What does 2×2^x really mean when xx is not an integer?

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 xN

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

bx=b×b×b××bx times

From this simple definition, you can observe two properties:

  • bx+y=bxby
  • bxy=(bx)y

For example:

  • 23+4=27=2222222=(222)(2222)=2324
  • 234=212=23232323=(23)4

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

When xZ

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

b0=byy=byby=1,b0
by=b0y=b0by=1by,b0

When xQ

If you assume that the multiplicate property of exponents holds for rationals, then (b1n)n=b1nn=b1=b. So b1n is a number whose nth power is b. In other words,

b1n=nb,b0

And bmn=(b1n)m=(nb)m.

For example, 40965/12=(124096)5=25=32.

When xR

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=102318.574187700290345
  • 23.14=2314/100=10023148.815240927012887
  • 23.141=23141/1000=1000231418.821353304551304
  • 23.1415=231415/1000=100002314158.824411082479122
  • 23.14159=2314159/10000=10000023141598.824961595059897

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)=xlogcb

And from the latter, you get bx=cxlogcb. 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”
  • e2.718281828459045, the base of the “natural logarithm” (ln), for its convenient properties in calculus.

So, if you wanted to calculate 2π, you’d actually calculate 10πlog102 or eπln2. And that would typically be done with the assistance of a logarithm table or a slide rule.

When xC

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=xx36+x5120x75040+=k=0(1)kx2k+1(2k+1)!
  • cosx=1x22+x424x6720+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+iθθ22iθ36+θ424+iθ5120θ6720iθ75040+θ840320+=(1θ22+θ424θ6720+θ840320)+i(θθ36+θ5120θ75040+)=cosθ+isinθ

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

ex+iy=exeiy=ex(cosy+isiny)

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

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