ACT® · Math: Number and Quantity
Rational and Irrational Numbers
A number is rational if it can be written as a ratio of two integers, even if its decimal goes on forever, as long as that decimal eventually repeats. Irrational numbers never repeat and never settle, because no such ratio exists for them.
1/3 equals 0.333... forever, but it is still rational because it comes straight from dividing two integers, 1 and 3. That is the whole test: can you write the number as integer over integer (with a nonzero denominator)? If yes, rational, no matter how messy the decimal looks. A decimal that repeats in a fixed block, like 0.333... or 22/7 = 3.142857142857... repeating '142857', is always rational, because any repeating decimal can be converted back into a fraction of integers. A decimal that never falls into a repeating block, like π = 3.14159265358979..., is irrational, and no fraction of integers ever equals it exactly.
Square roots give the cleanest examples. √49 = 7, a whole number, so it's rational. √20 is irrational, because 20 isn't a perfect square: there's no integer that squares to it, so its decimal runs forever with no repeating pattern. The rule is simple: √n is rational only when n is a perfect square (1, 4, 9, 16, 25, ...); otherwise it's irrational, full stop.
Mixing rational and irrational numbers under addition or subtraction almost always gives an irrational result, because the tidy fraction can't cancel out the endless non-repeating decimal. This matters on expressions built from √b terms: when you multiply conjugates like (x + √b)(x - √b), the cross terms cancel and √b times itself just gives the integer b, wiping out the irrational part entirely. But in expressions where those cross terms don't cancel, like (a + √c)2 - (a - √c)2, the irrational part survives and shows up in the final answer. Spotting which setup you're in is the real skill being tested.
Key terms
- Rational number
- Any number that can be written as a ratio of two integers with a nonzero denominator.
- Irrational number
- A number whose decimal never repeats and can never be written as a ratio of two integers.
- Perfect square
- An integer that is the square of another integer, like 49 = 72.
- Repeating decimal
- A decimal that eventually settles into a fixed block of digits that repeats forever, signaling the number is rational.
- Conjugate pair
- Two expressions like x + √b and x - √b whose product cancels the irrational part.
Watch it explained
- How it worksHow does writing a number as a ratio of two integers settle whether it is rational?
- How it worksHow does the square root of an integer turn out rational only when that integer is a perfect square?
- Worked exampleFind which one is irrational among 0.75, the square root of 49, the square root of 20, and 22/7.
- Common mistakeWhy isn't π equal to 22/7 when every calculator agrees with that fraction for two decimal places?
- Going furtherWhat happens to a rational number when you add an irrational one to it?
Check yourself
End of topic.