What is a complex number?

A complex number has the form z = a + bi, where a and b are real numbers and i is the imaginary unit defined by i squared equals negative one. The value a is the real part and b is the imaginary part. Every real number is a complex number with b = 0.

What is the difference between imaginary and complex numbers?

An imaginary number has the form bi with no real part (like 3i or -7i). A complex number has both a real and imaginary part (like 2 + 3i). Every imaginary number is complex, but not every complex number is imaginary -- for example, 2 + 3i has a nonzero real part.

What are the different forms of a complex number?

Complex numbers can be written in algebraic form (a + bi), trigonometric form (r times cos theta plus i sin theta), or exponential form (r times e to the i theta). Algebraic form is best for addition and subtraction. Trigonometric and exponential forms simplify multiplication, division, and powers.

What is the modulus of a complex number?

The modulus of z = a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as the square root of a squared plus b squared. It generalizes the absolute value of real numbers to two dimensions and is always non-negative.

What is a complex conjugate and why is it useful?

The complex conjugate of z = a + bi is a - bi, obtained by negating the imaginary part. Multiplying a complex number by its conjugate gives a real number equal to the modulus squared. This property is essential for dividing complex numbers and finding multiplicative inverses.

Complex Numbers Terms and Definitions

About This Glossary

This glossary organizes 16 complex number terms into four categories that build from basic definitions through geometric and algebraic structure.

Foundations establishes the core vocabulary with 7 entries: complex number, imaginary unit, imaginary number, pure imaginary number, real part, imaginary part, and algebraic form. These terms define what complex numbers are and how they are written in standard notation.

Representations covers 5 entries on the different ways to express and visualize complex numbers: the complex plane (Argand diagram), modulus (absolute value), argument (angle), trigonometric form, and exponential form. Each representation suits different operations -- algebraic form for addition, trigonometric and exponential forms for multiplication and powers.

Operations & Structure addresses 4 entries on algebraic operations specific to complex numbers: the complex conjugate, additive inverse, multiplicative inverse, and roots of unity. These terms describe how complex numbers interact under arithmetic and how the unit circle connects to polynomial roots.

Each definition includes an intuitive explanation, key properties, notation conventions, and links to detailed lesson pages. Use the search bar or category filters above to navigate.

FoundationsOperations & StructureRepresentations

AAdditive InverseOperations & Structure AAlgebraic FormFoundations AArgumentRepresentations CComplex ConjugateOperations & Structure CComplex NumberFoundations CComplex PlaneRepresentations EExponential FormRepresentations IImaginary NumberFoundations IImaginary PartFoundations IImaginary UnitFoundations MModulusRepresentations MMultiplicative InverseOperations & Structure PPure Imaginary NumberFoundations RReal PartFoundations RRoots of UnityOperations & Structure TTrigonometric FormRepresentations

Foundations(7)

Operations & Structure(4)

Representations(5)

16 of 16 terms

Foundations

Complex Number Imaginary Unit Imaginary Number Pure Imaginary Number Real Part Imaginary Part Algebraic Form

Operations & Structure

Complex Conjugate Additive Inverse Multiplicative Inverse Roots of Unity

Representations

Complex Plane Modulus Argument Trigonometric Form Exponential Form

16 terms

Foundations

(7 items)

Complex Number

A number of the form

z = a + bi

, where

a, b \in \mathbb{R}

and

i

is the imaginary unit

Algebraic Form

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Imaginary Unit

The number

i

defined by

i^2 = -1

Imaginary Numbers

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Imaginary Number

A number of the form

bi

, where

b \in \mathbb{R}

and

b \neq 0

Imaginary Numbers

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Pure Imaginary Number

A complex number

z = a + bi

where

a = 0

and

b \neq 0

Imaginary Numbers

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Real Part

For

z = a + bi

, the real part is

\operatorname{Re}(z) = a

Algebraic Form

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Imaginary Part

For

z = a + bi

, the imaginary part is

\operatorname{Im}(z) = b

Algebraic Form

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Algebraic Form

The representation

z = a + bi

where

a = \operatorname{Re}(z)

and

b = \operatorname{Im}(z)

Algebraic Form

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Representations

(5 items)

Complex Plane

The two-dimensional plane in which each complex number

z = a + bi

corresponds to the point

(a, b)

Geometric Representation

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Modulus

|z| = \sqrt{a^2 + b^2}

for

z = a + bi

Absolute Value

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Argument

\arg(z) = \theta

where

\cos\theta = \frac{a}{|z|}

and

\sin\theta = \frac{b}{|z|}

, for

z = a + bi \neq 0

Trigonometric Form

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Trigonometric Form

z = r(\cos\theta + i\sin\theta)

where

r = |z|

and

\theta = \arg(z)

Trigonometric Form

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Exponential Form

z = re^{i\theta}

where

r = |z|

and

\theta = \arg(z)

Exponential Form

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Operations & Structure

(4 items)

Complex Conjugate

For

z = a + bi

, the conjugate is

\overline{z} = a - bi

Complex Conjugate

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Additive Inverse

For

z = a + bi

, the additive inverse is

-z = -a - bi

Additive Inverse

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Multiplicative Inverse

For

z \neq 0

, the multiplicative inverse is

z^{-1} = \frac{\overline{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}

Multiplicative Inverse

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Roots of Unity

The

n

th roots of unity are

\omega_k = e^{i \frac{2\pi k}{n}}

for

k = 0, 1, \ldots, n-1

Equations & Polynomials

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