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<!-- Some Definitions -->
\(
    \def\naturals{\mathbb{N}}
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    \def\reals{\mathbb{R}}
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<h1> Complex Numbers </h1>
    <p> Up till now, you have been told that you can only take the square root of a positive number, because every number was positive after you square it.</p>
    <p> However, after given a new defined number, you can take the square root of a negative number. We called this new defined number "i", which stands for "imaginary" (we know it does not exist). </p>

<hr>
<h3>Definition</h3> 
    <p> Complex numbers are the field of $\mathbb{C}$ numbers of $x +iy$, where x and y are real numbers and i is the imaginary unit. </p>
    <p> We write complex number in the form of (x,y), which means $x + iy$</p>
    <p> $i = \sqrt{-1}$, then $i^2 = -1$ </p>
    <P> '+' $\rightarrow (a,b) + (c,d) = (a+b, c+d)$ </P>
    <P> '$\times$' $\rightarrow (a,b) \times (c,d) = (ac-bd, ad+bc)$ </P>
    <P> $\sqrt{-9} = \sqrt{9(-1)} = \sqrt{9}i = 3i$</P>
    <P> $\sqrt{-18} = \sqrt{18(-1)} = \sqrt{18}i = 3\sqrt{2}i$</P>    
    
<hr>
<h3>Operations</h3>    
    <p> Additive identity: '0' = (0,0)</p>
    <p> Multiplication identity: '1' = (1,0)</p>
    <p> Multiplication inverse: $(\frac{a}{a^2+b^2}, -\frac{b}{a^2+b^2})$ </p>
    <p> $i^n = 1, n = 4k, k \in \mathbb{Z}$ </p>
    <p> $i^n = i, n = 4k + 1, k \in \mathbb{Z}$ </p>
    <p> $i^n = -1, n = 4k + 2, k \in \mathbb{Z}$ </p>
    <p> $i^n = -i, n = 4k + 3, k \in \mathbb{Z}$ </p>
    <P> $2i + 5i = (2+5)i = 7i$</P>
    <P> $(2i)(5i) = (2 \times 5)(i \times i) = 10(-1) = -10$</P>

<hr>
<h3>Conjugate</h3>    
    <p> If a and b are real, and $z = a + bi$, then the complex number $\bar{z} = a - bi$. We can write a = Re(z), and b = Im(z)</p>
    <p> $|\bar{z}| = |z|$</p>
    <p> $|zw| = |z||w|$</p>
    <p> $Re(z) < |z|+|w|$</p>

<hr>
<h3>Cauchy&ndash;Schwarz Inequality</h3>  
$$
\left( \sum_{k=1}^n a_k b_k \right)^2 \leq
\left( \sum_{k=1}^n a_k^2 \right)
\left( \sum_{k=1}^n b_k^2 \right)
$$       
        
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