Home  18.013A  Chapter 1  Section 1.2 


A complex number is the sum of a real number and another real number multiplied by
$i$
, where
$i$
is a square root of 1.
Thus it can be written as
$a+ib$
where
$a$
and
$b$
are real.
We can add two such numbers by adding their real and imaginary parts separately. Thus $(5+7i)+(23i)=7+4i$ .
We subtract them similarly: $(5+7i)(23i)=3+10i$ .
We can multiply them as follows: $(5+7i)*(23i)=10+(1415)i21{i}^{2}=31i$ .
To do division you make use of the fact that $(a+ib)*(aib)={a}^{2}{(ib)}^{2}={a}^{2}+{b}^{2}$ .
Thus you write $\frac{a+ib}{c+id}=\frac{(a+ib)(cid)}{{c}^{2}+{d}^{2}}$ .
It is common to represent complex numbers by points in the "complex plane". The real part of the complex number
$(a+ib)$
is
$a$
, its imaginary part is
$b$
. We represent it by the point with
$x$
coordinate
$a$
, and
$y$
coordinate
$b$
.
The
$x$
axis is, in this complex plane, called the real axis, and the
$y$
axis is the imaginary axis. Numbers on the real axis are ordinary real numbers and numbers on the imaginary axis are imaginary numbers.
You can represent a complex number alternatively, by its distance to the origin, usually written as
$r$
and called its magnitude, and the angle that a line from it to the origin makes with the
$x$
axis at the origin, usually called theta (
$\theta $
). To anticipate what we will later see, the relations between these quantities is
and
$x$ and $y$ can be expressed in terms of $r$ and $\theta $ by
and the wonderful fact
implies that we can write
Exercises: Evaluate
1. $\frac{4+i}{32i}$ .
2. $(3+3i)*(2i)$ .
3. Find $r$ given $x=3,y=4$ .
4. Find $\theta $ given $x=3,y=2$ .
5. Find $\theta $ given $x=2,y=4$ .

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