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SECTIONComplex Numbers18 subsectionsBROWSE ALL ↓INTERACTIVE1 TOOLComplex NumbersVisual Tools10 toolsJump to ↓fREFERENCE69 ITEMSfComplex NumbersFormulas andExamples53 itemsJump to ↓AaComplex NumbersTerms andDefinitions16 itemsJump to ↓§CORE TOPICS15 SUBSECTIONSModulus (Absolute Value) ofComplex NumbersJump to ↓Additive Inverse of ComplexNumbersJump to ↓Algebraic Form of ComplexNumbersJump to ↓Complex Numbers BasicsJump to ↓Complex Numbers Cheat SheetsJump to ↓Complex ConjugateJump to ↓De Moivre TheoremJump to ↓Complex Polynomial EquationsJump to ↓Exponential Form of ComplexNumbersJump to ↓Geometric Representation ofComplex NumbersJump to ↓Imaginary NumbersJump to ↓Multiplicative Inverse ofComplex NumbersJump to ↓Operations on ComplexNumbersJump to ↓Properties of ComplexNumbersJump to ↓Trigonometric Form ofComplex NumbersJump to ↓

Complex Numbers Visual Tools

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Visualize complex numbers on an interactive Argand plane. Drag points or enter coordinates to see real parts, imaginary parts, modulus, and conjugates. Watch the right triangle form as you explore different quadrants with real-time calculations and explanations.
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Complex Numbers Formulas and Examples

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Complete collection of complex numbers formulas with step-by-step examples. Covers polar form, modulus, conjugate, De Moivre
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Complex Numbers Terms and Definitions

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Complete glossary of complex number terms with definitions and examples. Covers foundations, representations, conjugates, modulus, argument, and roots of unity.
Foundations7
Complex NumberA number of the form z=a+biz = a + bi, where a,b∈Ra, b \in \mathbb{R} and ii is the imaginary unitRead more →Imaginary UnitThe number ii defined by i2=−1i^2 = -1Read more →Imaginary NumberA number of the form bibi, where b∈Rb \in \mathbb{R} and b≠0b \neq 0Read more →Pure Imaginary NumberA complex number z=a+biz = a + bi where a=0a = 0 and b≠0b \neq 0Read more →Real PartFor z=a+biz = a + bi, the real part is Re⁡(z)=a\operatorname{Re}(z) = aRead more →Imaginary PartFor z=a+biz = a + bi, the imaginary part is Im⁡(z)=b\operatorname{Im}(z) = bRead more →Algebraic FormThe representation z=a+biz = a + bi where a=Re⁡(z)a = \operatorname{Re}(z) and b=Im⁡(z)b = \operatorname{Im}(z)Read more →
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Modulus (Absolute Value) of Complex Numbers

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Learn the modulus of complex numbers: definition |z| = √(a² + b²), the identity z·z̄ = |z|², properties, the triangle inequality and its proof, reverse triangle inequality, and applications.
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Additive Inverse of Complex Numbers

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Learn the additive inverse of complex numbers: definition -z = -a - bi, geometric meaning as reflection through origin, properties, comparison with conjugate, connection to subtraction, and common mistakes.
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Algebraic Form of Complex Numbers

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Master the algebraic form z = a + bi of complex numbers. Learn about real and imaginary parts, Re(z) and Im(z) notation, equality conditions, and how conjugate identities classify numbers.
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Complex Numbers Basics

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A complete guided tour of complex numbers on one page: why the imaginary unit exists, the algebraic, trigonometric and exponential forms, arithmetic with conjugates, the complex plane, the field axioms, the Euler formula, the De Moivre theorem with n-th roots, and the Fundamental Theorem of Algebra — with reference tables for the powers of i, the four operations, the roots of unity, and the three forms side by side.
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Complex Numbers Cheat Sheets

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Compact reference sheets for complex numbers: the key definitions, the three forms, operation rules, conjugate and modulus identities, and the De Moivre machinery — everything condensed for quick lookup during homework and exams.
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Complex Conjugate

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Master the complex conjugate: definition, geometric meaning as reflection, properties, the identity z·z̄ = |z|², applications to division, and conjugate pairs in polynomial roots.
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De Moivre Theorem

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Master De Moivre
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Complex Polynomial Equations

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Learn how complex numbers solve all polynomial equations. Covers the Fundamental Theorem of Algebra, factoring over ℂ, Vieta
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Exponential Form of Complex Numbers

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Master the exponential form z = re^(iθ) of complex numbers. Learn Euler
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Geometric Representation of Complex Numbers

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Learn to visualize complex numbers on the complex plane (Argand diagram). Plot points, understand vectors, see conjugates as reflections, and discover why complex numbers can
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Imaginary Numbers

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Learn about imaginary numbers: the imaginary unit i, simplifying square roots of negatives, pure imaginary numbers, powers of i cycle, and the imaginary axis on the complex plane.
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Multiplicative Inverse of Complex Numbers

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Learn the multiplicative inverse of complex numbers: formula z⁻¹ = z̄/|z|², geometric meaning, properties, comparison with conjugate, connection to division, and why zero has no inverse.
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Operations on Complex Numbers

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Learn how to add, subtract, multiply, and divide complex numbers. Step-by-step examples, the conjugate method for division, multiplicative inverse, and common pitfalls to avoid.
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Properties of Complex Numbers

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Learn properties of complex numbers: field axioms, conjugate properties, modulus and argument rules, triangle inequality, algebraic closure, and why complex numbers cannot be ordered.
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Trigonometric Form of Complex Numbers

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Learn the trigonometric (polar) form of complex numbers: modulus, argument, principal argument, cis notation, and how to convert between algebraic and polar forms.
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