Rational Number
Table of Contents
What is a Rational Number?
A rational number is a number that can be expressed as the quotient or fraction \frac{a}{b} of two integers, where a is the numerator and b is the denominator (b\ne0). In other words, a rational number is any number that can be represented as a fraction of two integers.
Fractional Form
Rational numbers can be written in the for \frac{a}{b}, where a and b are integers and b\ne0.
Repeating or Terminating Decimal
The decimal representation of a rational number either terminates (ends) or repeats (has a repeating pattern) after a certain point.
Integer as a Special Case
Every integer is a rational number because it can be written in the form \frac{a}{1}.
Example of Rational Numbers
Whole Numbers:
- -3: This integer is a rational number because it can be written as \frac{-3}{1}.
Fractions:
- \frac{2}{3}: This is a rational number because it is the quotient of two integers, where the numerator is 2 and the denominator is 3.
Decimals with a Repeating Pattern:
- .333…: This decimal representation is rational because it has a repeating pattern (denoted by the ellipsis).
Decimals with a Terminating Pattern:
- 0.25: This decimal representation is rational because it terminates after a finite number of digits.
Negative Rational Numbers:
- -\frac{4}{5}: This negative fraction is a rational number because it is the quotient of two integers.
Mixed Numbers:
- 2\frac{1}{4}: This mixed number is rational because it can be converted to the fractional form \frac{9}{4}.
Not Rational Numbers
Irrational Numbers:
- \sqrt{2}: The square root of 2 is not a rational number because it cannot be expressed as the quotient of two integers, and its decimal representation is non-repeating and non-terminating.
Non-Repeating, Non-Terminating Decimals:
- \pi: The mathematical constant pi (pi) is not a rational number because its decimal representation goes on forever without repeating.
Variables:
- \frac{x}{y}: This algebraic expression is not a specific rational number unless values are assigned to x and y.
Related Links
Functions
Irrational Number
Real Number
Temperature