Number Systems
Rational vs Irrational Numbers
A rational number can be written as a ratio of integers with a nonzero denominator. An irrational number cannot; its decimal expansion neither terminates nor repeats.
Reviewed
Worked examples
Convert 0.272727... to a fraction
Let x equal the repeating decimal.
100x=27.272727...100x-x=2799x=27x=27/99
Result: 0.272727... = 3/11, so it is rational
Classify sqrt(18)
Simplify the radical.
sqrt(18)=sqrt(9 x 2)sqrt(18)=3sqrt(2)sqrt(2) is irrational
Result: sqrt(18) is irrational
Equivalent definitions
Every integer and terminating decimal is rational because it can be placed over a power of ten. Repeating decimals are also rational; algebra can convert the repeating pattern into an integer ratio.
Irrational numbers are still real numbers and occupy points on the same number line. Their decimal digits continue without a repeating block.
rational: p/q, where p and q are integers and q != 0Roots and irrationality
The square root of a nonnegative perfect square integer is rational. The square root of a positive integer that is not a perfect square is irrational.
Not every expression containing a root is irrational: sqrt(49)=7, and sqrt(8)/sqrt(2)=2 when the real-root rules apply.
Operations do not have one simple outcome
Adding a rational number to an irrational number is irrational, and multiplying a nonzero rational by an irrational is irrational. But two irrational numbers can add or multiply to a rational result.
For example, sqrt(2)+(-sqrt(2))=0 and sqrt(2)sqrt(2)=2.
Where this idea is used
- Choosing exact fraction or radical forms.
- Understanding why some decimal answers must be approximations.
- Classifying solutions to algebraic and geometric problems.
Common mistakes
- Assuming every long decimal is irrational; it may terminate later or repeat.
- Assuming every square root is irrational.
- Treating pi as the fraction 22/7; that fraction is only an approximation.
Questions about rational and irrational numbers
Is zero rational?
Yes. It equals 0/1.
Are pi and e irrational?
Yes. Neither can be expressed as a ratio of integers.
Can a calculator prove a decimal is irrational?
No finite decimal display can prove that. Irrationality requires mathematical reasoning about the number's definition.