2.2 Complex Numbers


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2.2 Contents


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2.2.1 Imaginary Number Definition

Square roots of negative numbers fail to yield real results. The imaginary unit satisfies the equation $i^2=–1$. $$i^2=-1 \therefore i=\sqrt{-1}$$

2.2.2 Integer Powers of Imaginary Units

Powers of $i$ always reduce using a modulus of $4$ $$i^x=i^{(x \bmod 4)}$$ Even powers simplify into real numbers $$i^{-2}=i^2=i^6=-1\qquad i^{-4}=i^0=i^4=1$$ Odd powers simplify into imaginary numbers $$i^{-3}=i^1=i^5=i\qquad i^{-1}=i^3=i^7=-i$$ It follows that imaginary numbers with real integer exponents are a periodic series of $i, –1, –i, 1$

2.2.3 Complex Addition

Complex numbers add the same way as expressions with variables $$(a+b·i)\pm(c+d·i)=(a\pm c)+(b\pm d·i)$$
Reference Plus or Minus Notation
Example
$$5-3·i+(-7+9·i)=-2+6·i$$

2.2.4 Complex Conjugation

Complex conjugation functions the same as real conjugation with the sign reversal on the imaginary part $$z=x+y·i\qquad z^*=x-y·i$$
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2.2.5 Magnitude of Complex Numbers

$$|z|^2=z·z^*=x^2+y^2$$

2.2.6 Complex Division

$$\frac{a+b·i}{c+d·i}=\frac{a+b·i}{c+d·i}·\frac{c-d·i}{c-d·i}$$ $$=\frac{a·c+b·d+(b·c-a·d)·i}{c^2+d^2}$$

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