We all know that 25 means 2×2×2×2×2=32, but what does 2π mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.

**Answer**

# When x∈N

You were probably taught that “exponentiation is repeated multiplication”:

bx=b×b×b×⋯×b⏟x times

From this simple definition, you can observe two properties:

- bx+y=bx⋅by
- bxy=(bx)y

For example:

- 23+4=27=2⋅2⋅2⋅2⋅2⋅2⋅2=(2⋅2⋅2)⋅(2⋅2⋅2⋅2)=23⋅24
- 23⋅4=212=23⋅23⋅23⋅23=(23)4

We can then definite exponentation over more general sets of numbers in a way that these two properties continue to hold.

# When x∈Z

From the above rule for addition of exponents, we obtain a rule for *subtraction* of exponents: bx−y=bxby, because then b(x−y)+y=bx−y⋅by=bxby⋅by=bx as expected. This lets us expand the domain of exponents to include zero and negative integers:

b0=by−y=byby=1,b≠0

b−y=b0−y=b0by=1by,b≠0

# When x∈Q

If you assume that the multiplicate property of exponents holds for rationals, then (b1n)n=b1n⋅n=b1=b. So b1n is a number whose nth power is b. In other words,

b1n=n√b,b≥0

And bmn=(b1n)m=(n√b)m.

For example, 40965/12=(12√4096)5=25=32.

# When x∈R

I still haven’t answered your question of what 2π means. But at this point, we can calculate 2x for x aribitrarily close to π.

- 23 = 8
- 23.1=231/10=10√231≈8.574187700290345
- 23.14=2314/100=100√2314≈8.815240927012887
- 23.141=23141/1000=1000√23141≈8.821353304551304
- 23.1415=231415/1000=10000√231415≈8.824411082479122
- 23.14159=2314159/10000=100000√2314159≈8.824961595059897

As x approaches π, 2x approaches a limit, which is approximately 8.824977827076287. For the sake of making 2x continuous, we define 2π to be *equal* to this limit.

(Note that there’s nothing special about decimal fractions. I could have used the sequence [3,227,333106,355113,…] of best rational approximations, but that would have been less obvious.)

However, taking the trillionth root of huge powers of a number isn’t very practical for calculation. A more useful method is to use logarithms.

logcy is defined as the number x such that cx=y. From the two basic properties of exponentation, you can obtain the identities:

- logc(ab)=logca+logcb
- logc(bx)=x⋅logcb

And from the latter, you get bx=cx⋅logcb. This means that if you have an exponential and logarithm function for *one* value of c, you can calculate them for *any* value for b.

Typical choice of c are:

- 2, for convenience in working with computers
- 10, the base of our number system, giving “common logarithms”
- e≈2.718281828459045, the base of the “natural logarithm” (ln), for its convenient properties in calculus.

So, if you wanted to calculate 2π, you’d actually calculate 10π⋅log102 or eπ⋅ln2. And that would typically be done with the assistance of a logarithm table or a slide rule.

# When x∈C

In Calculus, you’ll learn about Taylor series, and the well-known ones for ex, sine and cosine:

- ex=1+x+x22+x36+x424+x5120+⋯=∑∞k=0xkk!
- sinx=x−x36+x5120−x75040+⋯=∑∞k=0(−1)kx2k+1(2k+1)!
- cosx=1−x22+x424−x6720+x840320−⋯=∑∞k=0(−1)kx2k(2k+1)!

What happens when you plug x=iθ into the Taylor series for ex?

eiθ=1+iθ+(iθ)22+(iθ)36+(iθ)424+(iθ)5120+(iθ)6720+(iθ)75040+(iθ)840320+⋯=1+iθ+i2θ22+i3θ36+i4θ424+i5θ5120+i6θ6720+i7θ75040+i8θ840320+⋯=1+iθ−θ22−iθ36+θ424+iθ5120−θ6720−iθ75040+θ840320+⋯=(1−θ22+θ424−θ6720+θ840320−…)+i(θ−θ36+θ5120−θ75040+⋯)=cosθ+isinθ

This is called Euler’s formula, and it lets us extend exponentiation to the complex numbers:

ex+iy=ex⋅eiy=ex(cosy+isiny)

**Attribution***Source : Link , Question Author : David G , Answer Author : Michael Hardy*