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) + (2  3i) = 7 + 4i.
We subtract them similarly: (5 + 7i)  (2  3i) = 3 + 10i.
We can multiply them as follows: (5 + 7i) * (2  3i) = 10 + (14  15)i  21i^{2} = 31  i.
To do division you make use of the fact that (a + ib) * (a  ib) = a^{2}  (ib)^{2} = a^{2} + b^{2}.
Thus you write .
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 ().
To anticipate what we will later see, the relations between these quantities
is
x and y can be expressed in terms of r and by
and the wonderful fact
implies that we can write
Exercises: Evaluate
1. (4 + i) / (3  2i).
2. (3 + 3i) * (2  i).
3. Find r given x = 3, y = 4.
4. Find given x = 3, y = 2.
5. Find given x = 2, y = 4.

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