A 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.
Rational and irrational numbers never overlap, and together they make up every real number.
Formal definition and notation
The set of real numbers, written ℝ or R, is the union of the rational numbers and the irrational numbers. Every real number has a decimal expansion, and every point on a number line corresponds to exactly one real number. The real numbers are ordered (for any two, one is less than, equal to, or greater than the other) and are closed under addition, subtraction, multiplication and division by a nonzero number. Numbers of the form bi with b ≠ 0, where i2 = −1, are imaginary, not real.
Symbol
Read it as
Meaning
R
R, or the reals
the set of all real numbers; often printed as a double-struck R
x ∈ R
x is an element of R
x is a real number
Key facts
Each set fits inside the next: natural ⊂ whole ⊂integers⊂ rational ⊂ real. Irrational numbers are real but not rational.
Every real number is either rational or irrational, never both.
Whole numbers are 0, 1, 2, 3, …; they are a small part of the real numbers.
Quick check: 3 questions
Question 1 of 3
Which of these is NOT a real number?
Show answer and explanation
Answer: B.√−9 asks for a number whose square is −9; no real number works, so it is imaginary (3i). −√9 = −3 is real, and 0 and π are real.
Question 2 of 3
Between which two consecutive integers is √50?
Show answer and explanation
Answer: D.72 = 49 and 82 = 64, so √50 is between 7 and 8 (about 7.07). 24 and 25 halve 50, and 49 and 51 compare squares instead of roots.
Question 3 of 3
How is the repeating decimal 0.121212… classified?
Show answer and explanation
Answer: A. A repeating decimal can be written as a fraction, here 1299 = 433, so it is rational, and every rational number is real. Only non-repeating, non-ending decimals are irrational.
Real Number: frequently asked questions
Is zero a real number?
Yes. Zero is a real number: it is the point in the middle of the number line. It is also a whole number, an integer and a rational number, since 0 = 01. Zero is neither positive nor negative, and it separates the positive real numbers from the negative ones.
What numbers are not real numbers?
Numbers that are not on the number line. The main examples in school math are square roots of negative numbers, such as √−1 = i and √−25 = 5i, and complex numbers like 2 + 3i. Infinity is not a real number either; it describes growing without bound.
Are all irrational numbers real numbers?
Yes. Every irrational number, such as √2, π or e, sits at an exact point on the number line, so it is real. Irrational only means it cannot be written as a fraction of integers. The real numbers are exactly the rational numbers plus the irrational numbers.
Rational 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 1Irrational 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 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 6Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5NumberA 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 8Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · 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, PrecalculusWhole 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 8Counting NumbersCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Number System · 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 1Complex 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