Every number on the familiar number line is a real number. Real numbers contain two broad groups: rational and irrational numbers. The main distinction is whether a number can be expressed as an ordinary fraction.
Rational numbers
A number is rational if it can be written in the form a/b, where a and b are integers and b is not zero.
- Ordinary fractions: 3/5, −7/4, 11/2.
- Integers: 5 = 5/1, −3 = −3/1, 0 = 0/1.
- Terminating decimals: 0.25 = 1/4.
- Non-terminating recurring decimals: 0.333… = 1/3.
In other words, the decimal representation of a rational number either terminates or contains a group of digits that repeats indefinitely.
Irrational numbers
An irrational number cannot be expressed as a fraction a/b with integers a and b. Its decimal representation is both non-terminating and non-recurring: it does not contain a finite block of digits that repeats indefinitely.
Familiar examples include:
- √2, √3, and √5;
- the number π;
- the number e;
- the golden ratio, (1 + √5) / 2.
Not every square root is irrational. For example, √9 = 3, which is an integer and therefore rational. Always calculate or simplify the expression instead of classifying a number from the radical sign alone.
How to identify the type of a number
- An integer is always rational.
- An ordinary fraction with an integer numerator and denominator is rational as long as the denominator is not zero.
- A terminating or recurring decimal is rational.
- A non-terminating, non-recurring decimal is irrational.
- Simplify a square root first: √16 is rational, while √10 is irrational.
Real numbers
Rational and irrational numbers together form the set of real numbers. Every point on the number line corresponds to one real number, and every real number can be represented by a point on that line.
Why the distinction matters
- It helps define the permitted domain of an equation correctly.
- Irrational numbers appear throughout geometry, including in lengths and calculations involving circles.
- The type of a number affects how a result is written exactly or rounded.
- Classifying numbers connects arithmetic and algebra with later mathematical analysis.
Quick check
Classify the following numbers: −8, 0.125, 0.272727…, √7, and √49. The first three are rational; √7 is irrational; and √49 = 7, so it is rational as well.
Conclusion
A rational number can be written as a ratio of two integers; an irrational number cannot. Together, these two groups fill the number line and form the set of real numbers.