CHAPTER 12 KEY FORMULAS AND EQUATIONS

Chapter Equations for Complex Numbers
Imaginary unit	$j = \sqrt{- 1} and j^{2} = - 1$ (12.1) $\sqrt{- a} = j \sqrt{a} (a > 0)$ (12.2)
Basic operations	$(a + b j) + (c + d j) = (a + c) + (b + d) j$ (12.3) $(a + b j) - (c + d j) = (a - c) + (b - d) j$ (12.4) $(a + b j) (c + d j) = (a c - b d) + (a d + b c) j$ (12.5) $\frac{a + b j}{c + d j} = \frac{(a + b j) (c - d j)}{(c + d j) (c - d j)} = \frac{(a c + b d) + (b c - a d) j}{c^{2} + d^{2}}$ (12.6)
Complex number forms	$\begin{array}{l} Rectangular : & x + y j \\ Polar : & r (\cos θ + j \sin θ) = r ∠ \underline{θ} \\ Exponential : & {r e}^{j θ} \end{array}$ $x = r \cos θ y = r \sin θ$ (12.7) $r^{2} = x^{2} + y^{2} \tan θ = \frac{y}{x}$ (12.8) $x + y j = r (\cos θ + j \sin θ) = r / \underline{θ} = r e^{j θ}$ (12.12)
Product in polar form	$\begin{array}{rcl} r_{1} (\cos θ_{1} + j \sin θ_{1}) r_{2} (\cos θ_{2} + j \sin θ_{2}) & = & r_{1} r_{2} [\cos (θ_{1} + θ_{2}) + j \sin (θ_{1} + θ_{2})] \\ (r_{1} ∠ \underline{θ_{1}}) (r_{2} ∠ \underline{θ_{2}}) & = & r_{1} r_{2} ∠ \underline{θ_{1} + θ_{2}} \end{array}$

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Basic Technical Mathematics, 11th Edition by Allyn J. Washington, Richard Evans