Def/Complex number

   =Arithmetic= The complex numbers $\CC$ form a field, where the addition and multiplication are defined (assuming the operations on real numbers) as follows: if $(x_1+y_1 i)$ and $(x_2 + y_2 i)$ are two complex numbers, then:
 * $(x_1 + y_1 i) + (x_2 + y_2 i) = (x_1 + x_2) + (y_1 + y_2) i$.
 * $(x_1 + y_1 i) \cdot (x_2 + y_2 i) = (x_1 x_2 - y_1 y_2) + (x_1 y_2 + x_2 y_1) i$.

Subtraction can be computed by a similar formula:
 * $(x_1 + y_1 i) - (x_2 + y_2 i) = (x_1 - x_2) + (y_1 - y_2) i$.

=Conjugation= If $z = x + yi$ is a complex number, the defines::complex conjugate of $z$ is the complex number: $$\bar z = x - yi.$$

Complex conjugation is an automorphism of the field $\CC$. Galois theoretically, it is the unique nontrivial element of $Gal(\CC / \RR)$. Explicitly, complex conjugation satisfies the following properties:
 * If $z_1, z_2 \in \CC$, then $\bar z_1 + \bar z_2 = \overline{z_1 + z_2}$.
 * If $z_1, z_2 \in \CC$, then $\bar z_1 \cdot \bar z_2 = \overline{z_1 \cdot z_2}$.
 * If $z \in \CC$, then $\overline{\bar z} = z$; in other words, complex conjugation is an automorphism of order $2$.

Observe that $z = \bar z$ if and only if $z \in \RR$, i.e., $z = x + 0i$ for some $x \in \RR$.

When $z$ is a complex number, the number $z \cdot \bar z$ is called the defines::norm of $z$, and written $N(z)$. Formulaically, if $x + yi \in \CC$, then: $$N(x + yi) = (x+yi) \cdot (x - yi) = x^2 + y^2.$$ It follows that $N(x + yi) \in \RR$, and $N(x + yi) \geq 0$ (since real squares are nonnegative. Furthermore, $N(x + yi) = 0$ if and only if $x + yi = 0$.

The defines::absolute value of a complex number is defined in terms of the norm: $$\vert z \vert = \sqrt{N(z)},$$ where the above square root refers to the unique non-negative real square root of the non-negative real number $N(z)$.

=Division= If $z = x + yi$, and $z \neq 0$, then $z$ has a multiplicative inverse.

This can be seen computationally, as follows: $$z \cdot (\bar z N(z)^{-1}) = z \bar z N(z)^{-1} = N(z) N(z)^{-1} =.$$
 * Since $z \neq 0$, $N(z)$ is a nonzero real number.
 * Since $\RR$ is a field, $N(z)$ has a multiplicative inverse in $\RR$.
 * Since $\RR$ is a subfield of $\CC$, we find that:

Hence, the multiplicative inverse of $z$ is $\bar z N(z)^{-1}$.

This can be used to divide complex numbers computationally. If $z_1, z_2 \in \CC$, and $z_2 \neq 0$, then division may be computed via multiplication (and using multiplicative inverses of real numbers) as: $${ {z_1} \over {z_2}} = { {z_1 \bar z_2} \over {N(z_2)} }.$$

=Bad Joke= Life is complex: it has both real and imaginary components.

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