That question stood out to me as a particularly bad question. What is the answer supposed to be? I completely understand how multiplication of exponents works, but I have no idea how to describe the "reason." You can give a simple algebraic proof quite easily (especially if we're just dealing with integer exponents), but unless "reason" had a more specific mathematical meaning in that time, it seems like a very vague question to me.
I would have to brush up a bit, but when I was in school I wouldn't have had much trouble proving it. But that's not the real issue.
My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (things get much more complicated with rational and irrational exponents)? If you're allowed to assume the definition of exponentiation, then the behavior of multiplied exponents probably follows almost trivially.
To me, this question is equivalent to asking the "reason" that 2 plus 2 equals 4. Everyone "knows why," and understands it pretty well (and could even give an intuitive "proof" by counting), but the question is poorly specified.
Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent, n say, and then multiply, you have first multiplied by the base m times, then multiplied by the same base a further n times. In total you have multiplied by the base m + n times.
Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.
Not really; a fractional exponent n yields the quantity one would have to multiply 1/n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).
How do you explain irrational exponents this way, for example? What about complex exponents?
Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well.
I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.
I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason".
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
Being able to explain proofs intuitively is a valuable way to check how deeply you know them.
In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications:
2^4 == 2 * 2 * 2 * 2
When you multiply 2^4 * 2^4, that is short hand for:
2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8
Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.
It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the power notation is short hand. But that wasn't the point of my reply. The point was that stating things multiple ways assists us in understanding. Does this not match your experience?
It still works with irrational exponents (start with fractional numbers and work towards that).
So why e^pi * e^e = e^(pi + e)? Yes, it follows from the fact that it works for rational numbers, but in order infer this, you'd need to prove the continuity of exponential function, which is nontrivial at best.
Of course, if you define a^x to be the unique continuous function f: R -> R, such that f(1) = a and f(a)f(b) = f(a+b), as soon as you proved the existence and uniqueness of this function, this follows straight from definition.
There are also different definitions of exponential functions, like exp(x) = lim n->inf (1+x/n)^n, or exp(x) = sum_{n=0}^inf x^n/n! . How easy it is to prove now that exp(pi)exp(e) = exp(pi+e) ?
I would say that "reason" in mathematics is akin to "motivation" for a definition.
In this particular case, the property a^x a^y = a^(x+y) (plus some very weak technical condition, like Lebesgue measurability) uniquely defines exponential functions.
So, in hindsight, you can think of exponentials as arising in the classification of homomorphisms from the additive group to the multiplicative group of reals.
It actually goes deeper than that. You can extend the reasoning to complex numbers (as everyone knows), to matrices, to Lie algebras, and probably beyond.
1. a basis or cause, as for some belief, action, fact, event, etc. ...
3. the mental powers concerned with forming conclusions, judgments, or inferences.
So, its the difference between "reasoning", which we do in math and logic all the time, and "belief" or "motive". That is "reason" did indeed have a specific mathematical meaning at that time, and it still does today.
