What Is a Real Number?

What is a number? Well that depends. There are a variety of different kinds of numbers, each with their own particular properties. One sort of number, upon which statistics, probability, and much of mathematics is based upon, is called a real number.

To learn what a real number is, we will first take a brief tour of other kinds of numbers.

Types of Numbers

We first learn about numbers in order to count.

We began with matching the numbers 1, 2, and 3 with our fingers.  Then we and kept going as high as we could, which probably wasn't that high. These counting numbers or natural numbers were the only numbers that we knew about.

Later, when dealing with subtraction, negative whole numbers were introduced. The set of positive and negative whole numbers is called the set of integers. Shortly after this, rational numbers, also called fractions were considered. Since every integer can be written as a fraction with 1 in the denominator, we say that the integers form a subset of the rational numbers.

The ancient Greeks realized that not all numbers can be formed as a fraction. For example, the square root of 2 cannot be expressed as a fraction. These kinds of numbers are called irrational numbers. Irrational numbers abound, and somewhat surprisingly in a certain sense there are more irrational numbers than rational numbers.

Other irrational numbers include pi and e.

Decimal Expansions

Every real number can be written as a decimal. Different kinds of real numbers have different kinds of decimal expansions. The decimal expansion of a rational number is terminating, such as 2, 3.25, or 1.2342, or repeating, such as .33333.

. . Or .123123123. . . In contrast to this, the decimal expansion of an irrational number is nonterminating and nonrepeating. We can see this in the decimal expansion of pi. There is a never ending string of digits for pi, and what's more, there is no string of digits that indefinitely repeats itself.

Visualization of Real Numbers

The real numbers can be visualized by associating each one of them to one of the infinite number of points along a straight line. The real numbers have an order, meaning that for any two distinct real numbers we can say that one is greater than the other. By convention, moving to the left along on the real number line corresponds to lesser and lesser numbers. Moving to the right along the real number line corresponds to greater and greater numbers.

Basic Properties of the Real Numbers

The real numbers behave like other numbers that we are used to dealing with. We can add, subtract, multiply and divide them (as long as we don't divide by zero). The order of addition and multiplication is unimportant, as there is a commutative property. A distributive property tells us how multiplication and addition interact with one another.

As mentioned before, the real numbers possess an order.

Given any two real numbers x and y, we know that one and only one of the following is true:

x = y, x < y or x > y.

Another Property - Completeness

The property that sets the real numbers apart from other sets of numbers, like the rationals, is a property known as completeness. Completeness is a bit technical to explain, but the intuitive notion is that the set of rational numbers has gaps in it. The set of real numbers does not have any gaps, because it is complete.

As an illustration, we will look at the sequence of rational numbers 3, 3.1, 3.14, 3.141, 3.1415, . . . Each term of this sequence is an approximation to pi, obtained by truncating the decimal expansion for pi. The terms of this sequence get closer and closer to pi. However, as we have mentioned, pi is not a rational number. We need to use irrational numbers to plug in the holes of the number line that occur by only considering the rational numbers.

How Many Real Numbers?

It should be no surprise that there are an infinite number of real numbers. This can be seen fairly easily when we consider that whole numbers form a subset of the real numbers. We could also see this by realizing that the number line has an infinite number of points.

What is surprising is that the infinity used to count the real numbers is of a different kind than the infinity used to count the whole numbers. Whole numbers, integers and rationals are countably infinite. The set of real numbers is uncountably infinite.

Why Call Them Real?

Real numbers get their name to set them apart from an even further generalization to the concept of number. The imaginary number i is defined to be the square root of negative one. Any real number multiplied by i is also known as an imaginary number. Imaginary numbers definitely stretch our conception of number, as they are not at all what we thought about when we first learned to count.