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Disappointed by Definitions: Where's the Deduction?

Date: 10/17/2015 at 10:33:56
From: Nicolas
Subject: Is there a rigorous proof of negative exponents? 

Hello, 

I've read in several places that there is no real proof of negative
exponents. Rather, we just made them fit the pattern of positive
exponents.

This deficiency has always bothered me. I always just agreed with the
concept of negative exponents because I assumed higher level mathematics
proved it. But now I'm doubting that. The concept of a negative exponent
is never explained, except through some little pattern. In no way is this
a proof. The few other ways I've seen are not at all rigorous.

It really bothers me to know that we accept an entire math operation
"because, you know, it just fits a pattern and makes sense this way." If
exponents were only defined for positive integers, why define them for
negative and fractional numbers without rigorously proving that
definition?

This came up while I was working on differential calculus, which often
requires negative exponents. Now that I know that negative exponents were
"made up" by mathematicians, I feel like my calculus work means nothing.

In fact, I feel like all of math is a sham because of these definitions
that I can't find real proofs for, beginning with fractional and negative
exponents -- and that is just really depressing!

I do not desire yet another "fake" proof. I am strongly hoping I am wrong
and hope my question is answered with a resounding, rigorous proof of the
definition of negative exponents.



Date: 10/17/2015 at 11:22:43
From: Doctor Peterson
Subject: Re: Is there a rigorous proof of negative exponents? 

Hi, Nicolas.

The reason it can't be proved is not that there is a flaw, but that this
is a DEFINITION. You can't prove a definition; that is the starting point
for talking about something.

That is, a negative exponent has no meaning until we say that

  a^-n = 1/a^n

The original definition of exponents doesn't cover that case.

What we are doing is making the only definition that will allow the
properties of exponents to be retained. THIS can be proved!

So here's the rigor:

Given the definition of positive integer exponents:

   a^n = 1*a*...*a
         \______/
             n

We derive properties of such exponents, such as

   a^m * a^n = a^(m + n)

Now we ASSUME that these properties are true for an extended
exponentiation operation for which the exponent can be any integer. If
this is true (and so far we don't even have a definition of the new
operation), we can see first that

   a^0 = a^(n - n)
       = a^n / a^n
       = 1

So now we have a definition for a zero exponent (though the version of the
definition I gave above can actually be taken to apply here).

Next, we can see that

   a^n * a^-n = a^(n + -n)
              = a^0 
              = 1

Dividing both sides by a^n gives us

   a^-n = 1/a^n

NOW we have a definition for any integer as an exponent. We can't prove
that this definition is "correct," because there is nothing to compare it
to. But we have proved that it is CONSISTENT with the existing definition
and its properties -- so it makes sense to call this exponentiation.

Beyond this, we will have to extend the operation to cover fractional
exponents, and then irrational exponents, and even complex exponents and
bases, each of which will require some new thinking. In each case, it
would not be wrong to say that we are just making things fit a 
pattern -- but patterns are what math is all about; and a carefully 
defined pattern (called a property or an axiom) is not the weak thing you 
are imagining it to be.

Do you get the idea? It IS rigorously proved, just not as a proof of
something that already existed. We are creating something new, and proving
that it works well with what exists.

For a related discussion, see

  n^0 Power = 1: Defined or Proved?
    http://mathforum.org/library/drmath/view/62707.html 

By the way, if you find it depressing to invent something new because it
doesn't already exist, I supposed you were really discouraged when you
learned about imaginary numbers! The story there is just the same as for
exponents, but far more impressive.

Invention is a powerful thing.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/18/2015 at 12:30:32
From: Nicolas
Subject: Thank you (Is there a rigorous proof of negative exponents? )

Thank you, Dr. Peterson, for answering my question on negative exponents.
Your answer was thoughtful and detailed. It did give me a different way of
looking at the concept of negative exponents, but still left me with
doubts about continuing my major in mathematics.

I guess I naively thought that mathematics came from a few axioms, and the
rest was derived from those logically. The fact that so much is "made up"
diminishes mathematics, in my opinion.

In fact, I used to think mathematics was discovered, not created.
Something created is not nearly as fascinating as something discovered. If
mathematics is created and not discovered, what am I doing majoring in it?

To make matters worse, it's created with holes (inconsistencies) all over
the place: negative roots ... division by zero ... some integrals, etc.

When I was a kid, I was fascinated by math because I thought everything
flowed logically; everything about it was discovered. The idea that math
concepts are created for convenience thoroughly drains my interest in the
subject.

As for imaginary numbers, you were correct: I desperately asked my math
teachers to confirm for me that imaginary numbers were "real" 
numbers -- that "imaginary" was just a misnomer. I still haven't recovered
from that disappointment!

I must sound like some naive, confused person; but I was majoring in math
because I enjoyed its logic and rigor. Apparently, I've had the wrong idea
all along. Logic ends when things are created and defined just to fit a
pattern, or just for the sake of defining (as with imaginary numbers).

"Let's define negative roots -- which most definitely do not exist -- but
let's define them anyway.... Let's define the square root of negative 1 as
i...."

Oh, the humanity! (Hint of sarcasm, of course -- but still.)

I don't think I'll recover from all these disappointments math has turned
out to be for me.



Date: 10/18/2015 at 14:24:28
From: Doctor Peterson
Subject: Re: Thank you (Is there a rigorous proof of negative exponents? )

Hi, Nicolas.

I think you are looking at everything the wrong way. What you think you've
learned about math is not quite right -- and what you find discouraging is
precisely what makes math interesting!

My own perspective is that math is BOTH discovered and invented or created:

  Math Invented or Discovered?
    http://mathforum.org/library/drmath/view/52340.html 

  Was Mathematics Invented or Discovered?
    http://mathforum.org/library/drmath/view/52490.html 

We ask a question by inventing notation and terminology, defining what we
are asking about with some simple axioms. Then we explore the 
implications -- and find a whole world out there seemingly waiting for us
to discover. We create a seed and discover a tree. We give birth to a
child and then discover who he will be. Those things can be quite as
fascinating as a country you discover by landing there! Creating things is
not dull.

You are exactly right that mathematics comes about by deriving theorems
from axioms. But axioms can be either borrowed from the world (to define,
for example, what we think of as a plane, then study its geometry), or
just made up out of whole cloth ("Suppose we had a kind of 'number' for
which multiplication was not commutative ..."). It doesn't actually matter
whether the result exactly fits the real world. The earth, for example, is
not a plane; and in fact the universe is not "flat," but is bent out of
shape by gravity. Regardless, plane geometry -- built on a set of
assumptions that turn out not to be quite true of physics -- has a truth
of its own; and still it models most real-world geometry just fine. We
"invented" it by defining something we thought was true of the real world;
but in doing so, we made possible all sorts of "discoveries."

It also sounds like you expect to be in control of everything, even though
you think everything is being discovered. The existence of "problems" in
math is a result of the fact that it is NOT under our control, but is, to
that extent, discovered rather than invented! You might not *want* to
create a mathematical world in which you can't divide by zero, but that's
the reality you discover -- it arises from the axioms of the number
system. And when we found that there were numbers without square roots, it
was "discovered" that we can "invent" something that we misleadingly call
"imaginary" numbers (when really *any* number exists in our imagination,
rather than being made of any physical substance) -- and it works!

  History of Complex Numbers
    http://mathforum.org/library/drmath/view/69513.html 

That's exciting! As it turns out, imaginary numbers are no more imaginary
than real numbers -- and no less real. But it seemed that we "invented"
them; and in a very real sense, we did.

Moreover, complex numbers are not just things that don't exist but that we
pretend are there. In a sense, we define them into existence -- and
significantly, we can define them in a way that requires no
contrary-to-fact assumptions. They may be a construct of our minds, but
they are also a natural consequence of what real numbers are:

  Imaginary Numbers - History and Commentary
    http://mathforum.org/library/drmath/view/55747.html 

We do this in order to be able to apply logic to them. After all,
something that has no definition can't be studied logically!

So your desire to follow logic actually requires that we invent imaginary
numbers in this way, rather than just (pun intended) imagine them. This
invention gives us something amazing to study -- and then, as I mentioned,
it turns out to describe real things in physics. Is that uninteresting?

Open your eyes to what math really is, and you'll see that it is not a
disappointment, but BETTER than your youthful expectations.

Just for the sake of completeness, I'll point you to a different view of
whether math is discovered. It delves deeper into a philosophy that I
don't fully agree with. But at least you'll see that there's a lot one can
discuss in this area of "what math really is":

  Was Mathematics Invented or Discovered?
    http://mathforum.org/library/drmath/view/52291.html 

But if you end up giving up on math, it just means you aren't willing to
face reality.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/18/2015 at 16:22:01
From: Nicolas
Subject: Thank you (Is there a rigorous proof of negative exponents? )

I was not aware that I would get a reply, so I thank you for that.

I haven't given up on math, but I do fervently hope that the fascination
and wonder and enjoyment that it used to bring me take the place of this
frustration and disillusion very soon.

Perhaps, as you say, the issue rests with me and my "youthful
expectations" ...

Thank you again for further expanding on my concerns, and best wishes!




Date: 10/18/2015 at 16:35:48
From: Doctor Peterson
Subject: Re: Thank you (Is there a rigorous proof of negative exponents? )

Hi, Nicolas.

Earlier, I made a point about how the extension of exponents to negative
values must be developed in a very deliberate way. That kind of methodical
thinking stands in stark antithesis to what you and I alike might both
dismiss as random and arbitrary ("Let's define this because we feel like
it....").

The fact that we are defining something doesn't make it illogical! Math is
as logical as you ever hoped it to be. We are simply making logical
definitions.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/18/2015 at 16:55:54
From: Nicolas
Subject: Thank you (Is there a rigorous proof of negative exponents? )

At this point, I think I ought to explain why negative exponents bother me
so much. It has to do with what I was told an exponent was: a number that
tells you how many times you multiply the base times itself.

That is what even the early college level math textbooks tell you an
exponent is, right?

I loved that concept of an exponent.

As a child, I used to wonder about negative exponents a lot. For instance,
I thought that 3 to the negative two power meant that 3 is multiplied by
itself -2 times; similarly, 3 to the one-half power meant that 3 is
multiplied by itself one-half of a time. I remember years ago getting lost
in thought during many bus rides, pondering how 3 multiplied by itself
negative 2 times would be 1/9. What did it mean to multiply a number by
itself negative two times? I gave up trying to grapple with that question,
but I assumed that mathematicians knew why 3 multiplied by itself negative
2 times is 1/9.

Then I realized that the whole base-exponent relationship of positive
integer exponents did not apply to negative and fractional exponents.

That is what caused my disillusionment, and made me think, after all this
time, "Well, then ... 3^(-2) doesn't actually exist."

My issue was that I was being told that an exponent is a number
that indicates how many times to multiply the base by itself. That is
clearly not the definition of an exponent, as negative and fractional
powers have laid bare. 

So I guess I blame middle school and high school mathematics education. :)



Date: 10/18/2015 at 22:07:35
From: Doctor Peterson
Subject: Re: Thank you (Is there a rigorous proof of negative exponents? )

Hi, Nicolas.

So you were happy with an illogical statement about multiplying by itself
a negative or fractional number of times, but are not happy seeing that
the basic idea can be logically extended far beyond its origins? I'm not
sure you're being consistent when you say you love the logic of math.

Yes, students are taught many half-truths that they have to later outgrow,
because teachers don't think they can handle reality. One that has been
the subject of a lot of discussion in recent years is that multiplication
is repeated addition -- essentially the same idea at a lower level, and
causing all sorts of confusion when multiplying by a fraction or negative
number. But it's just a description of the simplest case, not an outright
lie. It's only a failure to grow past the basics that is a problem.

Learning that there is more to it than you were taught at first should not
be a source of disillusionment, but rather a fount of excitement. Reality
is bigger and better than you knew!

On the other hand, it is interesting to think about how the basic
definitions can still be seen hidden in the extended versions. Multiplying
by a -2 can be thought of as "adding -2 times", meaning "taking away 2
times"; and the -2 power can be thought of as multiplying -2 
times -- meaning divide twice.

Most interesting to me, and of relevance to our present conversation: just
as multiplying a number by -1 rotates the number line 180 degrees,
multiplying by -1 to the 1/2 power can be thought of as rotating the
number line by 90 degrees -- making imaginary numbers.

Everything ends up being consistent. And *that* is fascinating to me.

I've been sprinkling my answers with asides and links to such beautiful
details in the hopes that they may give you a taste of what you are
missing.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/18/2015 at 22:45:30
From: Nicolas
Subject: Thank you (Is there a rigorous proof of negative exponents? )

Wow ... okay, I agree: that actually was fascinating!

And do not fear: I have been paying attention to the questions you have
linked, as well as the details. In fact, I've read your answers several
times. It is all fascinating. I'm very grateful that you have taken the
time to thoughtfully and thoroughly respond to my concerns. Really, thank
you.

Who knows, maybe you'll be the person responsible for setting me off on a
career in mathematics.

... so I'll be sure to thank you when I win that Fields Medal. (I kid, of
course) :)

But I am beginning to rapidly change my initial opinions ...
Associated Topics:
High School About Math
High School Exponents
High School Imaginary/Complex Numbers

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