Real numbers

Real numbers are one of the broadest categories of numbers. Real numbers are divided into rational numbers and irrational numbers, which include all positive and negative integers, 0, and all the fractional and decimal values in between (fractions, decimals, transcendental numbers, etc.)

Real numbers were created to distinguish the set of real numbers from imaginary numbers. Imaginary numbers are the result of trying to take the square root of a negative number.

The set of real numbers is indicated using this symbol: ℝ. Below are a few examples of real numbers.

The above is just a small sample of the various types of numbers that make up the real numbers.

Rational vs irrational numbers

A rational number is a number that can be expressed as a fraction where the numerator and denominator are integers, the ratio of which results in a terminating decimal, or a non-terminating decimal that repeats. Decimals that repeat are indicated by writing a horizontal bar above the portion of the decimal that repeats. For example, ⅓ repeats indefinitely:

⅓ = 0.33333333... = 0.3

An irrational number is made up of all the real numbers that are not rational numbers: non-terminating decimals that do not repeat. Examples include π, Euler's number e, and the golden ratio.

Integers vs real numbers

Real numbers and integers can be compared using number lines. The number line below represents integers shown using red points to show that only whole values (not fractional or decimal values) are included in the set of integers.

The number line below represents all real numbers. The blue line shown on top of the number line shows that all the values between the integers are included as well, not just their individual points.

These number lines show that all integers are real numbers, but not all real numbers are integers.