Basic Math and Pre-Algebra For Dummies (106 page)

The set of rational numbers is closed under the Big Four operations. In other words, if you add, subtract, multiply, or divide any two rational numbers, the result is always another rational number.

In a sense, the irrational numbers are a sort of catchall; every number on the number line that isn't rational is irrational.

By definition, no
irrational number
can be represented as a fraction, nor can an irrational number be represented as either a terminating decimal or a repeating decimal (see Chapter
11
for more about these types of decimals). Instead, an irrational number can be approximated only as a
nonterminating, nonrepeating decimal:
The string of numbers after the decimal point goes on forever without creating a pattern.

The most famous example of an irrational number is π, which represents the circumference of a circle with a diameter of 1 unit. Another common irrational number is
, which represents the diagonal distance across a square with a side of 1 unit. In fact, all square roots of nonsquare numbers (such as
,
, and so forth) are irrational numbers.

Irrational numbers fill out the spaces in the real number line. (The
real number line
is just the number line you're used to, but it's continuous; it has no gaps, so every point is paired with a number.) These numbers are used in many cases where you need not just a high level of precision, as with the rational numbers, but the
exact
value of a number that you can't represent as a fraction.

Irrational numbers come in two varieties:
algebraic numbers
and
transcendental numbers.
I discuss both types of numbers in the sections that follow.

To understand
algebraic numbers,
you need a little information about polynomial equations. A
polynomial equation
is an algebraic equation that meets the following conditions:

• Its operations are limited to addition, subtraction, and multiplication. In other words, you don't have to divide by a variable.
  
• Its variables are raised only to positive, whole-number exponents.
  

You can find out more about polynomials in
Algebra For Dummies,
by Mary Jane Sterling (Wiley). Here are some polynomial equations:

Every algebraic number shows up as the solution of at least one polynomial equation. For example, suppose you have the following equation:

You can solve this equation as
. Thus,
is an algebraic number whose approximate value is 1.4142135623... (see Chapter
4
for more information on square roots).

A
transcendental number,
in contrast to an algebraic number (see the preceding section), is
never
the solution of a polynomial equation. Like the irrational numbers, transcendental numbers are a sort of catchall: Every number on the number line that isn't algebraic is transcendental.

The best-known transcendental number is π, whose approximate value is 3.1415926535…. Its uses begin in geometry but extend to virtually all areas of mathematics. (See Chapter
16
for more on π.)

Other important transcendental numbers come about when you study
trigonometry,
the math of right triangles. The values of trigonometric functions — such as sines, cosines, and tangents — are often transcendental numbers.

Another important transcendental number is
e,
whose approximate value is 2.718281828459…. The number
e
is the base of the natural logarithm, which you probably won't use until you get to pre-calculus or calculus. People use
e
to do problems on compound interest, population growth, radioactive decay, and the like.

The set of
real numbers
is the set of all rational and irrational numbers (see the earlier sections). The real numbers comprise every point on the number line.

Like the rational numbers (see “Knowing the Rationale behind Rational Numbers,” earlier in this chapter), the set of real numbers is closed under the Big Four operations. In other words, if you add, subtract, multiply, or divide any two real numbers, the result is always another real number.

An
imaginary number
is any real number multiplied by
.

To understand what's so strange about imaginary numbers, it helps to know a bit about square roots. The
square root
of a number is any value that, when
multiplied by itself, gives you that number. For example, the square root of 9 is 3 because 3 × 3 = 9. And the square root of 9 is also –3 because –3 × –3 = 9. (See Chapter
4
for more on square roots and multiplying negative numbers.)

The problem with finding
is that it isn't on the real number line (because
isn't in the set of real numbers). If it were on the real number line, it would be a positive number, a negative number, or 0. But when you multiply any positive number by itself, you get a positive number. And when you multiply any negative number by itself, you also get a positive number. Finally, when you multiply 0 by itself, you get 0.