If multiplication is not repeated addition, what is it? How do you define multiplication?
All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to think of it as repeated addition?
Voodoo magic?
The first thing you should consider is that there are, in some sense, different "hierarchies" of numbers. At each stage, we enlarge the class of numbers, and try to do so in a way that leaves everything we could do before still the same, but that now we can do more.
A common way of doing this is to start with the natural numbers (sometimes called "counting numbers" or positive integers). We start with $1$, $2$, $3,\ldots$.
Here, we do define multiplication as repeated addition. For example, one way to define multiplication is to assume we know how to add, and then define it by saying: $$\begin{align*} n\times 1 &= n\\ n\times (k+1) &= (n\times k) + n % need to edit at least 6 characters \end{align*}$$ Using mathematical induction, we can show this defines multiplication for all positive integers, and that it has the usual properties we know (commutative, so that $n\times k = k\times n$ for all positive integer $n$ and $k$, distributes over the sum, is associative, etc).
Then we have two choices for "expanding our universe of numbers": we can now define negative integers, by considering things that would help us solve all equations of the form $a+x=b$ with $a$ and $b$ positive integers; or we can introduce positive rationals (fractions) by considering all things that would help us solve all equations of the form $ax = b$. Let's do the latter, since that came first historically.
So, we had the positive integers, and we knew how to add and multiply them. Now we are going to have more numbers: now, for every pair of positive integers $a$ and $b$, we will have a number "$\frac{a}{b}$", which is a number that satisfies the property that $$b\times\left(\frac{a}{b}\right) = a.$$ We also say that $\frac{a}{b}$ is "the same fraction" as $\frac{c}{d}$ if and only if $ad=bc$ (here we are comparing products of positive integers, so that's fine).
We also notice that our old positive integers can also be considered fractions: the positive integer $a$ is a solution to $1x = a$, so $a$ corresponds to the fraction $\frac{a}{1}$.
Now, how do we add two of these numbers? Since $\frac{a}{b}$ represents the solution to $bx=a$, and $\frac{r}{s}$ represents the solution to $sx=r$, then $\frac{a}{b}+\frac{r}{s}$ represents the solution to something; to what? A bit of algebra will tell you that it is the solution to precisely $(bs)x = (as+br)$. So we define $$\frac{a}{b}+\frac{r}{s} = \frac{as+br}{bs}.$$ There's a bit of work that needs to be done to ensure that if you write the fractions differently, the answer comes out the same (if $\frac{c}{d}=\frac{a}{b}$, and if $\frac{t}{u}=\frac{r}{s}$, does $\frac{cu+td}{du} = \frac{as+br}{bs}$? Yes). And we also notice that if we add positive integers as if they were fractions, we get the same answer we did before: $$\frac{a}{1} + \frac{c}{1} = \frac{a1+c1}{1} = \frac{a+c}{1}.$$ That's good; it means we are enlarging our universe, not changing it.
How about products? If $\frac{a}{b}$ represents the solution to $bx=a$, and $\frac{r}{s}$ represents the solution to $sy=r$, their product will be the solution to $(bs)z = ar$. So we define $$\frac{a}{b}\times\frac{r}{s} = \frac{ar}{bs}.$$ And then we notice that it extends the definition of multiplication for integers, since $\frac{a}{1}\times\frac{b}{1} = \frac{a\times b}{1}$. And we check to see that multiplication and addition still have the properties we want (commutativity, associativity, etc).
(There are other ways to figure out what multiplication of fractions "should be", on the basis of what we want it to do. For example, we want multiplication to extend multiplication of integers, so $\frac{a}{1}\times\frac{b}{1}$ should be $\frac{ab}{1}$; and we want it to distribute over the sum, so we want $$\frac{a}{1} = \frac{a}{1}\times \frac{1}{1} = \frac{a}{1}\times\left(\underbrace{\frac{1}{b}+\frac{1}{b}+\cdots+\frac{1}{b}}_{b\text{ summands}}\right) = \underbrace{\left(\frac{a}{1}\times\frac{1}{b}\right) + \cdots + \left(\frac{a}{1}\times\frac{1}{b}\right)}_{b\text{ summands}}.$$ So $\frac{a}{1}\times \frac{1}{b}$ should be a fraction which, when added to itself $b$ times, equals $a$; that is, a solution to $bx=a$; that is, $\frac{a}{b}$. And so on).
Then we move on from the positive rationals (fractions) to the positive reals. This is more complicated, as it involves "filling in gaps" between rationals. It is very technical. But what it turns out is that for every real number you can find a sequence of rationals $q_1,q_2,q_3,\ldots$ that get progressively closer to each other and to $r$ (we say the sequence "converges to $r$"); it won't hurt too much if you think of the $q_i$ as being progressive decimal approximations to $r$ (they don't have to be, and ahead of time you don't have any notion of decimal approximation, but you can think of it that way for our purpose). So then the way we define multiplication of real numbers $r$ and $s$ is to find a sequence of rationals $q_1,q_2,q_3,\ldots$ giving the approximation to $r$, and one $p_1,p_2,p_3,\ldots$ giving the approximation to $s$, and we define $r\times s$ to be whatever it is that the sequence $$p_1\times q_1,\ p_2\times q_2,\ p_3\times q_3,\ \ldots$$ approximates. This ensures that if you take rational numbers and multiply them as if they were reals you get the same thing as if you multiply them as rationals, and likewise for integers.
So multiplication of positive reals is really a series of approximations made up by multiplying rationals; and multiplication of rationals is really a way to codify solutions to certain equations with integers; and it is only multiplication of (positive) integers that really corresponds to "repeated addition".
Finally, once you have the positive reals, you can introduce the negative real numbers. We consider solutions to equations of the form $a+x=b$ with $a$ and $b$ positive real numbers. Some of these already have solutions, some don't. This gives us "zero" and "negative reals". We then extend the definition of multiplication to "zero" and to "the negative reals" in a way that makes sense relative to this definition. Turns out we need to make $0\times r = 0$ for all $r$, and have to respect the "rules of signs" to make sure everything still works. So we define it that way to make sure everything works and what we had before still works exactly the same.
$n(a+b)=\displaystyle\sum\limits_{i=1}^n a+b= \displaystyle\sum\limits_{i=1}^n a + \displaystyle\sum\limits_{i=1}^n b = na + nb$?
If we consider all the variables as positive integers to simplify the derivation, of course.
– Curiosity Sep 15 '11 at 02:45At Willie Wong's suggestion, I post my comment as an answer.
Keith Devlin has written a few columns at the MAA site in recent years explaining why he thinks it's a bad idea to introduce multiplication to children as repeated addition. See this and the other essays cited there.
Multiplication of positive integers is defined as repeated addition. However, this doesn't help us define multiplication by a negative number, or a rational, or anything else -- you can't add three to itself minus four times, or half of a time. This means we have to start from whole-number multiplication and then "fill in the gaps" to define multiplication for larger classes of numbers.
I gave a long response to this question as part of an answer to another question. The basic point I want to get across is that repeated addition does not generalize in the way that multiplication generalizes; for example you'd be hard-pressed to think of multiplication of complex numbers or matrices as repeated addition. That is because what they really are are compositions of functions.
In particular, multiplication of real numbers is (in my opinion) better thought of as compositions of scalings of the real line. This generalizes immediately to multiplication of complex numbers, which are compositions of scalings-and-rotations of the plane.
(Another point I want to get across is that it is completely unnecessary, even when dealing with natural numbers, to define multiplication as repeated addition.)
To understand multiplication, discover it yourself (see 2nd section below)!
Starting point: The natural numbers $\mathbb N$ with only one binary operation called addition.
Before long, you'll find it both useful and convenient to define multiplication. For example, you store many cans of Campbell's soup in your ultra-organized pantry.
You see that two cans are in the front and you have three rows. Without the use of a calculator, you 'know' that you have six cans. You like this so much that you set up a notation for what it means to multiply two natural numbers $a$ and $b$:
${\displaystyle a\times b=\underbrace {b+\cdots +b} _\text{a-times}}$
You call the 'thing being multiplied', $b$, the multiplicand, and the other 'thing', $a$, the multiplier. You are not bothered at all if either $a$ or $b$ is equal to $0$ - you see the result as a big fat zero (no soup).
You like to think about things in an abstract way, and you discover something that 'knocks your socks off' - multiplication distributes over addition:
$\tag 1 a \times (b + c) = (a \times b) + (a \times c)$
You now think that perhaps multiplication can be viewed as a binary operation in its own right. You then ask if the distributivity law 'gives you back' multiplication as a uniquely defined arithmetic operation. You decide to add the 'law', $\, 1 \times 1 = 1$, and testing some more,
$\quad 2 \times 2 = 2 \times (1 + 1) = $
$\qquad 2 \times 1 + 2 \times 1 = $
$\qquad (1 + 1) \times 1 + (1 + 1) \times 1 =$
$\qquad 1 \times 1 + 1 \times 1 + 1 \times 1 + 1\times 1 = $
$\qquad 1 + 1 + 1 + 1 = 4$
This is a great start! You realize that for your 'laws of arithmetic' it will be both convenient and useful to include multiplication, an operation that is 'tied to the hip' with addition via (1).
The link
What Exactly is Multiplication? \ Keith Devlin
that Gerry Myerson provides contains the quote
multiplication is complex and multi-faceted
and the author suggests that to understand multiplication, you need to form a personal 'mental amalgam' with a heuristic, like 'multiplication is scaling'.
I like to look at the original geometrical sources: A non-negative number is the length of a line segment, the sum of two numbers is the combined lengths of the segments, and the product is the area of the rectangle with the segments as the sides. I know this has its problems (it's surprisingly hard to get the integers from the reals), but I like to look at things from different points of view.
Devlin is profane - do not trust him!
Multiplication is repeated addition ($N = n·u= u+u+u+...$, unit is preserved) , but there is another thing called product.($u×u = u^2$, unit is changed).
Euclid had problems with this issue too, but clearly he was aware of it. Euclid speaks of product $A\times B$ as plane number, and of product $ A \times B \times C$ as solid number.
René Descartes resolved the issue completely.
These terms is often(almost always) misused, because $|A × B| = |A| · |B|$.
Hermann Hankel and George Peacock raised some theory called "The Principle of Permanence of Equivalent Forms", or just the "The Principle of Permanence".
Below are some quote from here:
An explicit formulation of the principle of permanence appears in George Peacock's 1833 “Report on the recent progress and present state of certain branches of analysis”, where it is expressed in the following way: “Whatever form is algebraically equivalent to another when expressed in general symbols, must continue to be equivalent, whatever those symbols denote.” This “Principle of Permanence of Equivalent Forms”, as Peacock called it, also stipulates the following: “Whatever equivalent form is discoverable in arithmetical algebra considered as the science of suggestion, when the symbols are general in their form, though specific in their value, will continue to be an equivalent form when the symbols are general in their nature as well as in their form.”2 For example, if m, n and a denote any integers, as they do in what Peacock called arithmetical algebra, then it is the case that
ma + na = (m+n)a. Arithmetically equivalent forms like this one are said to have a “necessary” existence (Peacock, 1833, 199), for they are true in virtue of the definitions of the basic operations (like addition and multiplication). Permanence demands thatma + na = (m+n)aremains an equivalent form in what Peacock called symbolic algebra, wherem,nandamay denote anything whatsoever.
So, based on the above theory, the result and laws we obtained from the experience of counting, should still hold true in other symbolic scenarios.
As to multiplication, both addition and multiplication are just some operations that map two elements to one element. I think you can define any operations on a set to be addition or multiplication. As long as the operations respect the laws you get from arithmetic because of the "Principle of Permanence". The x and + are just two symbols to represent some operations.
When we do multiplicaton for natural numbers, we need to remember all the results of combinations of 1~9. That's all we need to remember to carry out multiplication in decimal system. If you change a system, you may need to remember a totally different set of results.
Kronecker once declared that “Natural numbers were created by God, everything else is the work of men” - the point being that all else is constructed so that it 'fits'.
Addition follows naturally from combining sets, and then it isn't hard to see that you can take part of a set away, which gives you subtraction. But how do you take more away than there is? So we invented negative numbers because it made it possible to subtract a larger number from a smaller.
And multiplication starts naturally with repeated addition, of course, but then you observe that if you have a set consisting of a whole number of smaller sets, you can also go the other way - but that gives you the problem that you can't always do this; thus we invented the rational numbers. And so on. The rules for multiplication of rational numbers, distributive laws etc are invented to make everything work nicely.
The real and complex numbers are 'merely' thinking this through and refining the concepts in the extreme.
This can be summed up as 'algebraic convenience': all operations 'want' to have an inverse, and we end up with groups theory, ring theory etc., and in the end something like the category theoretical definition of the product for multiplication and the coproduct being addition, after which there isn't much hope left.
Especially when we have 0.3 * 0.2, This is basically a fraction multiplied with a fraction... But how to interpret the operation of multiplication here? As scaling? Between 0 and 1 it could be thought as division... Am I making any sense?
I am just quite confused.
– Curiosity Sep 14 '11 at 13:35