z = 2i on the imaginary axis
Complex Numbers Basics: Core Concepts and Forms › The Imaginary Unit (iii) and Pure Imaginary Numbers
z and z̄ mirrored in the real axis, |z| = 3.6
Complex Numbers Basics: Core Concepts and Forms › Algebraic Representation and the Complex Conjugate
z = 2 + 3i plotted as the point (2, 3)
Complex Numbers Basics: Core Concepts and Forms › Geometric Representation: The Complex Plane
z = 3 + 2i: r = 3.61 and the angle θ
Complex Numbers Basics: Core Concepts and Forms › Trigonometric Representation: Radius and Argument
e^(iπ) on the unit circle
Complex Numbers Basics: Core Concepts and Forms › Exponential Form and Euler's Formula
z, z², z³, z⁴: the argument multiplies by n
Complex Numbers Basics: Core Concepts and Forms › De Moivre's Theorem and nnn-th Order Roots
Four roots for a degree-4 equation
Complex Numbers Basics: Core Concepts and Forms › Complex Equations and Polynomial Theory
|z| = 3.6, the length of the arrow
Modulus: Absolute Value & Triangle Inequality › Definition of Modulus
A pure imaginary number and its modulus
Modulus: Absolute Value & Triangle Inequality › Special Cases
z₁ + z₂: the diagonal is shorter than the two sides together
Modulus: Absolute Value & Triangle Inequality › The Triangle Inequality
Distance as the modulus of a difference
Modulus: Absolute Value & Triangle Inequality › Applications
−z₂ is z₂ turned through 180°
Additive Inverse: Negation of Complex Numbers › Additive Inverse Notation
z and −z through the origin
Additive Inverse: Negation of Complex Numbers › Geometric Interpretation
z̄ reflects in the real axis; −z would reflect through the origin
Additive Inverse: Negation of Complex Numbers › Additive Inverse vs. Complex Conjugate
z = 2 + 3i: Re(z) = 2 is the horizontal coordinate
Algebraic Form: Standard Form a + bi › The Real Part
A negative imaginary part
Algebraic Form: Standard Form a + bi › The Imaginary Part
z and z̄ mirrored in the real axis, |z| = 3.6
Algebraic Form: Standard Form a + bi › The Complex Conjugate
z and z̄ mirrored in the real axis, |z| = 3.6
Complex Conjugate: Definition & Properties › Conjugate Notation
z · z̄ lands on the positive real axis
Complex Conjugate: Definition & Properties › Conjugate and Modulus
Dividing by the conjugate clears the denominator
Complex Conjugate: Definition & Properties › Applications to Division
z and z²: the argument doubles
Demoivre Theorem › De Moivre's Formula
z through z¹⁰: a spiral of powers
Demoivre Theorem › Applying De Moivre's Theorem to Powers
The three cube roots of 8
Demoivre Theorem › Introduction to nnn-th Roots
Three linear factors, three roots
Polynomial Equations & Fundamental Theorem › Factoring Polynomials over C\mathbb{C}C
The n + 2 solutions for n = 2
Polynomial Equations & Fundamental Theorem › Solving zn=zˉz^n = \bar{z}zn=zˉ
e^(iθ) at θ = 45°: the point (cos θ, sin θ)
Exponential Form › Euler's Formula
e^(iπ) = −1
Exponential Form › Euler's Identity
z = 2e^(iπ/3)
Exponential Form › The Exponential Form of a Complex Number
z₁z₂: arguments add, moduli multiply
Exponential Form › Multiplication in Exponential Form
Dividing subtracts the angles
Exponential Form › Division in Exponential Form
z and z²: (re^(iθ))² = r²e^(2iθ)
Exponential Form › Powers in Exponential Form
z = −3 + 2i plotted as (−3, 2)
Complex Plane & Argand Diagram: Visualize ℂ › Plotting Complex Numbers
z₁ + z₂ as the diagonal of a parallelogram
Complex Plane & Argand Diagram: Visualize ℂ › Complex Numbers as Vectors
z and z̄ mirrored in the real axis, |z| = 3.6
Complex Plane & Argand Diagram: Visualize ℂ › Visualizing the Conjugate
z₁z₂: arguments add, moduli multiply
Complex Plane & Argand Diagram: Visualize ℂ › Visualizing Operations
z = 2i: a point on the imaginary axis
Imaginary Numbers: Definition, Powers of i › Defining Pure Imaginary Numbers
Im(z) is a signed real number
Imaginary Numbers: Definition, Powers of i › Critical Distinction: The "Imaginary Part" is Real
The powers of i: a cycle of four
Imaginary Numbers: Definition, Powers of i › The Cyclic Nature of the Powers of iii
z = 3 + 4i and z⁻¹ = 0.12 − 0.16i
Multiplicative Inverse: Reciprocal of ℂ › Multiplicative Inverse Notation
Dividing by zero: nothing to draw
Multiplicative Inverse: Reciprocal of ℂ › Why Zero Has No Multiplicative Inverse
z₁/z₂: arguments subtract, moduli divide
Multiplicative Inverse: Reciprocal of ℂ › Connection to Division
z₁ + z₂ as the diagonal of a parallelogram
Operations on Complex Numbers: Add, Multiply, Divide › Addition of Complex Numbers
z₁ − z₂ = z₁ + (−z₂)
Operations on Complex Numbers: Add, Multiply, Divide › Subtraction of Complex Numbers
z₁z₂: arguments add, moduli multiply
Operations on Complex Numbers: Add, Multiply, Divide › Multiplication of Complex Numbers
z₁/z₂: arguments subtract, moduli divide
Operations on Complex Numbers: Add, Multiply, Divide › Division of Complex Numbers
|z| = |z̄| = 3.6
Properties of Complex Numbers: Field Axioms › Properties of the Modulus
z = −4 + 3i: r = 5, argument in the second quadrant
Properties of Complex Numbers: Field Axioms › Properties of the Argument
z = 3 + 2i: r = √13 ≈ 3.61
Trigonometric Form of Complex Numbers › Modulus (Radius)
z = −4 + 3i: the angle θ measured from the positive real axis
Trigonometric Form of Complex Numbers › Argument (Angle)
A negative principal argument
Trigonometric Form of Complex Numbers › The Principal Argument
z = −3 − 4i: arctan(b/a) alone gives the wrong angle
Trigonometric Form of Complex Numbers › Quadrant Considerations
z₁z₂: arguments add, moduli multiply
Trigonometric Form of Complex Numbers › Multiplication and Division in Trigonometric Form
Addition mode — (3+i) + (1+3i), frozen
Complex Addition & Subtraction Visualizer › The Parallelogram Rule for Addition
Subtraction mode — (3+i) − (1+3i), frozen
Complex Addition & Subtraction Visualizer › Subtraction and the Difference Vector
Both mode — the opening pair, frozen
Complex Addition & Subtraction Visualizer › Both Mode — Comparing Addition and Subtraction
Mirror pair — (2+2i) & (−2+2i), frozen
Complex Addition & Subtraction Visualizer › The Mirror Pair Preset
Axis pair — 4 & 3i, frozen
Complex Addition & Subtraction Visualizer › The Axis Pair Preset
Mixed signs — (−1+3i) & (2−i), frozen
Complex Addition & Subtraction Visualizer › The Mixed-Signs Preset
Conjugate pair — (3+2i) & (3−2i), frozen
Complex Addition & Subtraction Visualizer › Conjugate Pairs in Addition and Subtraction
(2+i)(−1+2i), frozen
Complex Multiplication Visualizer › Reading the Three Angle Arcs
2(−3+4i), frozen
Complex Multiplication Visualizer › Multiplication by a Pure Real Number
3 × 2i, frozen
Complex Multiplication Visualizer › Multiplication by Pure Imaginary Numbers
(1+i)(1−i), frozen
Complex Multiplication Visualizer › Conjugate Pair Multiplication
i × i, frozen
Complex Multiplication Visualizer › Why i² = −1 Makes Geometric Sense
(4+2i)/(1−i), frozen
Complex Division Visualizer › The Three Angle Arcs — Subtraction in Action
6/3, frozen
Complex Division Visualizer › Division of Pure Real Numbers
4i/2i, frozen
Complex Division Visualizer › Division of Pure Imaginary Numbers
1/i, frozen
Complex Division Visualizer › Dividing by i — The −90° Rotation
(3+4i)/(3−4i), frozen
Complex Division Visualizer › Conjugate Pair Division
(−2+6i)/(1+2i), frozen
Complex Division Visualizer › The Clean-Division Preset
z₂ = 0, frozen
Complex Division Visualizer › The Divide-by-Zero State
z = 3 + 2i, frozen
Complex Conjugate and Modulus Visualizer › The Starting Point: 3 + 2i
z = −1 + 4i, frozen
Complex Conjugate and Modulus Visualizer › Negative Real Part: −1 + 4i
z = 3i, frozen
Complex Conjugate and Modulus Visualizer › Purely Imaginary: 3i
z = 4, frozen
Complex Conjugate and Modulus Visualizer › Purely Real: 4
z = −2 − 3i, frozen
Complex Conjugate and Modulus Visualizer › Third Quadrant: −2 − 3i
z = 0, frozen
Complex Conjugate and Modulus Visualizer › The Origin: z = 0
(−2+i) & (3+3i), frozen
Complex Distance & Midpoint Calculator › The Right Triangle and Distance Segment
(1+4i) & (1−2i), frozen
Complex Distance & Midpoint Calculator › The Vertical Pair
−4 & 4, frozen
Complex Distance & Midpoint Calculator › The Horizontal Pair
(−3−2i) & (3+2i), frozen
Complex Distance & Midpoint Calculator › Symmetric Points and Midpoint at the Origin
0 & (3+4i), frozen
Complex Distance & Midpoint Calculator › Distance from the Origin — Modulus as a Special Case
z₁ = z₂ = 2 + i, frozen
Complex Distance & Midpoint Calculator › Coincident Points — Distance Zero
z = 0, frozen
Complex Number Explorer › The Origin
z = 3, frozen
Complex Number Explorer › Purely Real Numbers
z = 2i, frozen
Complex Number Explorer › Pure Imaginary Numbers
z = 2 + 3i, frozen
Complex Number Explorer › Quadrant I: Upper Right
z = −3 + 2i, frozen
Complex Number Explorer › Quadrant II: Upper Left
z = −2 − 3i, frozen
Complex Number Explorer › Quadrant III: Lower Left
z = 3 − 2i, frozen
Complex Number Explorer › Quadrant IV: Lower Right
z = 3 + 2i, frozen
Polar-Rectangular Complex Number Converter › Quadrant I Baseline: 3 + 2i
z = −4 + 3i, frozen
Polar-Rectangular Complex Number Converter › Quadrant II: −4 + 3i
z = −3 − 4i, frozen
Polar-Rectangular Complex Number Converter › Quadrant III: −3 − 4i
z = 5 − 5i, frozen
Polar-Rectangular Complex Number Converter › Equal-Component States and Special Angles
z = 5i, frozen
Polar-Rectangular Complex Number Converter › On the Imaginary Axis: 5i
z = −6, frozen
Polar-Rectangular Complex Number Converter › On the Negative Real Axis: −6
θ = π/6, frozen
Euler's Formula Explorer › The 30 Degree Landmark
θ = π/4, frozen
Euler's Formula Explorer › The 45 Degree Landmark
θ = π/3, frozen
Euler's Formula Explorer › The 60 Degree Landmark
θ = 0, frozen
Euler's Formula Explorer › The Zero Angle
θ = π/2, frozen
Euler's Formula Explorer › The Quarter Turn to i
θ = 3π/2, frozen
Euler's Formula Explorer › The Three-Quarter Turn to −i
r = 2, θ = π/3, frozen
Euler's Formula Explorer › Adjusting the Radius
θ = π, frozen
Euler's Formula Explorer › Euler's Identity — The Special Case at θ = π
i¹⁰⁰ — remainder 0, frozen
Powers of i Calculator › Remainder 0: Full Cycles Vanish
i¹⁷ — remainder 1, frozen
Powers of i Calculator › Remainder 1: The Cycle Restarts
i⁸² — remainder 2, frozen
Powers of i Calculator › Remainder 2: The Definition Itself
i³²³ — remainder 3, frozen
Powers of i Calculator › Remainder 3: The Opening Example
(1+i)², frozen
De Moivre's Theorem Visual Calculator › The Squaring Baseline
(1+i)⁴, frozen
De Moivre's Theorem Visual Calculator › The Outward Spiral — When |z| > 1
(1+i)⁸, frozen
De Moivre's Theorem Visual Calculator › Escaping the Window
(0.5+0.5i)⁶, frozen
De Moivre's Theorem Visual Calculator › The Inward Spiral — When |z| < 1
i³, frozen
De Moivre's Theorem Visual Calculator › Unit Circle Rotation — When |z| = 1
(3+4i)⁻¹, frozen
De Moivre's Theorem Visual Calculator › Negative Exponents — Reciprocals and Reversal
2¹⁰, frozen
De Moivre's Theorem Visual Calculator › Pure Real Base — No Spiral, Just Scaling