Imaginary Number (i) Calculator

Compute powers of i, perform complex number arithmetic, and explore the imaginary unit cycle.

Select Operation

Enter any integer (positive, negative, or zero). The cycle repeats every 4: i, -1, -i, 1.

Result

i7
-i
imaginary unit
i0
1
i1
i
i2
-1
i3
-i
Exponent7
n mod 43
Cycle Position4th (i^3 = -i)

Step-by-Step Solution

i^7 = i^(4*1 + 3) = (i^4)^1 * i^3 = 1 * (-i) = -i

Understanding the Imaginary Unit i

The imaginary unit i is defined as the square root of -1, written as i = √(-1). It was introduced to extend the real number system to the complex number system, allowing solutions to equations like x2 + 1 = 0 that have no real solutions. Complex numbers take the form a + bi, where a is the real part and b is the imaginary part.

The Powers of i Cycle

The powers of i repeat in a cycle of 4. This cyclic pattern is fundamental to working with imaginary numbers:

i0 = 1

Any nonzero number raised to the power of 0 equals 1.

n mod 4 = 0 => i^n = 1

i1 = i

The imaginary unit itself. The square root of -1.

n mod 4 = 1 => i^n = i

i2 = -1

By definition, i squared equals negative one.

n mod 4 = 2 => i^n = -1

i3 = -i

i cubed equals i squared times i, which is -1 times i = -i.

n mod 4 = 3 => i^n = -i

Complex Number Operations

Addition and Subtraction

To add or subtract complex numbers, combine the real parts and imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i. This works just like combining like terms in algebra.

Multiplication

Use the FOIL method and remember that i2 = -1: (a + bi)(c + di) = ac + adi + bci + bdi2 = (ac - bd) + (ad + bc)i.

Division

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) = (a + bi)(c - di)/((c + di)(c - di)) = ((ac + bd) + (bc - ad)i)/(c2 + d2).

Applications of Complex Numbers

  • Electrical engineering: AC circuit analysis uses complex impedance.
  • Signal processing: Fourier transforms rely on complex exponentials.
  • Quantum mechanics: Wave functions are complex-valued.
  • Control theory: Transfer functions use complex poles and zeros.
  • Fractal geometry: The Mandelbrot set is defined using complex numbers.