Irrational numbers

Definition

Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. Their decimal expansions are non-repeating and non-terminating.

5 Must Know Facts For Your Next Test

1. The square root of any prime number is an irrational number.
2. Common examples include $\pi$ and $e$.
3. The sum or product of a rational number and an irrational number is always irrational.
4. The set of irrational numbers is uncountable, meaning there are infinitely many of them.
5. Irrational numbers can be approximated by rational numbers but never exactly represented.

Review Questions

• What is the primary characteristic that distinguishes an irrational number from a rational number?
• Give two examples of commonly known irrational numbers.
• How does the decimal expansion of an irrational number differ from that of a rational number?

Related terms

Rational Numbers:

Numbers that can be expressed as the ratio of two integers, where the denominator is not zero.

$\pi$: An irrational number representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.

$e$: An irrational number known as Euler's number, approximately equal to 2.71828, and base of natural logarithms.