An 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.
In Spanish
número irracional
Also called
non-repeating decimal, irrational, irrationals
Grade level
6-8
Grade 8Algebra 1CCSS 8.NS.A.1CCSS 8.NS.A.2SATACT
Irrational Number in a picture
The diagonal of a 1-by-1 square is √2, a real length whose decimal never ends or repeats. Irrational numbers still have exact places on the number line, between the rational numbers.
Formal definition and notation
A real numberx is irrational if there are no integers p and q, with q ≠ 0, such that x = pq. Equivalently, its decimal expansion neither terminates nor eventually repeats. Every real number is either a rational number or irrational, never both. Examples include √n for every whole numbern that is not a perfect square, π, and e. Irrational numbers can be located on the number line and approximated as closely as you like by rational numbers, such as 1.4 < √2 < 1.5.
Symbol
Read it as
Meaning
√2
the square root of two
the positive number whose square is 2, about 1.414
A square root can be rational or irrational. √36 = 6 is rational; √35 is irrational. The root sign alone does not decide it.
Irrational Numbervs.Repeating decimal
A repeating decimal such as 0.45 goes on forever but has a pattern, so it is rational (511). An irrational decimal goes on forever with no repeating block.
Quick check: 3 questions
Question 1 of 3
Which number is irrational?
Show answer and explanation
Answer: C. 7 is not a perfect square, so √7 is irrational. √25 = 5, 0.444… = 49 repeats, and 227 is a fraction of integers (it only approximates π).
Question 2 of 3
Between which two whole numbers is √30?
Show answer and explanation
Answer: B.52 = 25 and 62 = 36, and 30 is between them, so √30 is between 5 and 6. Choice C halves 30 instead of taking the root, and choice D brackets 30 itself.
Question 3 of 3
Which number is rational?
Show answer and explanation
Answer: D.√36 = 6, a whole number. π and √3 are irrational, and adding the rational number 1 to the irrational √2 still gives an irrational number.
Irrational Number: frequently asked questions
Is pi an irrational number?
Yes. π, the ratio of a circle’s circumference to its diameter, is irrational: its decimal 3.14159265… never ends and never repeats. The values 3.14 and 227 that you use in class are rational approximations. That is why answers are often left in terms of π, as in 10π centimeters, to keep them exact.
How can you tell if a number is irrational?
Check its form. A fraction of integers, a terminating decimal or a repeating decimal is rational. The square root of a whole number that is not a perfect square is irrational, and so are π and numbers like 3 + √5. Seeing only the first few digits on a calculator is not enough to decide, because a long repeating block can look random.
Are irrational numbers real numbers?
Yes. The real numbers are all the numbers on the number line, and they split into two groups: rational and irrational. √2 has an exact position between 1.41 and 1.42. Numbers like √−4 are not real at all; they are imaginary, which is a different idea from irrational.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.NS.A.1, CCSS 8.NS.A.2
Cite this entry
Effortless Math Team. “Irrational Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/irrational-number/
Flashcard
TermIrrational Number√2, π, √50
DefinitionA number that cannot be written as a fraction of two integers; its decimal never ends or repeats. √2 = 1.41421…
Real 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 1Rational NumberA 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.Number System · Grade 7, Grade 8, Algebra 1Whole 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 5Perfect 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 8Number 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 6NumberA 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 2SquareA 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 3RatioA 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 8CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Square 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 1IntegerAn 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 8DecimalA 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 6Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1PiPi (π) is the number you get when you divide any circle’s circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 227.Geometry · Grade 7, Grade 8, GeometryImaginary 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