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SECTIONComplex Numbers18 subsectionsBROWSE ALL ↓INTERACTIVE1 TOOLComplex NumbersVisual Tools11 toolsJump to ↓fREFERENCE69 ITEMSfComplex NumbersFormulas andExamples53 itemsJump to ↓AaComplex NumbersTerms andDefinitions16 itemsJump to ↓§CORE TOPICS15 SUBSECTIONSKey TermsJump to ↓Additive Inverse of ComplexNumbersJump to ↓Algebraic Form of ComplexNumbersJump to ↓Forms of a complex numberJump to ↓Complex Numbers Cheat SheetsJump to ↓ConjugateJump to ↓De Moivre TheoremJump to ↓Key TermsJump to ↓Exponential Form of ComplexNumbersJump to ↓The Lack of Order in$\\mathbb{C}$Jump to ↓Natural numbersJump to ↓Multiplicative InverseJump to ↓Complex arithmeticJump to ↓Key TermsJump 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} (0=0i0 = 0i is included; some textbooks require b≠0b \neq 0)Read more →Pure Imaginary NumberA complex number z=a+biz = a + bi where a=0a = 0 (00 is included; some textbooks require b≠0b \neq 0)Read 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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Key Terms

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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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Forms of a complex number

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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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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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Key Terms

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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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The Lack of Order in $\\mathbb{C}$

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Natural 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

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Complex arithmetic

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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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Key Terms

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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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