- Topics At a Glance
- Different Types of Numbers
- Natural Numbers
- Whole Numbers
- Integers and Negative Numbers
- Integers and Absolute Value
- Rational Numbers
- Irrational Numbers
- Real Numbers and Imaginary Numbers
- Different Ways to Represent Numbers
- Fractions
- Equivalent Fractions
- Mixed Numbers
- Reducing Fractions
- Comparing Fractions
- Least Common Denominator
- Addition and Subtraction of Fractions
- Multiplication of Fractions
- Equivalent Fractions and Multiplication by 1
- Multiplication by Clever Form of 1
- Multiplicative Inverses
- Division of Fractions
- Multiplication and Division with Mixed Numbers
- Decimals
- Converting Fractions into Decimals
- Converting Decimals into Fractions
- Comparing Decimals
- Adding and Subtracting Decimals
- Multiplication and Division by Powers of 10
- Multiplying Decimals
- Dividing Decimals
- Infinite Decimals
- Percents
- Portion of the Whole
**Things to Do with Real Numbers**- Addition and Subtraction of Real Numbers
- Properties of Addition
- Subtraction
- Multiplication
**Division**- Long Division Remainder
- Exponents and Powers - Whole Numbers
- Properties of Exponents
- Prime Factorization
- Order of Operations
- Even and Odd Numbers
- Infinity
- Sequences
- is Irrational
- Counting Rational Numbers
- Counting Real Numbers?
- Counting Irrational Numbers
- In the Real World
- Decimals in Use
- How to Solve a Math Problem
- I Like Abstract Things: Summary

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.

Exercise 1

6.8 ÷ (-0.02) =

Exercise 2

Exercise 3

-6.4 ÷ 8 =

Exercise 4