The definition of a real number is so broad that it encompasses almost all numbers in the mathematical universe. Whole numbers and integers are a subset of real numbers, as are both rational and irrational numbers. The real number set is denoted by the symbol ℝ.
Whole Numbers and Integers
The numbers that we typically use for counting are known at the natural numbers (1, 2, 3...). When you include zero, you have a group known as whole numbers (0, 1, 2, 3...). Integers are the set of numbers that includes all whole numbers along with the negative versions of the natural numbers. The integer number set is represented by ℤ.
Numbers that we normally think of as fractions make up the set of rational numbers. A fraction is a number represented as a ratio between two integers, a and b, of the form a / b, where b is not equal to zero. A fraction with zero on the right side of its ratio is undefined or indeterminate. A rational number can also be represented in decimal form. The decimal expansion of a rational number will always either terminate or have a pattern of numbers that repeats to the right of the decimal point. All integers are rational numbers since any integer can be represented by the ratio a / 1. The rational number set is represented by ℚ.
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The set of numbers that cannot be represented as a ratio between integers are called irrationals. When represented in decimal form, an irrational number is non-terminating and has a non-repeating pattern of numbers to the right of the decimal point. There is no standard symbol for the set of irrational numbers. The set of rational and irrational numbers is mutually exclusive, which means that all real numbers are either rational or irrational, but not both.
Real Numbers and the Number Line
The real number set represents an ordered set of values that can be represented on a number line that is drawn horizontally, with increasing values to the right and decreasing values to the left. Every real number corresponds to a discrete point on this line, known as its coordinate. The number line extends to infinity in both directions, which means the real number set has an infinite number of members.
There are some mathematical equations for which the solution is not a real number. An example is a formula that includes the square root of a negative number. Since squaring two negative numbers always results in a positive number, the solution seems impossible. A set of numbers known as complex numbers includes imaginary numbers such as the square root of a negative number. The complex number set is separate from the real number set and is represented by the standard symbol ℂ.