Author Topic: Raising a number to a non-integer power.  (Read 2580 times)

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Offline Circlotron

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Raising a number to a non-integer power.
« on: May 27, 2020, 06:00:11 am »
Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?
 

Online greenpossum

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Re: Raising a number to a non-integer power.
« Reply #1 on: May 27, 2020, 06:09:31 am »
It's just the extension in the real number domain of taking the exponent. Mathematics is like that, analogies can only go so far.

But if you like, think of it as 4^(39/10) so take the 39th power of 4 and then the 10th root. Things get tricky to imagine when the exponent is irrational.
« Last Edit: May 27, 2020, 06:12:29 am by greenpossum »
 
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Offline Ian.M

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Re: Raising a number to a non-integer power.
« Reply #2 on: May 27, 2020, 06:14:52 am »
I'm no mathematician so the terminology here is going to give a real mathematician fits.  Without getting into logarithms, or huge powers of four, break that down to:
 43 x root10(4)9

root10(n) is the tenth root of n, i.e, the function where if:
 m=root10(n)
then:
 m10=n

That approach is valid for any rational real number.

As GreenPossum points out, its not any help with irrational numbers. so letting X be the exponent, although it gives you a function with an infinite number of points along any interval of the X axis, it isn't continuous.   To get the  'in-between' irrational values you have to consider exponentiation as multiplying the logarithm of n by the power X, then taking the antilog, which works in any base of logarithms, and is a continuous function, numerically identical to the 'classic' multiplying numbers by them-selves X times concept for any real rational X.

Where it gets gnarly to conceptualise is when you start extending exponentiation YX into the complex number domain for both Y and X.   I cant wrap my brain around the number of dimensions required to visualise that plot - its bad enough visualising the complex result of a function operating on a single real number parameter, which is a 3D plot.
« Last Edit: May 27, 2020, 06:38:44 am by Ian.M »
 

Offline RoGeorge

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Re: Raising a number to a non-integer power.
« Reply #3 on: May 27, 2020, 07:51:55 am »
Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?

Short answer, the exponential function is defined as a polynomial.  Explanation starts at minute 7:00, at about 7:50 is exactly your question.

 
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Offline tom66

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Re: Raising a number to a non-integer power.
« Reply #4 on: May 27, 2020, 08:57:22 am »
Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?

Because log10(222.861)/log10(3.9) = ~4.  And that is the division of two base10 logarithms.  You can easily define log10(n), and you have inferred what the solution to your non-integer power is with nothing more than division.

Mathematical operations often have an inverse.

Try fractional and negative factorial if you -really- want to blow your mind.
 

Offline magic

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Re: Raising a number to a non-integer power.
« Reply #5 on: May 27, 2020, 09:00:17 am »
How can you multiply a number by itself a non-integer amount of times?
You don't. But you can say that you wish to have a function with the following convenient properties, arbitrarily chosen to mach the behavior of normal exponentiation:

an = n times a·a·a·..·a
ax+y = ax·ay
ax·y = (ax)y
be a continuous function over all real numbers because why not

The first condition defines ax for positive integers.

The second condition forces to choose the following for nonpositive integers:
a0 = 1
a-n = 1 / an

The third condition forces the following on rationals:
a1/n = nth root of a

The fourth condition leaves little choice for the reals because the rationals are dense in the reals:
ax = lim (axn), for any infinite sequence xn of rationals such that lim (xn) = x

Simple :phew:
 :-DD

Short answer, the exponential function is defined as a polynomial.
Certainly not. The exponential rises asymptotically faster than any polynomial.
x100 is not exponential growth, contrary to what most journalists would want you to believe these days.

TL;DW but your video either discusses infinite sums of polynomials which aren't really polynomials anymore, locally valid approximations or is simply wrong.
 
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Offline Kleinstein

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Re: Raising a number to a non-integer power.
« Reply #6 on: May 27, 2020, 09:43:56 am »
One can use the logarithm and exponential to rewrite the power:
 A^B =  exp( B * ln(A))

In this form there is no problem having B with a non integer value. It just gets messy if A is negative as this would lead to complex numbers.
 
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Offline Brumby

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Re: Raising a number to a non-integer power.
« Reply #7 on: May 27, 2020, 10:25:42 am »
This is a good example of how such an expression actually does make sense:
But if you like, think of it as 4^(39/10) so take the 39th power of 4 and then the 10th root.

You can then take this to the extreme for any numbers.  For example: 3.839^5.776 would be the 1/1000th root of 3.839^5776.  Each of these steps is pretty straightforward, even if laborious to solve.

By following the principles involved, it becomes apparent that exponentiation encompasses a broad scope where integer exponents are just a special case.
 

Offline T3sl4co1l

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Re: Raising a number to a non-integer power.
« Reply #8 on: May 27, 2020, 11:38:22 am »
Understood more generally, and perhaps apparent if you've watched the 3b1b video -- we use an algebraic expression (exponentiation of ordinary numbers) to imply an algebraic (or higher level, e.g. transcendental) equation, which may have zero, one or many solutions.

Of note, taking the 10th root implies there are 10 possible, equivalent, solutions: using the 10 roots of unity.  Remember that sqrt(4) = +/- 2, not just +2!  If we constrain our answer to the real domain, only two of those remain; but in the complex domain, all ten are equally valid.

When working with general analytical functions, we often must be mindful of this, not just to finite numbers of answers, but whole spaces of them -- for example there are infinite "branches" of the ln(z) function, because e^(i x) is periodic.  (When this fact isn't interesting, we usually choose the "main branch" (x ∈ (-pi, pi]), denoted Ln(z).)

Similar answers pop up for expressions like x^x, or i^i.  While you can easily solve these for a superficial curve or value, the functions they are equivalent to lie in much more interesting spaces, and there are many equivalent answers, not just one!  (I think one of the follow-up videos in the 3b1b series covers exactly this, if you're interested.)

Tim
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Bringing a project to life?  Send me a message!
 

Online nctnico

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Re: Raising a number to a non-integer power.
« Reply #9 on: May 27, 2020, 12:42:24 pm »
#include "math.h"

float a=pow(4.0, 3.9);
There are small lies, big lies and then there is what is on the screen of your oscilloscope.
 

Online SiliconWizard

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Re: Raising a number to a non-integer power.
« Reply #10 on: May 27, 2020, 03:05:05 pm »
I guess already explained above.
It's the exponentiation, and is explained there: https://en.wikipedia.org/wiki/Exponentiation
An integer exponent is a particular case of exponentiation.
 

Offline mark03

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Re: Raising a number to a non-integer power.
« Reply #11 on: May 27, 2020, 03:12:31 pm »
What is this number i that you speak of?
 

Offline T3sl4co1l

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Re: Raising a number to a non-integer power.
« Reply #12 on: May 27, 2020, 03:17:43 pm »
What is this number i that you speak of?

You must be 'j'oking ;D

Tim
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Electronic design, from concept to prototype.
Bringing a project to life?  Send me a message!
 
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Offline TimFox

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Re: Raising a number to a non-integer power.
« Reply #13 on: May 27, 2020, 07:11:36 pm »
The best joke I encountered in a physics lecture was by the late Professor Ugo Fano at the University of Chicago (ca. 1976).
He was demonstrating the quantum calculation of dielectric polarization, expressing the macroscopic result as the expectation value of the quantum calculation.
He used quantum perturbation theory to relate the polarization to the applied E field, expressing the Hamiltonian (essentially the energy) as a function of E and the harmonic binding of electrons.
He then performed a Fourier decomposition of E(t) into frequency components, an integral over w (I can't find lower-case omega here) of E(w) ei wt dw .
One of the theory weenies in the first row objected, "Professor Fano, that Hamiltonian is not Hermitian!", by which he meant that the energy must be real and not complex.
Prof. Fano erased the "i", replacing it by "j" and declared the Hamiltonian was now Hermitian.
 
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Offline TimFox

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Re: Raising a number to a non-integer power.
« Reply #14 on: May 27, 2020, 07:14:17 pm »
I also learned that the formal definition of exponentiation in modern mathematics is the one involving logarithms, which is required when the exponent is not a rational fraction.  It also allows raising a variable by a complex exponent.  It reduces to the usual expression when the exponent is an integer or fraction.
 

Offline Alex Eisenhut

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Re: Raising a number to a non-integer power.
« Reply #15 on: May 27, 2020, 07:18:20 pm »
For that matter, how can you multiply a number by itself a negative number of times?
*Except AC/DC adapters on eBay. Avoid them all!
 

Offline RoGeorge

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Re: Raising a number to a non-integer power.
« Reply #16 on: May 27, 2020, 07:53:31 pm »
(I can't find lower-case omega here)

EEVblog forum can render LaTex notation.  Lower case Omega can be written as
Code: [Select]
\$\omega\$
That will be displayed as \$\omega\$.

Offline daqq

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Re: Raising a number to a non-integer power.
« Reply #17 on: May 27, 2020, 08:18:44 pm »
What is this number i that you speak of?

You must be 'j'oking ;D

Tim
He probably just imagined it.
Believe it or not, pointy haired people do exist!
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Offline Kleinstein

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Re: Raising a number to a non-integer power.
« Reply #18 on: May 27, 2020, 08:33:17 pm »
For that matter, how can you multiply a number by itself a negative number of times?

This one is easy:  the negative exponents give the inverse. So  A^(-B)  =  1 / A^B.
So multiply -2 times is the same as dividing 2 times.
 

Offline magic

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Re: Raising a number to a non-integer power.
« Reply #19 on: May 27, 2020, 09:24:55 pm »
One can use the logarithm and exponential to rewrite the power:
 A^B =  exp( B * ln(A))
Yeah, I guess you can do that, but then you need to define exp(x) as something other than ex to avoid circular definition.

It can be done, there are weird-ass formulas which resolve to ex without explicitly mentioning e itself and taking powers of it.
But actually teaching it that way is an exercise in applied obfuscation.

I also learned that the formal definition of exponentiation in modern mathematics is the one involving logarithms, which is required when the exponent is not a rational fraction.
Not required. I have defined it without a single stinkin' logarithm ;)
 

Offline TimFox

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Re: Raising a number to a non-integer power.
« Reply #20 on: May 28, 2020, 02:54:58 am »
No, your definition only works for real numbers, using the limit of a sequence of rational numbers to deal with the irrational case.
The logarithm method also works for complex numbers.
 

Offline Nominal Animal

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Re: Raising a number to a non-integer power.
« Reply #21 on: May 28, 2020, 03:25:48 am »
Actually, the form ab = 2b log2 a is useful, as it is something a computer can do in binary directly, and quite efficiently for base-2 floating point numbers. Details.
 

Online vk6zgo

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Re: Raising a number to a non-integer power.
« Reply #22 on: May 28, 2020, 05:12:50 am »
From 30+ years ago (& it was revision then), the classic formula for fractional indices is a^p/q = the qth root of  a^p
I was hoping to use the calculator, but the HP22S doesn't seen to have the capability of powers or roots to other than base 2 or e.

I could use the logs, but to hell with it!
 

Offline magic

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Re: Raising a number to a non-integer power.
« Reply #23 on: May 28, 2020, 12:57:38 pm »
No, your definition only works for real numbers, using the limit of a sequence of rational numbers to deal with the irrational case.
The logarithm method also works for complex numbers.
Fair enough, off the top of my head I can't provide a simple algebraic argument why complex exponents should be defined the way they are. That doesn't necessarily mean no such argument exists. I don't know what reasoning originally led to the Euler identity and things like that - perhaps the Taylor series, maybe something more direct and straightforward.

That being said, I totally expected OP to stop reading at the first limit if he even made it that far, so who cares about C ;)
 

Offline TimFox

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Re: Raising a number to a non-integer power.
« Reply #24 on: May 28, 2020, 01:25:36 pm »
In my youth, if I needed more accuracy than from my slide rule, I used 5-place logarithm tables (and log-trig tables) to do my serious computations, so the logarithm definition comes naturally to me.   We referred to the Chemical Rubber Company "Handbook of Mathematical Tables" as the Rubber Bible.  (Formally, the "Mathematical Tables from the Handbook of Chemistry and Physics")  7-place tables were available, but that was overkill.
 


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