Complex Numbers Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Simplify i2i^2, i3i^3, and i4i^4.

Solution

  1. 1
    i2=−1i^2 = -1 by definition of the imaginary unit.
  2. 2
    i3=i2⋅i=(−1)⋅i=−ii^3 = i^2 \cdot i = (-1) \cdot i = -i.
  3. 3
    i4=i3⋅i=(−i)⋅i=−i2=−(−1)=1i^4 = i^3 \cdot i = (-i) \cdot i = -i^2 = -(-1) = 1.

Answer

i2=−1,i3=−i,i4=1i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
The powers of ii cycle with period 4: i,−1,−i,1,i,−1,−i,1,…i, -1, -i, 1, i, -1, -i, 1, \ldots Knowing this cycle allows rapid simplification of any power of ii by finding the remainder when the exponent is divided by 4.

About Complex Numbers

Numbers of the form a+bia + bi where a,ba, b are real and i=−1i = \sqrt{-1}; they extend the real numbers to solve x2=−1x^2 = -1.

Learn more about Complex Numbers →

More Complex Numbers Examples

Example 2 medium

Multiply [formula] and write the result in standard form [formula].

Example 3 easy

Add [formula].

Example 4 medium

Simplify [formula].

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