Polar Functions: Background You’ll Need 2

  • Compute operations with complex numbers
The imaginary number i s[latex]i=\sqrt{-1}[/latex]. A complex number is written in the form: [latex]a+bi[/latex] where [latex]a[/latex] is the real part and [latex]b[/latex] represents the coefficient of the imaginary part.
  1. [latex](3 + 2i) + (5 - 7i) = 8 - 5i[/latex]
  2. [latex](2 + 3i)(4 + i)[/latex]
  3. [latex]\dfrac{3 + 2i}{1 - i}[/latex]

Squaring a Complex Number

To find the square of a complex number [latex](a + bi)^2[/latex], apply the distributive property:

[latex]\begin{align*} (a + bi)^2 &= (a + bi)(a + bi) \\ &= a^2 + abi + abi + b^2i^2 \\ &= a^2 + 2abi + b^2(-1) \\ &= (a^2 - b^2) + 2abi \end{align*}[/latex]

So, the square of [latex](a + bi)[/latex] is another complex number
[latex](a^2 - b^2) + 2abi[/latex].

The TI-84+ series can perform operations with complex numbers.

  1. Press MODE → choose a + bi format.

  2. Enter your expression: (3 + 4i) ^ 2

  3. Press ENTER to calculate.
    Your calculator should display –7 + 24i.