Maths for Economics 10: Complex Numbers In Finance And Economics - Topic 1: Complex Number Fundamentals

Question 1

State De Moivre’s Formula

Include when it was developed

Answer 1

1707, 1722 - (cos theta + i sin theta )^n = cos ntheta + i sin ntheta

Question 2

State Eueler’s Formula

Include when it was developed and its use

Answer 2

1743, 1748 - exp (itheta) = cos theta + i sin theta.

The
trigonometric functions are periodic.
Euler’s Formula tells us – among many other things - that complex
numbers can be used to study cyclical processes like seasonal
effects

Question 3

What’s a complex number?

Answer 3

A complex number is a number of the form a+bi, where a and b
are real numbers. For instance 4 - 5i is complex

Question 4

Give clarity on what numbers are complex, real or imaginary?

Answer 4

Note that real numbers are also complex, since for instance we can
write 2 = 2 + 0i.
An imaginary number is a number of the form bi, where b is real. For
instance root -6 = i root 6, is imaginary. 0 is both real and imaginary.
Numbers like 4 - 5i are complex, but not imaginary.
We sometimes say ‘not purely imaginary’, to emphasize the
distinction.
If z=4-5i, then the real part of z, written Re z, is 4, and the
imaginary part of z, written Im z, is -5.
Note that the imaginary part of a complex number is therefore real,
not imaginary (!)

Question 5

Describe & explain complex numbers as vectors

Answer 5

Complex numbers can be represented as points on the plane: the set
R^2, the set of all 2x1 vectors.

The idea that complex numbers are like 2x1 vectors is correct – up to
a point - and useful. It forms the insight behind the Argand diagram.
But it omits an important aspect of complex numbers. Unlike 2x1 vectors, complex numbers can be multiplied and divided by each
other, to produce new complex numbers.
The possibility of multiplication and division makes the theory of
complex numbers much richer than the theory of 2x1 vectors.