Math Help: How to Find the Real Number System

What Is the Real Number System?

The real number system, which is also called the set of real numbers, includes all of the numbers that are usually labeled on a number line, like zero, 20 and -11. It also includes all of the decimals, like 3.14, and the fractions, like 4/15. It even includes almost all of the square roots, like radical two. Any number that represents some real quantity, no matter how small or large, is a real number.

But if all of the numbers that we use on a daily basis are 'real,' then what do we need this category for? Why don't we just call it 'the set of all numbers?' We need it because there are actually imaginary numbers out there as well as real ones. You don't need to worry about them until you begin to study more advanced math. For now, it's enough to know that sometimes mathematicians need to make up numbers to solve problems, and the numbers they make up are called imaginary numbers.

Subsets of Real Numbers

Since there are so many different kinds of real numbers, mathematicians have divided them up into smaller categories, or subsets. Knowing these subsets helps you compare and contrast the properties of different numbers. All of the numbers in the subsets below are also part of the set of real numbers.

Natural Numbers

Natural numbers are the numbers that you use when you're counting things. They start with one and go up to infinity. Zero isn't included among the natural numbers because we start counting with one, not zero. Negative numbers are also not natural numbers.

Whole Numbers

Whole numbers are all of the natural numbers, plus zero. For instance, one, 72 and 1,000,000,000 are all whole numbers. Negative numbers aren't included among whole numbers.

Integers

Integers include all of the whole numbers, as well as the negatives of all of the natural numbers. Any number that is typically labeled on a number line is an integer. The list of integers is infinite in both the positive and negative directions. Examples of integers include -354, zero and seven.

Rational Numbers

The rational numbers are all of the numbers that can be written as fractions. This subset includes all integers, since any integer can be written as a fraction by putting it over one (e.g., 4/1 = 4 and -14/1 = -14). A number doesn't have to actually be a fraction in order to be a rational number; it just needs to be possible to write it as a fraction.

Keep in mind that a fraction represents division of the top number, or numerator, by the bottom number, or denominator. Sometimes this division results in an integer, but it often results in a decimal. For example, 9/5 = 1.8, because 9 ÷ 5 = 1.8. Since 1.8 can be written as the fraction 9/5, it is a rational number. Even repeating decimals that can be written as fractions are rational numbers. An example is 0.3 repeating, which is rational because it can be written as 1/3.

Many times, radicals can also be written as fractions. Some radicals, like radical nine, are equal to integers. The square root of nine is three, and three can be written as 3/1. This fraction is equal to radical nine, so radical nine can be written as a fraction. This makes it a rational number. However, not all radicals are rational numbers.

Mathematicians have discovered that some radicals, when written as decimals, are infinite. Not only that, but they do not continue in a pattern like 0.3 repeating. This means that they cannot be written as fractions, and so they are not rational numbers. Radical two and radical three are examples of these infinite decimals. The decimals represented by pi and e are also examples of decimals that can't be written as fractions because they never end.

That was a lot of information, so let's recap. Here's a basic summary of the different kinds of rational numbers:

• Integers
• Fractions
• Decimals that can be written as fractions
• Radicals that can be written as fractions

Irrational Numbers

Irrational numbers are the last subset of the real number system. They include all of the numbers that are not rational because they cannot be written as fractions. When written as decimals, irrational numbers continue forever and follow no particular pattern. There are just a handful of numbers in this category. Here are a few examples:

• Pi, or 3.1415926535897932384626433832795…
• Euler's number (e), or 2.7182818284590452353602874713527…
• Radical three, or 1.7320508075688772935274463415059…
• The golden ratio, or 1.61803398874989484820...