i Calculator (Imaginary Unit)

Calculate powers of i, convert between complex number forms, and explore imaginary numbers

Complex Number Calculator

Calculation Mode

Power of i (n in i^n)

Enter any integer (positive, negative, or zero)

Results

i¹ = i

Powers of i of the form 4k+1 equal i

Pattern of Powers of i

i⁰ = 1

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

Pattern repeats every 4 powers: i, -1, -i, 1, i, -1, -i, 1, ...

Example Calculations

Powers of i Pattern

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

i⁵ = i

i⁶ = -1

i⁷ = -i

i⁸ = 1

Rule: i^n = i^(n mod 4)

Complex Number Example

Given: z = 3 + 4i

Magnitude: |z| = √(3² + 4²) = √25 = 5

Phase: φ = arctan(4/3) ≈ 53.13°

Polar form: z = 5e^(i × 53.13°)

Quick Reference

Definition of i

i = √(-1)

i² = -1

Powers of i Cycle

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

Then repeats...

Complex Number Forms

Rectangular: a + bi

Polar: r × e^(iφ)

Trigonometric: r(cos φ + i sin φ)

Complex Number Properties

Magnitude

|a + bi| = √(a² + b²)

Phase Angle

φ = arctan(b/a)

Conjugate

conj(a + bi) = a - bi

Euler's Formula

e^(iφ) = cos φ + i sin φ

Understanding the Imaginary Unit i

What is i?

The imaginary unit i is defined as the square root of -1. While this might seem impossible with real numbers, i extends our number system to include complex numbers, which are essential in mathematics, physics, and engineering.

Key Properties

•i = √(-1), therefore i² = -1
•Powers of i follow a cycle of 4: i, -1, -i, 1
•Complex numbers have the form a + bi
•i enables solutions to equations like x² + 1 = 0

Applications

Complex numbers are used extensively in many fields including electrical engineering (AC circuits), quantum mechanics, signal processing, and computer graphics. They provide a powerful mathematical framework for describing oscillations and rotations.

Powers of i Pattern

i⁰ = 1

i¹ = i

i² = -1

i³ = -i

i⁴ = 1 (cycle repeats)

Rule: For any integer n, i^n = i^(n mod 4)

Key Formulas

Basic Operations

i = √(-1)

i² = -1

i³ = i² × i = -i

i⁴ = i² × i² = 1

Complex Number Conversions

|z| = √(a² + b²)

φ = arctan(b/a)

a = |z| cos φ

b = |z| sin φ

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