Imagine you are in a library and you are asked to count how many books are there. After a whole day working with your assignment, you are ready to answer:
"book, book, book, book, book, book, book, ... (half an hour later) ..., book, book, and book!"
Maybe your count was correct, but who cares? It is not practical.
When entities to be counted belong to the same category being counted, we can abridge such long enumerations by using multiplication:
"3207 books."
That's much better! So we can write
book + book = 2 ⨯ book book + book + book = 3 ⨯ book book + book + book + book = 4 ⨯ book,
and so on. The symbol ⨯ indicates multiplication and it is equivalent to say "times". For example, 2 ⨯ 4 is "two times 4".
In this case, the ⨯ symbol is not strictly necessary, so we could have written
book + book = 2 book book + book + book = 3 book book + book + book + book = 4 book,
or, of you prefer, books.
We can use letters instead of numbers, and nothing really changes:
a ⨯ 20 -----> 20a b ⨯ 351 -----> 351a ,
where we usually place the number before the letter.
If the number is just a 1, there is no need to explicitly write it,
1 ⨯ c = 1c = c .
If there are more than two members multiplying, we can collect them as
2 ⨯ 3a = (2 ⨯ 3)a = 6a 5x ⨯ 7 = (5 ⨯ 7)x = 35x .
Since the letter x is so often used in algebra, if you intend to use the multiplication symbol ⨯, try to write it in a smaller size than the letter. So instead of writing a mess like
x x x ,
try to write
x ⨯ x .
We can multiply several letters, so that we may want to sort them alphabetically, although this is by no means necessary.
b ⨯ a = ab = ba b ⨯ q ⨯ p = bpq = bqp = qpb = etc.
Usually, in algebra, we don't use two-letter symbols for a number, so if you see fg it will usually mean f times g. If you are working with computers, take into account that you may want to write f*g, otherwise fg would be interpreted as a single box. The symbol * (asterisk) is what computers use for multiplication.
Multiplication is *commutative*, meaning that a ⨯ b = b ⨯ a, so that the order in which we multiply letters and numbers does not matter. Euler says that the truth of this is self-evident, but I don't think it is so.
Imagine we have 3 ⨯ 7, which means 3 times a group of 7 elements each:
_ _ _ _ _ _ _ _ _ |_|_|_|_ |_|_|_|_ |_|_|_|_ |_|_|_|_| |_|_|_|_| |_|_|_|_|
And now consider 7 ⨯ 3, which means 7 groups of 3 elements each:
_ _ _ _ _ _ _ |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_|
In both cases, if we count one by one, we get 21 squares. How evident is that the two cases give the same result? Perhaps if draw the first set with vertical columns as well...
_ _ _ |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_|
I think it is still not evident that the two are the same. Let's draw both sets in a more packed way.
_ _ _ |_|_|_| |_|_|_| |_|_|_| |_|_|_| _ _ _ _ _ _ _ |_|_|_| |_|_|_|_|_|_|_| |_|_|_| |_|_|_|_|_|_|_| |_|_|_| |_|_|_|_|_|_|_| 3 ⨯ 7 7 ⨯ 3
OK, now the evidence starts to arise. If we turn one of the rectangles by 90 degrees, we will obtain the other.
We know the area of a rectangle is bh, meaning base ⨯ height. We also know that if we turn it by 90°, the base becomes the height and the height becomes the base, but no change in the area will be produced.
This is a geometrical approach to an algebraic property. I think it is always worthy to develop a geometrical view of algebra, and also an algebraic view to geometry!
We have seen how between numbers we are forced to place a multiplication symbol, like 3 ⨯ 4. However, between letters, it is not necessary to place any symbol, like in xy, although you can use the symbol if you want.
The reason we don't use 34 as equivalent of 3 ⨯ 4 is evident: we don't want to think this is number thirty-four.
But, if you don't like the symbol ⨯ you can use a middle dot instead, as in 3·4. It is equivalent to write 3·4 and 3⨯4.
My experience with these symbols can be summarized as follows:
I tend to use the dot even when writing in computer text like this one.
It is important to keep elegance in multiplication expressions. But what is elegant or not depends on the context. For example, c·d·5·a·7·b·8 is never elegant. It would be better to write 280·abcd if you want a compact expression. However, if you want to cancel out numbers (as we will see in division), you may find more elegant to keep 5·7·8·abcd.
In 5·7·8·abcd, each of the numbers and letters is called a *factor*, while the multiplication of all factors gives as a quantity which is called the *product*.
What about negative numbers? Can they be used in multiplications? Of course! But they are tricky, and they lead to questions that lead to the most fruitful and powerful mathematics ever seen (that of complex numbers).
Let's multiply -3 by +3. A first good thing to do is to avoid the plus signs and to always enclose negative numbers into parentheses, as in (-3)·3, or (-3)·(+3) if you wish, but not necessary.
Another good practise is to separate the multiplication of the absolute values and the multiplication of the signs. We could write it like this:
(sign of A)A · (sign of B)B = [(sign of A)·(sign of B)] |A|·|B| .
A trivial but illustrative example would be
(+2)·(+3) = (+·+) 2·3 = +6 .
We already know how to multiply numbers, but how to multiply signs? The rules are
+ · + = - · - = + + · - = - · + = - .
But how ugly is to give rules, as if they were God given laws! Let's try to understand them.
Euler proposes to think again in terms of economics. If we own +5 apples and we multiply our possessions by two, then we will own still more apples, so (+2)·(+5)= +10. This makes it clear that + · + = +.
What if we have a debt of five apples, so that we have (-5) apples? Imagine now we make our debt twice what was before, so that it is multiplied by (+2). Then, (+2)·(-5) = -10, as our intuition says: a debt increased leads to more debt, or a debt with more depth.
What about having (+5) apples and multiplying it by (-2)? Then, our economics intuition starts to weaken. What Euler suggests is to think that, since we separate multiplication of signs and numbers, (-2)·(+5) must give (sign)10. The sign can be either - or +. But we know that (+5)·(+2) and (+5)·(-2) must be different. Then, as the former gives the + sign, the latter case will render a negative sign, and (+5)·(-2) = -10. Quite an indirect path.
We still have the case of multiplying (-2)·(-5). Euler again makes an indirect argument to conclude that (-)·(-) = +.
But I think we have had enough of economics intuition. The geometrical perspective is way more powerful. Recall how we represented integer numbers as a horizontal line in which negatives are on the left and positives on the right.
If we take a compass and pin it at 0 and draw a circle, since the numbers +3 and -3 are at the same distance to the zero, both numbers will belong to the circumference of the circle:
_,...._
,' `.
,' `.
/ \
-3 -2 -1 0 1 2 3--------> integers
| |
\ /
`. ,'
`.._ _,,'
`''
Notice how in order to go from 3 to -3 we need to rotate the compass 180°. In order to go from -3 to 3 we need to rotate another 180°.
In order to stay at +3 we need to rotate 0° or 360° (or a whole number of full turns). The same for staying at -3.
Then, our powerful geometrical intuition tells us that:
Let's use this intuition for multiplying (-2)·(+3)·(-4)·(-5). We first deal with absolute values of numbers, and see that 2·3·4·5 = ... In which order will you multiply them? If you go 2·3=6, then 6·4=24, we get 24·5 as the final operation. Is there a better way for you? What about 5·4=20, then 20·2=40 and 40·3=120?
Once we know that absolute numbers give 120, let's consider the signs: (-)(+)(-)(-). Notice how between parentheses we don't need to place a multiplication symbol!
Let's begin at the number +120, since it is what the absolute numbers produce as a factor. Now take the compass, pin it at 0 and extend the other arm to +120. Then produce a 180° turn for every (-) and stay where you are for every (+). We see how the events are (turn)(stay)(turn)(turn) = turn which implies that our final result is -120.
If you are in elementary school, you may be happy remembering that equal signs multiplied give a +, while different signs multiplied give a -.
If you are someone with deeper questions, you can ask:
Why cannot get a negative number when multiplying a number by itself?
(+1)·(+1) gives +1, and (-1)(-1) gives +1. If we use letters, we can write a·a = + |a|·|a|. But isn't there some a such that a·a < 0? The answer is that YES, there exists such a! There is a mysterious number called i such that i·i = -1. Isn't this fantastic? As a clue, I can tell suggest that you consider the circle we drew before and take into account turns that are not only of 180° or 0°. Consider, for example, a turn of 90° and you will start to understand these weird numbers. But this is not the place to talk more about this.
We can not just multiply numbers and letters: we can also make products of products. For example (ab)·(cd) give abcd.
What about 36·12? Now you can tell me: "This is a product of numbers, not of products!" But if you try to multiply them, it is rather difficult. We can convert this into (3·13)·(3·4), and now it is a product of products and also an easier operation. We can do 3·13=36 and then 36·3=108. Now it is quite easy to do 108·4=432. There is a lot of subjectivity in this. Different people will prefer different approaches and all of them should be equivalent.
I hope you approach will not be "let the calculator/computer do it". Never underestimate the mind power you can get by exercising yourself in these matters. Maybe the effort seems absurd, since a machine will calculate this faster and without error, but then your mind will become weaker.
As final examples, multiply (3cd)·(5ab) = 15abcd. Also, (12pqr)(7xy) = 84pqrxy.
You may have noticed how sloppy my criterion for placing or not placing a dot is. It has been on purpose. Sometimes you want to me more redundant, other times more compact. Sometimes you have elementary school children in front of you, sometimes advanced mathematician colleagues.
I would advise to never go to very compact forms just to appear cool. Redundancy may seem to slow your calculation, but it can also prevent many mistakes.