Algebra 2 - Simplifying Complex Numbers

Complex Numbers

  • We know that 25=±5, but what if we want to find the value of -25. Is it possible to compute its value?
  • The answer is YES. But, the result will be a complex number.
  • -25=25×-1=52×-1=5-1=5i where i=-1 is also called an imaginary unit.
  • In general, a complex number is represented as z=a+ib, where a=Re{z}=real part and b=Im{z}=imaginary part.
  • An example of a complex number written in standard form is z=2-3i.

Powers of Imaginary Unit [i]

  • We know that i=-1.
    • i2=-1
    • i3=-i
    • i4=1
    • i4k=1, where k is any integer.
    • i4k+1=i
    • i4k+2=-1
    • i4k+3=-i

Solved Examples

Example 1: Simplify i33.

Solution: i33=i4×8+1=i4×8·i=1·i=i


Example 2: Simplify i2034.

Solution: i2034=i4×508+2=i4×508·i2=1·-1=-1


Example 3: Simplify i-87.

Solution: i-87=1i87=1i4×21+3=1i4×21·i3=i41·i3=i


Example 4: Simplify 5i.

Solution: 5i=5×1i=5×i4i=5×i3=5×-i=-5i

Cheat Sheet

  • When 'i'  is raised to the power of any integer, the result can be 1, i, –1, or – i.
  • Simplifying complex numbers means transforming the given complex number into the standard form a+ib.

Blunder Areas

  • i0ii00. Rather i0=1.