5 ms·

That's the point -- you don't understand it well enough to explain it.

barrkel 15y ago
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.
- dlss 15y ago
  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.
  - baddox 15y ago
    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.
    - dlss 15y ago
      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?
      - xyzzyz 15y ago
        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) ?
  - gbaygon 15y ago
    Well explained, but 2^4 * 2^4 doesn't take into account the "different powers" part.
    - lobo_tuerto 15y ago
      2^3 == 2 * 2 * 2 2^2 == 2 * 2 2^3 * 2^2 == 2 * 2 * 2 * 2 * 2 == 2^5
      - gbaygon 15y ago
        The explanation is trivial, what i meant is that using 2^4 * 2^4 as an example will make you fail this point of the exam.
- officemonkey 15y ago
  I think "reason" here is implying "show me how you think (that is, your reasoning). In fact, I was impressed at how open-ended these questions were. For instance: "Leonidas, Pausanias, Lysander" is about as open-ended as you can get. I for one, would just create a list of interesting anagrams. (eg: Paranoia Saleslady Snide Sun) I figure it would show my moxie.
- capriotti 15y ago
  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.
baddox 15y ago
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.
- wbhart 15y ago
  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.
  - Peaker 15y ago
    That explanation is somewhat problematic with fractional exponents. The generalization of exponentiation makes the "multiply N times" explanation fail.
    - nitrogen 15y ago
      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).

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.

dlss 15y ago
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.
- baddox 15y ago
  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.
  - dlss 15y ago
    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?
    - xyzzyz 15y ago
      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) ?
- gbaygon 15y ago
  Well explained, but 2^4 * 2^4 doesn't take into account the "different powers" part.
  - lobo_tuerto 15y ago
    2^3 == 2 * 2 * 2 2^2 == 2 * 2 2^3 * 2^2 == 2 * 2 * 2 * 2 * 2 == 2^5
    - gbaygon 15y ago
      The explanation is trivial, what i meant is that using 2^4 * 2^4 as an example will make you fail this point of the exam.
officemonkey 15y ago
I think "reason" here is implying "show me how you think (that is, your reasoning). In fact, I was impressed at how open-ended these questions were. For instance: "Leonidas, Pausanias, Lysander" is about as open-ended as you can get. I for one, would just create a list of interesting anagrams. (eg: Paranoia Saleslady Snide Sun) I figure it would show my moxie.
capriotti 15y ago
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.

baddox 15y ago

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.

wbhart 15y ago
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.
- Peaker 15y ago
  That explanation is somewhat problematic with fractional exponents. The generalization of exponentiation makes the "multiply N times" explanation fail.
  - nitrogen 15y ago
    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).