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
Classify by representation, not appearance
A decimal display can be misleading because a calculator always shows a finite approximation. The defining question is whether the exact value equals a ratio of integers.
| Property | Rational | Irrational |
|---|---|---|
| Integer ratio | Can be written p/q with q not equal to 0 | Cannot be written as any integer ratio |
| Decimal form | Terminates or eventually repeats | Never terminates and never repeats |
| Examples | -7, 3/8, 0.125, 0.272727... | sqrt(2), pi, e |
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 != 0Worked example 1
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
Classification checks
Keep the value exact while deciding; a rounded decimal is not evidence of rationality.
1. Is sqrt(81) rational?
Hint: Simplify the square root before classifying.
Answer: Yes. sqrt(81) = 9.
2. Is 0.101001000100001... rational?
Hint: Look for an eventually repeating block.
Answer: No. The gaps between 1s keep changing, so the decimal does not repeat.
3. Is pi/2 rational?
Hint: A nonzero rational multiple of pi remains irrational.
Answer: No, pi/2 is irrational.
Roots 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.
Questions about rational and irrational numbers
Is zero rational?
Yes. It equals 0/1.