## Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

### Part VI. The Part of Tens

### Chapter 25. Ten Important Number Sets to Know

*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!

*Counting on Counting (or Natural) Numbers*

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.*

*Identifying Integers*

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.

*Knowing the Rationale behind Rational Numbers*

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.)

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.

*Making Sense of Irrational Numbers*

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.

*Absorbing Algebraic Numbers*

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).

*Moving through Transcendental Numbers*

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.

*Getting Grounded in Real Numbers*

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.

*Trying to Imagine Imaginary Numbers*

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.

If isn't on the real number line, where is it? That's a good question. For thousands of years, mathematicians believed that the square root of a negative number was simply meaningless. They banished it to the mathematical nonplace called *undefined,* which is the same place they kept fractions with a denominator of 0. In the 19th century, however, mathematicians began to find these numbers useful and found a way to incorporate them into the rest of math.

Mathematicians designated with the symbol *i.* Because it didn't fit onto the real number line, *i* got its own number line, which looks a lot like the real number line. Figure __25-1__ shows some numbers that form the imaginary number line.

*Illustration by Wiley, Composition Services Graphics*

**Figure 25-1:** Numbers on the imaginary number line.

Even though these numbers are called imaginary, mathematicians today consider them no less real than the real numbers. And the scientific application of imaginary numbers to electronics and physics has verified that these numbers are more than just figments of someone's imagination.

*Grasping the Complexity of Complex Numbers*

A *complex number* is any real number (see “Getting Grounded in Real Numbers,” earlier in this chapter) plus or minus an imaginary number (see the preceding section). Consider some examples:

· 1 + *i* 5 – 2*i* –100 + 10*i*

**Getting inside subsets**

Many sets of numbers actually fit inside other sets. Mathematicians call these nesting sets *subsets*. For instance, the set of integers is called for short. Because the set of counting or natural numbers (represented as ) is completely contained within the set of integers, is a subset, or part, of *.*

The set of rational numbers is called *.* Because the set of integers is completely contained within the set of rational numbers, and are both subsets of .

stands for the set of real numbers. Because the set of rational numbers is completely contained within the set of real numbers, , *,* and are all subsets of .

The set of complex numbers is called *.* Because the set of real numbers is completely contained within the set of complex numbers, *,**,**,* and are all subsets of *.*

The symbol means “is a subset of” (see Chapter __20__ for details on set notation). So here's how the sets fit inside each other:

You can turn any real number into a complex number by just adding 0*i* (which equals 0):

· 3 = 3 + 0*i* –12 = –12 + 0*i* 3.14 = 3.14 + 0*i*

These examples show you that the real numbers are just a part of the larger set of complex numbers.

Like the rational numbers and real numbers (check out the sections earlier in this chapter), the set of complex numbers is closed under the Big Four operations. In other words, if you add, subtract, multiply, or divide any two complex numbers, the result is always another complex number.

*Going beyond the Infinite with Transfinite Numbers*

The *transfinite numbers* are a set of numbers representing different levels of infinity. Consider this for a moment: The counting numbers (1, 2, 3, ...) go on forever, so they're infinite. But there are *more* real numbers than counting numbers.

In fact, the real numbers are *infinitely more infinite* than the counting numbers. Mathematician Georg Cantor proved this fact. He also proved that, for every level of infinity, you can find another level that's even higher. He called these ever-increasing levels of infinity *transfinite,* because they transcend, or go beyond, what you think of as infinite.

The lowest transfinite number is ℵ_{0} (aleph null), which equals the number of elements in the set of counting numbers ({1, 2, 3, 4, 5, ...}). Because the counting numbers are infinite, the familiar symbol for infinity (∞) and ℵ_{0}mean the same thing.

The next transfinite number is ℵ_{1} (aleph one), which equals the number of elements in the set of real numbers. This is a higher order of infinity than ∞.

The sets of integers, rational, and algebraic numbers all have ℵ_{0} elements. And the sets of irrational, transcendental, imaginary, and complex numbers all have ℵ_{1} elements.

Higher levels of infinity exist, too. Here's the set of transfinite numbers:

· {ℵ_{0}, ℵ_{1}, ℵ_{2}, ℵ_{3}, ...}

The ellipsis tells you that the sequence of transfinite numbers goes on forever — in other words, that it's infinite. As you can see, on the surface, the transfinite numbers look similar to the counting numbers (in the first section of this chapter). That is, the set of transfinite numbers has ℵ_{0} elements.