolate chip cookies and some of them appear to have nibbles taken out of them. There's also the matter of signs, which we didn't have to worry about with whole numbers.

When dividing one positive real number into another, we just do the division: 4.2 ÷ 2.1 = 2

When one of the real numbers (either the divisor or the dividend) has a negative sign, perform the division while ignoring the signs, and then afterward reflect your answer across 0 on the number line. You'll always put those negative signs back on as the finishing touch. The icing on the cake. The Hershey's syrup on the Cracklin' Oat Bran. (We haven't actually tried that but we're guessing it would be delicious.)

Sample: -4.2 ÷ 2.1 = -2

= 4.2 ÷ -2.1 = -2

If both the divisor and dividend have negative signs, perform the division ignoring the signs. The answer needs to be reflected across 0 twice, which gets us the same answer as if both the divisor and dividend were positive. In this case, we can simply ignore the negative signs entirely. Which is rude, we guess, but whatever.

-4.2 ÷ -2.1 = 2.

Remember that division is an abbreviation for multiplying by the multiplicative inverse of a number. This will probably be most useful when fractions are involved.

Sample: .

Division is not commutative.

Sample:  is different from . Both are correct, but they're saying different things. Like your parents when they argue. See, that's the problem.

Division is not associative.

Sample: 1 ÷ 2 ÷ 3

If we evaluate 1 ÷ 2 first, we get one thing:

But if we evaluate 2 ÷ 3 first, we get something else:

Be Careful: Division by zero is undefined, as discussed in the section on rational numbers. If you ever feel tempted to divide by zero even though you know it's wrong, we're pretty sure there's a hotline you can call.