A 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.
In Spanish
número complejo
Also called
complex numbers, a + bi, number with a real and imaginary part
A complex number is a point in a plane: real part across, imaginary part up. Its conjugate is its mirror image across the real axis.
Formal definition and notation
The complex numbers are all expressions a + bi with a, b real and i2 = −1. Two complex numbers are equal when their real parts and imaginary parts are both equal. They are added, subtracted and multiplied with the usual commutative, associative and distributive properties, replacing i2 by −1. Every real number is complex with b = 0.
the complex conjugate: if z = a + bi, then z = a − bi
|a + bi|
the modulus of a plus b i
the distance from 0 in the complex plane, √a2 + b2
Key facts
Powers of i repeat every 4:i1 = i, i2 = −1, i3 = −i, i4 = 1. So i10 = i2 = −1.
√−n = i√n for n > 0: √−25 = 5i and √−8 = 2i√2.
A number times its conjugate is real: (a + bi)(a − bi) = a2 + b2.
Every quadratic equation has two solutions among the complex numbers, counting repeats: x2 + 4 = 0 gives x = ± 2i.
Plot a + bi as the point (a, b) in the complex plane: real part across, imaginary part up.
Real numbers (b = 0) and imaginary numbers (a = 0, b ≠ 0) are both special cases of complex numbers.
Operations with complex numbers
Operations with complex numbers
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
a, c
the real parts
b, d
the imaginary parts
−bd
comes from bi · di = bdi2 = −bd
To add or subtract, combine real parts and imaginary parts separately: (2 + 5i) + (4 − 3i) = 6 + 2i. To divide, multiply the top and bottom by the conjugate of the denominator.
An imaginary number has real part 0, like 4i. A complex number can have both parts, like 3 + 4i. Every imaginary number is complex, but 3 + 4i is not purely imaginary.
An irrational number such as √2 is still real. A number like √−2 = i√2 is not real at all; it is imaginary.
Quick check: 3 questions
Question 1 of 3
What is i3?
Show answer and explanation
Answer: C.i3 = i2 · i = −1 · i = −i. −1 is i2, 1 is i4, and i is i1.
Question 2 of 3
Simplify (4 − i)(4 + i).
Show answer and explanation
Answer: B. Conjugates multiply to a2 + b2: 16 − i2 = 16 + 1 = 17. 15 treats i2 as 1, and 16 − i2 is correct but not simplified.
Question 3 of 3
Which is √−49 written as a complex number?
Show answer and explanation
Answer: B.√−49 = √49 · √−1 = 7i. −7 squares to 49, not −49; 49i forgets the square root; −7i2 equals 7.
Complex Number: frequently asked questions
Is every real number a complex number?
Yes. Any real number a can be written a + 0i, so it is a complex number with imaginary part 0. For example, 5 is 5 + 0i and −12 is −12 + 0i. The complex numbers contain the real numbers the way the real numbers contain the integers.
What is the difference between a complex number and an imaginary number?
A complex number has the form a + bi, with any real a and b. An imaginary number (or pure imaginary number) is the special case with a = 0 and b ≠ 0, such as 6i. So 6i is both imaginary and complex, while 2 + 6i is complex but not purely imaginary.
What are complex numbers used for?
In algebra they give every quadratic equation solutions, even when the discriminant is negative. Outside school, engineers use them to describe alternating current, and they appear in signal processing, physics and computer graphics, because multiplying by a complex number rotates and scales points in the plane.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSN-CN.A.1, CCSS HSN-CN.A.2
Cite this entry
Effortless Math Team. “Complex Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/complex-number/
Flashcard
TermComplex Numbera + bi
DefinitionA number with a real part and an imaginary part, where i2 = −1. Example: 3 + 2i; also √−9 = 3i.
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 1Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7NumberA 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 2Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Imaginary 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, PrecalculusSimplifySimplify 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 6Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2DiscriminantThe discriminant of a quadratic equation ax2 + bx + c = 0 is the number b2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.Algebra · Algebra 1, Algebra 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 3Number 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 6Irrational 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 1Square 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 1VectorA vector is a quantity that has both a size, called its magnitude, and a direction. It is drawn as an arrow from an initial point to a terminal point and written in components, such as v = ⟨ 4, 3 ⟩: 4 units right and 3 units up. Velocity and force are vectors; speed and mass are not.Trigonometry · PrecalculusRational 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 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 8Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7