Ten Important Number Sets to Know - The Part of Tens - Basic Math & Pre-Algebra For Dummies

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

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

arrow Discovering imaginary and complex numbers

arrow 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, ...}

remember_4c.eps 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 9781118791981-eq25001_fmt, you get 9781118791981-eq25002_fmt, which is a fraction.

technicalstuff_4c.eps 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 9781118791981-eq25003_fmt, 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 9781118791981-eq25005_fmt 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 9781118791981-eq25006_fmt, which represents the diagonal distance across a square with a side of 1 unit. In fact, all square roots of nonsquare numbers (such as 9781118791981-eq25007_fmt, 9781118791981-eq25008_fmt, 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 9781118791981-eq25011_fmt. Thus, 9781118791981-eq25012_fmt 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 9781118791981-eq25013_fmt .

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 9781118791981-eq25014_fmt is that it isn't on the real number line (because 9781118791981-eq25015_fmt 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.

technicalstuff_4c.eps If 9781118791981-eq25016_fmt 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 9781118791981-eq25017_fmt 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 – 2i –100 + 10i

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 9781118791981-eq25018_fmt for short. Because the set of counting or natural numbers (represented as 9781118791981-eq25019_fmt) is completely contained within the set of integers, 9781118791981-eq25020_fmt is a subset, or part, of 9781118791981-eq25021_fmt.

The set of rational numbers is called 9781118791981-eq25022_fmt. Because the set of integers is completely contained within the set of rational numbers, 9781118791981-eq25023_fmt and 9781118791981-eq25024_fmt are both subsets of 9781118791981-eq25025_fmt.

9781118791981-eq25026_fmt stands for the set of real numbers. Because the set of rational numbers is completely contained within the set of real numbers, 9781118791981-eq25027_fmt , 9781118791981-eq25028_fmt, and 9781118791981-eq25029_fmt are all subsets of 9781118791981-eq25030_fmt.

The set of complex numbers is called 9781118791981-eq25031_fmt. Because the set of real numbers is completely contained within the set of complex numbers, 9781118791981-eq25032_fmt,9781118791981-eq25033_fmt,9781118791981-eq25034_fmt, and 9781118791981-eq25035_fmt are all subsets of 9781118791981-eq25036_fmt.

The symbol 9781118791981-eq25037_fmt 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 0i (which equals 0):

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

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 ℵ0 (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 ℵ0mean the same thing.

The next transfinite number is ℵ1 (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 ℵ0 elements. And the sets of irrational, transcendental, imaginary, and complex numbers all have ℵ1 elements.

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

· {ℵ0, ℵ1, ℵ2, ℵ3, ...}

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 ℵ0 elements.