Rational and Irrational Numbers

Rational and irrational numbers are two categories of real numbers. A rational number is any value that can be expressed as a fraction of two integers with a nonzero denominator. The word rational refers to a ratio, not whether a value seems reasonable. Integers, fractions, terminating decimals, and repeating decimals are all rational numbers.

Recognize rational numbers

The integer −7 is rational because it can be written as −7/1. The fraction 3/8 is already a ratio of integers. The decimal 0.625 terminates and equals 5/8, so it is rational. The repeating decimal 0.272727… equals 27/99, or 3/11, so it is rational too. A decimal need not stop to be rational; it can repeat a fixed block forever.

A finite decimal always represents a rational number. To see why, write the digits over a power of ten: 0.37 = 37/100 and 2.125 = 2125/1000 = 17/8. Then simplify if useful. Fractions may produce terminating or repeating decimals depending on their denominators after reduction.

A reduced fraction has a terminating decimal precisely when its denominator's prime factors are only 2s and 5s, because powers of ten are built from 2 × 5. For example, 3/40 terminates because 40 = 2³ × 5, while 2/7 repeats because 7 cannot divide any power of ten. This is a useful explanation for why a rational decimal either terminates or repeats: long division has only finitely many possible remainders, so a remainder must eventually be zero or repeat.

Recognize irrational numbers

An irrational number cannot be expressed as a ratio of integers. Its decimal expansion continues forever without a repeating block. The number π is irrational, as is √2. A calculator can display only a finite approximation, such as 1.4142 for √2, but the displayed digits do not terminate the actual decimal; they are rounded or truncated for the screen.

A square root of an integer is rational when the integer is a perfect square: √49 = 7 and √(1/4) = 1/2. The square root of a positive integer that is not a perfect square is irrational: √3 and √10 do not simplify to ratios of integers. Simplify the radical first; for example, √12 = 2√3 remains irrational because √3 is irrational.

A quick decimal test

A decimal that ends is rational. A decimal that eventually repeats a fixed sequence is rational. A decimal that continues without repeating is irrational. For instance, 0.181818… repeats '18,' so it is rational. A long calculator display such as 0.101001000100001… appears to add a growing number of zeros and has no fixed repeating block; that pattern is irrational.

You cannot prove a number irrational just because its decimal looks long. A repeating pattern may begin after many digits. For school-level questions, use known facts, exact fractions, perfect-square tests, or a clearly stated decimal pattern rather than guessing from a short approximation.

Operations can change the category

Adding, subtracting, or multiplying rational numbers produces another rational number. Dividing rational numbers also produces a rational number when the divisor is not zero. For example, 1/3 + 1/4 = 7/12, still rational. Multiplying a rational number by an irrational number usually gives an irrational result when the rational factor is nonzero, but multiplying by zero gives zero, which is rational.

Two irrational numbers can combine to make a rational number. √2 × √2 = 2, and √2 + (−√2) = 0. Therefore, do not assume that an operation involving an irrational number must produce an irrational answer. Simplify the expression and use the definitions.

The reverse caution matters too: adding a rational number to an irrational number stays irrational. If √2 + 3 were rational, subtracting the rational 3 would make √2 rational, contradicting the fact that √2 cannot be expressed as a ratio of integers. For multiplication, a nonzero rational factor preserves irrationality for the same reason: if 5√2 were rational, dividing by 5 would make √2 rational. Zero is the exception because 0 times any real number is 0.

Where the categories sit on the number line

Every rational and irrational number is a real number and can be represented by a point on the number line. Rational numbers are dense: between any two different real numbers, there is a rational number. Irrational numbers also occur throughout the line. The two categories do not form separate intervals; they are interspersed.

Zero is rational because it equals 0/1. Negative fractions are rational, and negative square roots of non-perfect squares are irrational. A square root symbol alone does not tell you the category: evaluate or simplify the radicand to see whether it is a perfect square or a rational square.

The number sets nest: natural counting numbers are integers, integers are rational numbers, and rational numbers are real numbers. Irrational numbers are also real, but they sit outside the rational set. A decimal approximation does not change the exact number's category. For example, π may be approximated by 3.14 for a calculation, but π itself remains irrational; 3.14 is a nearby rational decimal, not the exact value of π.

Common errors

• Thinking a rational number must be a whole number.
• Calling every infinite decimal irrational; repeating decimals are rational.
• Treating a rounded calculator display as the exact number.
• Assuming every square root is irrational, including √64 = 8.
• Assuming sums or products of irrational numbers are always irrational.
• Forgetting that the denominator in a fraction cannot be zero.

Exam takeaway

Ask whether the number can be written as a ratio of integers. Terminating and repeating decimals are rational; nonterminating, nonrepeating decimals are irrational. Simplify roots and expressions before classifying, and remember that both kinds of numbers lie on the real number line.

Check the rule in context

A rational number can be written as a ratio of integers with a nonzero denominator. This includes terminating decimals such as 0.375 and repeating decimals such as 0.2727…, since both represent fractions. An irrational decimal neither terminates nor repeats, as with √2 or π. A square root is rational when the number under the radical is a perfect square, but √12 is irrational and can be simplified to 2√3 without changing that fact. The sum or product of two irrationals is not automatically irrational: √2 × √2 = 2 is rational, while 1 + √2 is irrational. Check the actual expression instead of relying on a blanket rule.