A rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.
Whole numbers are integers, and integers are rational numbers, so each box sits inside the next. Irrational numbers such as √2 and π are the real numbers outside the rational box.
Formal definition and notation
A numberr is rational if r = ab for some integersa and b with b ≠ 0. Every integer is rational, because n = n1, and so is every fraction and its opposite. A number is rational exactly when its decimal form terminates or eventually repeats; numbers whose decimals never end and never repeat, such as √2 and π, are irrational. Adding, subtracting, multiplying or dividing two rational numbers (except dividing by 0) always gives a rational number.
Symbol
Read it as
Meaning
ab
a over b
the form every rational number can take, with integers a, b and b ≠ 0
0.3
zero point three repeating
the bar shows digits that repeat forever: 0.333… = 13
Q
Q
the usual name for the set of all rational numbers (from quotient)
Key facts
Every integer and every whole number is rational: 7 = 71 and 0 = 01.
A terminating decimal is rational: 2.35 = 235100 = 4720.
A repeating decimal is rational: 0.6 = 23 and 0.27 = 2799 = 311.
The square root of a perfect square is rational (√49 = 7); the square root of any other whole number, such as √50, is irrational.
Between any two rational numbers there is another one, such as their average: between 13 and 12 lies 512.
Test for a rational number
Test for a rational number
r = ab, with a and b integers and b ≠ 0
a
any integer: positive, negative or 0
b
any integer except 0
If you can find such a fraction, the number is rational. Terminating decimal: 0.125 = 1251000 = 18. Repeating decimal: 0.4 = 49.
Try it: place a rational number
Interactive
Drag the point or use the arrow keys. It moves in steps of one fourth, and the readout names the rational number as a fraction, a decimal and a mixed number; try the negative side too.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Rational Number examples, step by step
Example 1: A terminating decimal
Warm-up
Show that 2.35 is a rational number.
2.35 has two decimal places, so write it over 100: 2.35 = 235100.
An irrational number cannot be written as a fraction of integers; its decimal never ends and never repeats, like √2 = 1.41421…. A rational number’s decimal stops or repeats.
Real numbers are all the numbers on the number line: the rational numbers together with the irrational numbers.
Quick check: 3 questions
Question 1 of 3
Which number is not rational?
Show answer and explanation
Answer: C. 50 is not a perfect square, so √50 ≈ 7.071… never ends or repeats: irrational. −73 is a fraction of integers, √49 = 7, and 0.8 = 89.
Question 2 of 3
Which fraction equals 0.3?
Show answer and explanation
Answer: B. If x = 0.333…, then 10x − x = 3, so x = 39 = 13. Choices A and C are 0.3 and 0.33, which stop; D is 0.03.
Question 3 of 3
Which statement about −12 is true?
Show answer and explanation
Answer: C.−12 is an integer, and every integer is rational because −12 = −121. Being negative does not stop a number from being rational.
Rational Number: frequently asked questions
Is 0 a rational number?
Yes. 0 can be written as 01, 05 or 0 over any nonzero integer, so it fits the definition. What is not allowed is 0 in the denominator: 50 is undefined, not a rational number. 0 is also a whole number and an integer.
Are all fractions rational numbers?
Every fraction made of two integers, with a nonzero bottom, is rational: 38, −114, 93. But an expression that only looks like a fraction need not be: √22 and π4 have irrational parts, so they are irrational. Check that the top and bottom are integers.
How can I tell if a decimal is rational?
Look at how it ends. If it stops, like 0.875, it is rational (78). If a block of digits repeats forever, like 0.272727…, it is rational (311). If it goes on forever with no repeating block, like 0.1010010001… or π, it is irrational. A calculator display cannot always show which case you have.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.NS.A.2.d, CCSS 8.NS.A.1
Cite this entry
Effortless Math Team. “Rational Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/rational-number/
Flashcard
TermRational Numberab, integers, b ≠ 0
DefinitionA number that can be written as a fraction of two integers. −5, 34, 0.75 and 0.3 are rational; √2 is not.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7, Grade 8AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7Imaginary NumberAn imaginary number is a real number multiplied by i, the imaginary unit, where i = √−1 and i2 = −1. Examples are 3i, −12i and √−25 = 5i. Imaginary numbers give square roots of negative numbers a value, so equations like x2 = −9 have solutions: x = ± 3i.Number System · Algebra 2, PrecalculusComplex NumberA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, with i2 = −1. The real part is a and the imaginary part is b. Examples: 3 + 2i and −4i. Complex numbers let you take square roots of negatives, such as √−9 = 3i.Number System · Algebra 2, Precalculus