annual (nominal) rate – Basically, this is the rate before it is compounded.
compounded rate – Rate after it has been compounded.

8 per cent interest compounded semi-annually equals what annual (nominal) rate?
We know the annual (nominal) rate is 8 per cent so:

Compounded Interest Rate = (1 + [.08 ÷ 2])² -1

Compounded Interest Rate = (1 + .04)² -1

Compounded Interest Rate = 1.0816 -1

Compounded Interest Rate = .0816

which equals 8.16 per cent.

You probably have concluded that:
n = 4 for quarterly compounded interest
n = 12 for monthly compounded interest and
n = 365 for daily compounded interest.

If we have a monthly compounded interest rate of .072290080856235 (or 7.2290080856235%), what was the rate before compounding?
(Or what is the annual (nominal) rate?)

Since we are dealing with monthly compounding, n=12.
Putting the numbers into the formula, we see that the annual (nominal) rate equals:

12 * [(1 + .072290080856235)^{(1 ÷ 12)}-1)]

= 12 * [(1.072290080856235)^{(.08333333…)}-1)]

= 12 * (1.00583333333333… -1)

= 12 * (.00583333333333)

annual (nominal) rate = .07 or 7 per cent

For the first time, we have a formula that uses the mathematical constant e which equals 2.71828182845904523536….
So, let’s comptue the continuously compounded rate of an annual (nominal) rate of 9 per cent.

continuously compounded rate = e^r -1

continuously compounded rate = (2.71828…)^.09-1

continuously compounded rate = 1.09417428370521 -1

continuously compounded rate = .09417428370521

continuously compounded rate = 9.417428370521%

A bank account yields 6% interest when compounded continuously.
What is the annual (nominal) interest rate?

annual (nominal) rate = natural log [1 + .06]

annual (nominal) rate = 0.0582689081239758

rate = 5.82689081239758% before compounding