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What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we need to memorize? I understand that things like $\sqrt2$ could just be memorize as the thing, that when applied to itself, gives two... But this makes a lot less sense to me then thinking about things like $2+2=4$. Logarithms are a function that express a number as a power of some base, but when you get fractions for the logarithms, I don't really know what that means-What is $2^{1.2}$, or any decimal power for that matter.

Upon rereading the link posted below, I think I can repose my question a bit. If these operations are axiomatic, as I felt they were, the idea of a square root is the inverse of exponentiation. How do I grasp exponentiation intuitively? Multiplication and addition have natural roots in our minds, if exponentiation is also fundamental, what is it? Also when is exponentiation not just repeated multiplication?

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    It ain't no repeated addition. – Git Gud Nov 20 '13 at 23:10
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    Does your question more center on how the idea of multiplication as repeated addition jive with exponentiation to arbitrary real numbers? – Hayden Nov 20 '13 at 23:10
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    It's still not repeated addition. – Git Gud Nov 20 '13 at 23:14
  • I understand it isn't repeated addition. That being said, is multiplication simple another "abstract" concept we can imagine? Addition is luckily something we understand, is multiplication another? And along the same lines, is exponentiation repeated multiplication, or something more grand? –  Nov 20 '13 at 23:23
  • I just want to point out a small thing that you may not have thought about: while multiplication and addition are commutative, exponentiation is not. This means that exponentiation has two inverses $i_1$ and $i_2$. The first, $i_1$ is the operation of taking the root of a number, and second $i_2$ involves taking the logarithm. – Karl Kroningfeld Nov 20 '13 at 23:43
  • Woah... Why didn't I learn these things in grade school. –  Nov 20 '13 at 23:48
  • Although, logarithms are the inverse of putting a number in a power, while roots are the inverse of raising a number to some other number... right? –  Nov 20 '13 at 23:49
  • Exponentiation is a binary operation--it has two inputs. We could look at the curve of all $(x,y)$ pairs which solve the equation $x^y=a$, but it is normal to hold one of the inputs fixed and solve for the other to get an inverse function. The $y$th root of $a$ is the possibly nonexistent number $x$ which solves that equation. The logarithm in base $x$ of $a$ is the number $y$ which solves that equation. – Karl Kroningfeld Nov 21 '13 at 00:01
  • I see. Thanks Karl. –  Nov 21 '13 at 00:11
  • If you're comfortable with multiplication of rational numbers (are you?), then the only remaining step is to understand multiplication of real numbers. This is really just part of the broader problem of understanding real numbers in general. – Jack M Nov 21 '13 at 00:25

3 Answers3

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First, you shouldn't find expressions like $\sqrt{2}$ weird. Saying that $\sqrt[a]{b}$, where $a$ and $b$ are positive integers, is the number whose $a^\text{th}$ power is $b$ is perfectly fine. Just think about the graph of $x^a$. This is fudging a little bit, but we know $x^a$ is continuous, meaning it's a solid line, and we know it goes up to $\infty$ in the positive direction. So we know that the lines $y = x^a$ and $y = b$ intersect at some point. When you write down $\sqrt[a]{b}$ that's just notation for the $x$-coordinate of that point of intersection.

Now if you're fine with $\sqrt[a]{b}$ then you should be fine with raising a number to a fraction. When we write $x^\frac{a}{b}$ we just mean $\sqrt[b]{(x^a)}$ or $(\sqrt[b]{x})^a$ (they're the same).

After you're fine with raising numbers to fractions you might ask what does $x^a$ mean if $a$ is $\pi$ or some other number that's not a fraction. This is maybe farther than you want to go at the moment, so I won't give you a detailed answer (I'm sure this questions exists elsewhere on the site). I'll just say that $x^a$ is the number that $x^b$ gets closer to as you choose fractions $b$ that get closer and closer to $a$.

Jim
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The answer you learns at your mother's knee, that it's "repeated addition", is part of the truth. That is one manifestation of multiplication, and should probably be regarded as the most important one.

You should NEVER consider anything in mathematics just another thing you need to memorize.

$3\times4$ is the sum of three $4$s; it is $4+4+4$.

$4\times3$ is the sum of four $3$s; it is $3+3+3+3$.

There is also this item.

You can do with real numbers what the item above does with complex numbers. Here we see $4\cdot3$. $$ \begin{array}{cccc} \hline \\ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 0 & & & 1 & & & 2 & & & 3 & & & 4 \end{array} $$ Here we see $4\cdot(-3)$: $$ \begin{array}{cccc} \hline \\ -12 & -11 & -10 & -9 & -8 & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 \\ \phantom{-}4 & & & \phantom{-}3 & & & \phantom{-}2 & & &\phantom{-} 1 & & & 0 \end{array} $$ Here we see $(-2)\cdot(-3)$ and $2\cdot(-3)$: $$ \begin{array}{cccc} \hline \\ -6 & -5 & -4 & -3 & -2 & -1 & 0 & \phantom{-}1 & \phantom{-}2 & \phantom{-}3 & \phantom{-}4 & \phantom{-}5 & \phantom{-}6 \\ \phantom{-}2 & & &\phantom{-} 1 & & & 0 & & & -1 & & & -2 \end{array} $$ In the same way we could look at $(-1.5)\cdot(-3)$, etc.

Michael Hardy
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Exponents are a way of showing multiplication when what is being multiplied is being multiplied by itself. Saying $4^{1.2}$ is just saying that you are multiplying $4$ by itself $1.2$ times.

Vladhagen
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No one
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