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Raising a number to a non-integer power.
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Nominal Animal:
I understand your point, magic, but I disagree; not because I have a different opinion, but because I think your approach leads to problems when trying to understand  more complex mathematical "stuff".

If it was just a matter of opinion or different experiences, different opinions would be just useful, because people do think and learn in different ways.

First, a couple of points:

--- Quote from: magic on June 02, 2020, 10:21:28 pm ---What I mean is that one definition essentially is an algorithm (fastest or not), the other is some differential equation which I am supposed to know how to solve before even thinking about calculating anything.
--- End quote ---
Using a ratio to approximate a real number, then calculating the power as the denominator'th root of the nominator'th power of the value, is itself an algorithm.  Calculating an N'th root of a value is nontrivial; quite a lot of work, really.


--- Quote from: magic on June 02, 2020, 10:21:28 pm ---Even if you make the obvious fix and simply define exp as the usual power series
--- End quote ---
No, I did not, and would not.  The intuitive or real-world definition of the exponential function is "the curve which has value 1 at x=0, and slope x at x".  The power series is just one way to apply it or evaluate it: to calculate a specific point on that curve.

Now, back to the core disagreement.

OP asked how to intuitively grasp ab, when b is not an integer.  They intuitively grasp the case where b is an integer, as multiplying a by itself b times.

If I understood you correctly, your point is to just approximate b with a ratio, c/d, so that ab≃ac/d, in which case ab is approximately equal to the d'th root of ac.

I disagree with that, because it gives an incorrect intuition about the continuity and other properties of exponentials and real powers; intuitions, that will cause difficulty in understanding more in-depth mathematical concepts.

(My objection is similar to the one when teachers tell kids that electrons orbit nuclei like planets orbit the sun.  They do not.  Electrons do have properties like angular momentum and orbital radius that make the orbit model one that gives a good intuitive grasp of the properties of such electrons, but the fact is, they're delocalized in a region around the nuclei in manner better examined using quantum mechanics, and are definitely not just whizzing around like a rock around a gravity well.  It is an analog that works in one specific situation – when considering electron angular momentum and orbital properties – but is a hindrance when trying to understand anything else about atoms and molecules.  Physicists like me don't get weirded about this, because we learn to use different analogs depending on the situation, and understand that those are just tools to help us think, and not a representation of reality.)

My own suggestion is basically this:

The "best" one (that makes any further math easier to integrate to ones understanding and mathematical toolbox), is to just consider non-integer powers as an "extension" of the integer ones, with exactly the same rules and behaviour.  That is, to understand, that not every mathematical tool has an intuitive real-world analog; that requiring such intuition can hinder ones use of math.  In math, it is perfectly okay to multiply a by itself 2.1276352 times, because fractional "numbers of times" are just an extension of integer number of times, and have the exact same properties.  The fact that 2.1276352 is not countable – that is, you cannot have 2.1276352 items, because it is not a natural integer – is just completely irrelevant in this context.

The answer to exactly how to multiply something by itself a non-integer number of times, is via a mathematical identity: ab = eb loge a, where ex ≝ exp(x) is a curve that has slope x at x=x, and exp(0) = 1; and where loge x is a curve that has slope 1/x at x=x, and loge 1 = 0.  We have several different tools for calculating any point on those curves.

In fact, when you tell a current computer to calculate ab for you (in C, pow(a, b); in Python, a**b; and so on), it actually uses base-two exponent and logarithm: ab = 2b log2 a.  Mathematically, 2x = eC x and log2 y = C loge y, where C is a constant, C = 1 / loge 2.  It turns out there are very fast and efficient techniques, or algorithms, to calculate base-2 exponentials and logarithms, when numbers are expressed in binary floating-point format, m·2b.  The IEEE-754 standard defines two, Binary32 and Binary64, that use exactly this form, and these are used by almost all current computer architectures (as "float" and "double" real types, typically).  Intel x86 and AMD64 processor architectures include machine instructions that do these operations in hardware, and have had these for decades.

See?  I understand why one would see the root-of-power approach better, even more powerful, but I don't like it because of how it can affect ones further understanding of math.  I like to separate the what and the how, with an approach for understanding the what that isn't likely to bite oneself in the ass later on.

Of course, if one can take both models, and simply switch between them, they're way ahead of either of us already (since we're still here discussing which one to use), and can use such analogs themselves as tools, switching them as they need – if they need such analogs at all.  But I suspect that those people are good at math anyway, and don't need our help!
magic:

--- Quote from: Nominal Animal on June 03, 2020, 07:42:51 am ---The "best" one (that makes any further math easier to integrate to ones understanding and mathematical toolbox), is to just consider non-integer powers as an "extension" of the integer ones, with exactly the same rules and behaviour.
--- End quote ---
I agree, plus the particular rules and behaviors you want to generalize ought to be listed so that there is no doubt and confusion.

Moreover, in any remotely serious mathematics, it is also considered a good idea to convince yourself and the reader that such a function is even possible in the first place, and there is no better way than defining it and verifying that it meets all the requirements.


--- Quote from: Nominal Animal on June 03, 2020, 07:42:51 am ---The answer to exactly how to multiply something by itself a non-integer number of times, is via a mathematical identity: ab = eb loge a, where ex ≝ exp(x) is a curve that has slope x at x=x, and exp(0) = 1; and where loge x is a curve that has slope 1/x at x=x, and loge 1 = 0.  We have several different tools for calculating any point on those curves.
--- End quote ---
Yes, you can, but there is a problem: you typically cannot calculate ex exactly unless x comes in the form x=ln(y). This complicates some simplest examples possible, like 4½. At some point it's convenient to just admit that fractional powers are simply roots.

So yes, you have defined the function and provided the "how", but it's not even clear if it meets your own "what" criteria without bringing in some calculus heavy artillery, which no beginner would understand if you actually tried to explain. That's the Arduino approach to mathematics, I guess I'm just not a fan of Arduino :P

You've got me on one point: I utterly glossed over the problem of continuity / monotonicity / etc. Why would my function be continuous on the rationals? Is it even true that a½<a⅔<a¾? I believe those problems are solvable by elementary means (like converting the fractions to a common denominator, and others), which means that if you want a truly honest definition, with no cheating, handwaving and appeals to authority, this looks like a promising direction.

And yes, it will take limits to get to the irrationals. But limits are either explicit or implicit in pretty much everything dealing with the irrationals. The irrationals are pure cancer IMO and one shouldn't feel bad about treating them like they are some kind of cancer. Most of the time you too don't deal with the irrationals, you just compute e3.1415926536 instead and call it a day.

And no, I certainly didn't say to approximate an irrational with a fraction and leave it at that. That's not what limits are :P
GlennSprigg:
To me, simple INTEGERS are enough to bamboozle most!  ;D
10^24     Approx number of atoms in an average adult human being.
10^50     Approx number of atoms in our Earth.
10^80     Approx number of atomic particles in the 'known' Universe!
10^124   Approx number of Protons, if the 'known' Universe was a SOLID!!
10^100   Was named a 'Googol' just for fun, by a scientists son.
A 'GoogolPlex' was then decided to be 10^Googol...  That's just beyond stupid!  :-DD
Circlotron:
Googolplex in popular culture:

--- Quote ---"Clara was one in a million... One in a billion... One in a googolplex... The woman of my dreams, and I've lost her for all time."
—Doc Brown
--- End quote ---
T3sl4co1l:
This looks like a good point to add my "'real numbers' are the least real number system" rant...  :blah: :-DD

To know a truly real number, is to know the infinite madness that is darkness itself... to know the wisdom of The Old Ones...

Rational numbers, within reason, we can conceive of -- as long as we can put a number to it, or an approximate range, that's fine.  The algebraic numbers are a little more tricky, with more operations involved in their creation -- but still only finitely many operations, be it exponentiation (aha, see it's still on topic!) or what.

The computables are even more difficult, with such monsters as Graham's number cropping up even from relatively simple criteria.  Obviously the large numbers are quite sparse in the computables, but vastly more sparse is any other random real number -- to be able to compute a number, we must have some code, algorithm, proof, or even just proposition, which implies the existence of the number in question -- no such criterion is needed for a general real number.

The few real numbers we know, are all constructed based on their relationships.  Pi is very useful because it has many equivalent relationships, from geometry to complex analysis and beyond.  It is well known, of course never exactly as we cannot enumerate all the decimals of a transcendental real number -- but its strength is only rarely in its numerical value, and more fundamentally among the many facets of mathematics it connects.

Similarly for e, gamma, etc.

One of the few truly real, non-computable numbers that we can even begin to grasp, is: https://en.wikipedia.org/wiki/Chaitin%27s_constant  Which, because we know [upper bounds] on the first few BB(n) numbers, we have an approximate value for already -- but it's patently clear that we can't know, really much more than that, like, even a decimal more, but perhaps proofs will be developed that give just a few and then it's provable that further development (even given universe-sized recursive proofs) is realistically impossible.

Now consider that, for as many numbers as a googolplex, there are far more numbers than that, just among the rationals between 0 and 1; but compared to the number of uncomputable reals in the same range, even that infinity has measure zero, and not just laughably so, but trivially so!

P.S. My spell check wants to correct "uncomputable" to "uncomfortable".  That is understandable.  :scared:

Tim
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