Simplifying Complex Numbers

Input

Exponent n (integer)

Computes iⁿ in a+bi form using the i-cycle (period 4).

Examples

Radicand (negative integer or decimal)

Computes √n where n < 0. Simplifies the radical and writes result as bi.

Examples

Expression

Enter expressions with i, i^n, sqrt(-n). Use ^ for powers.

Examples

Result in a + bi Form

Select a mode, enter your expression, and press Simplify to see the result in standard a + bi form.

Standard Form

—

a (Real Part)

b (Imaginary Part)

i-cycle reference

i¹ = i

i² = −1

i³ = −i

i⁴ = 1

Powers of i — Period-4 Cycle

i¹ = i | i² = −1 | i³ = −i | i⁴ = 1

The powers of i repeat with period 4. To find iⁿ for any integer n, compute n mod 4:

n mod 4 = 0 → i⁴ = 1 | n mod 4 = 1 → i¹ = i

n mod 4 = 2 → i² = −1 | n mod 4 = 3 → i³ = −i

Example: i¹⁵ → 15 mod 4 = 3 → i³ = −i

For negative exponents: i⁻¹ = −i, i⁻² = −1, i⁻³ = i, i⁻⁴ = 1 (same cycle, same rule applies).

Square Roots of Negatives & Standard Form

√(−a) = i√a for a > 0

A complex number in standard form is a + bi where:

a = real part | b = imaginary part | i = √(−1)

To simplify √(−n): factor out √(−1) = i, then simplify √n by pulling out perfect square factors.

Example: √(−12) = i√12 = i√(4·3) = 2i√3