How to add two imaginary numbers:
(A+Bi)+(C+Di)=(A+C)+(B+D)i
How to multiply two imaginary numbers:
We need to use a simple bit of algebra:
(A+Bi)x(C+Di)=(A x C)+(A x Di)+(Bi x C)+(Bi x Di)
Now we want this to look like a new imagainry number. The trick is to remember that the number i defined to be the number such that i x i =-1. So Bi x Di = BxDxixi=-BxD. Hence:
(A+Bi)x(C+Di)=(A x C - B x D)+(A x D+B x C)i
There is much more geometric way to understand multiplying imaginary numbers as you can find out by clicking here.

What about raising 10 to the power of an imaginary number These seems completely meaningless at first glance, until we realise that raising 10 to the power of number can be done by calcuting infinite polynomials or power series. This is how logarithms were calculated.

The infinite expression that one needs to calculate is given as follows:
10^N = 1+(cN)+(cN)²/2!+(cN)³/3!+… +(cN)ⁿ/n!+…
N can be any number now as this converges for any choice of N. Again the number c depends on the base that you are raising numbers to. For base 10 you must take c approximately 2.3, or more precisely c=log_e10. Again the best base is e because then c=1:
e^N = 1+N+N²/2!+N³/3!+… +Nⁿ/n!+…

The point is that these infinite sums can also be used to calculate e to the power of an imaginary number because we know how to do multiplication of imaginary numbers and this is all that is involved.
For example
e^pi =1+ i+ (i) ²/2!+i) ³/3!+… +(i) ⁿ/n!+…

We are going to use the only thing we know about this imaginary number i, that i²=-1. From this we can deduce that i³=-i, i⁴=1, i⁵=i and then we are back to the beginning again.

Armed with this information about i let's calculate the exponential e^{p I}
e^pi = 1+(i) + (i) ²/2!+ (i) ³/3!+ (i) ⁴/4!+…
=1 -²/2! +⁴/4!…
+i( -³/3!+…)

The curious thing is that by putting in imaginary numbers into the exponential series, we see two different series emerging, one giving the real part of the answer, the other giving the imaginary part. But these aren’t any old infinite series. In fact these are the series used for calculating sine and cosine.
sin(x)=x-x³/3!+x⁵/5!…
cos(x)=1-x²/2!+x⁴/4!-…
Just as distance can be measured in terms of miles or kilometres, angles can be measured not only in degrees but also something called radians. 180 degrees is equal to radians. This infinite polynomial gives the sine function when fed with angles measured in radians rather than degrees. So if you feed in the number , adding up more and more terms of the infinite polynomial, the sum will home in eventually on the answer 0.
Hence
e^pi = cos() + sin()i = -1.
This is often heralded as one of the most beautiful formulas of mathematics unifying as it does many of the most important constants of mathematics: e, , i and –1.
In general we get the following relationship:
e^A+Bi= e^A(cos(B)+sin(B)i)

This formula is the key to why Riemann got music when he combined the zeta function and imaginary numbers as we shall see.

Newton liked to think of the infinite sums as generalizations of the decimal expansion of a real number. Euler worked with them like they were just big polynomials.