= Complex number
{title2=$\mathbb C$}
{wiki}
A <complex number> is $z=x+iy$, where $x,y$ are <real numbers> and the <imaginary unit> satisfies $i^2=-1$. The <real part> is $x$ and the <imaginary part> is $y$: for $3+4i$ they are $3$ and $4$, respectively. Equality requires both coordinates to agree. Thus the <complex plane> gives each <complex number> a concrete position, while ordinary <addition> and <multiplication> become geometric operations.
For $z=3+4i$ and $w=1+i$, <addition> works coordinate by coordinate, whereas <multiplication> uses $i^2=-1$:
$$
z+w=4+5i,\qquad zw=(3+4i)(1+i)=3+7i-4=-1+7i.
$$
These answers encode different motions: <addition> by $w$ is a <translation>, while <multiplication> by $w$ is a <rotation> combined with a uniform change of size. Keeping these pictures distinct prevents a common beginner's confusion.
The <complex numbers> form a <field>: every nonzero $z$ has a <multiplicative inverse>, computable using its <complex conjugate> and <modulus>. They also supply the missing solutions of a <quadratic equation> such as $u^2+1=0$, namely $u=\pm i$. The surprising payoff is that extending arithmetic also makes <Euclidean geometry> easier to calculate. For a systematic introduction to these definitions, see https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf[Jeremy Orloff, MIT 18.04, topic 1, sections 1.3–1.5].
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