Basic Math and Pre-Algebra For Dummies (105 page)

In the positive numbers, the factor is always the
smaller
of the two numbers and the multiple is always the
larger.

For more on factors and multiples, see Chapter
8
.

Math teachers usually request (or force!) their students to use the smallest-possible version of a fraction — that is, to reduce fractions to lowest terms.

To reduce a fraction, divide the
numerator
(top number) and
denominator
(bottom number) by a
common factor,
a number that they're both divisible by. For example, 50 and 100 are both divisible by 10, so

The resulting fraction,
, can still be further reduced, because both 5 and 10 are divisible by 5:

When you can no longer make the numerator and denominator smaller by dividing by a common factor, the result is a fraction that's reduced to lowest terms.

See Chapter
9
for more on reducing fractions.

Adding and subtracting fractions that have the same denominator is pretty simple: Perform the operation (adding or subtracting) on the two numerators and keep the denominators the same.

When two fractions have different denominators, you can add or subtract them without finding a common denominator by using cross-multiplication, as shown here:

For more on adding and subtracting fractions, see Chapter
10
.

To multiply fractions, multiply their two numerators to get the numerator of the answer, and multiply their two denominators to get the denominator. For example:

To divide two fractions, turn the problem into multiplication by taking the
reciprocal
of the second fraction — that is, by flipping it upside-down. For example:

Now multiply the two resulting fractions:

For more on multiplying and dividing fractions, see Chapter
10
.

Everything in algebra is, ultimately, for one purpose: Find
x
(or whatever the variable happens to be). Algebra is really just a bunch of tools to help you do that. In Chapter
21
, I give you these tools. Chapter
22
focuses on the goal of finding
x.
And in Chapter
23
, you use algebra to solve word problems that would be much more difficult without algebra to help.

The main idea of algebra is simply that an equation is like a balance scale: Provided that you do the same thing to both sides, the equation stays balanced. For example, consider the following equation:

• 8
  x
  – 12 = 5
  x
  + 9
  

To find
x,
you can do anything to this equation as long as you do it equally to both sides. For example:

Each of these steps is valid. One, however, is more helpful than the others, as you see in the next section.

For more on algebra, see Chapters
21
through
23
.

The best way to find
x
is to
isolate it
— that is, get
x
on one side of the equation with a number on the other side. To do this while keeping the equation balanced requires great cunning and finesse. Here's an example, using the equation from the preceding section:

As you can see, the final step isolates
x,
giving you the solution:
x
= 7.

For more on algebra, see Chapters
21
through
23
.

Chapter 25

In This Chapter

Identifying counting numbers, integers, rational numbers, and real numbers

Discovering imaginary and complex numbers

Looking at how transfinite numbers represent higher levels of infinity

The more you find out about numbers, the stranger they become. When you're working with just the counting numbers and a few simple operations, numbers seem to develop a landscape all their own. The terrain of this landscape starts out uneventful, but as you introduce other sets, it soon turns surprising, shocking, and even mind blowing. In this chapter, I take you on a mind-expanding tour of ten sets of numbers.

I start with the familiar and comfy counting numbers. I continue with the integers (positive and negative counting numbers and 0), the rational numbers (integers and fractions), and real numbers (all numbers on the number line). I also take you on a few side routes along the way. The tour ends with the bizarre and almost unbelievable transfinite numbers. And in a way, the transfinite numbers bring you back to where you started: the counting numbers.

Each of these sets of numbers serves a different purpose, some familiar (such as accounting and carpentry), some scientific (such as electronics and physics), and a few purely mathematical. Enjoy the ride!

The
counting numbers
— also called the
natural numbers
— are probably the first numbers you ever encountered. They start with 1 and go up from there:

• {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...}
  

 The three dots (or ellipsis) at the end tell you that the sequence of numbers goes on forever — in other words, it's infinite.

The counting numbers are useful for keeping track of tangible objects: stones, chickens, cars, cell phones — anything that you can touch and that you don't plan to cut into pieces.

The set of counting numbers is
closed
under both addition and multiplication. In other words, if you add or multiply any two counting numbers, the result is also a counting number. But the set isn't closed under subtraction or division. For example, if you subtract 2 – 3, you get –1, which is a negative number, not a counting number. And if you divide
, you get
, which is a fraction.

 If you place 0 in the set of counting numbers, you get the set of
whole numbers.

The set of
integers
includes the counting numbers (see the preceding section), the negative counting numbers, and 0:

• {..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...}
  

The dots, or ellipses, at the beginning and the end of the set tell you that the integers are infinite in both the positive and negative directions.

Because the integers include the negative numbers, you can use them to keep track of anything that can potentially involve debt. In today's culture, it's usually money. For example, if you have $100 in your checking account and you write a check for $120, you find that your new balance drops to –$20 (not counting any fees that the bank charges!).

The set of integers is
closed
under addition, subtraction, and multiplication. In other words, if you add, subtract, or multiply any two integers, the result
is also an integer. But the set isn't closed under division. For example, if you divide the integer 2 by the integer 5, you get the fraction
, which isn't an integer.

The
rational numbers
include the integers (see the preceding section) and all the fractions between the integers. Here, I list only the rational numbers from –1 to 1 whose denominators (bottom numbers) are positive numbers less than 5:

The ellipses tell you that between any pair of rational numbers are an infinite number of other rational numbers — a quality called the
infinite density
of rational numbers.

Rational numbers are commonly used for measurement in which precision is important. For example, a ruler wouldn't be much good if it measured length only to the nearest inch. Most rulers measure length to the nearest
of an inch, which is close enough for most purposes. Similarly, measuring cups, scales, precision clocks, and thermometers that allow you to make measurements to a fraction of a unit also use rational numbers. (See Chapter
15
for more on units of measurement.)