So far, we've been working a lot with the number line. Now it's time for some pictures in a different dimension. Imagine, if you will, a bunch of boxes.
If p and q are positive, the multiplication p × q means "take p groups of q things and see how many things you end up with." This can be pictured nicely with a rectangle divided into smaller boxes.
3 × 4
Here's one group of 4 things:
...and here's 3 groups of 4 things:
Count up all the things (each of the small boxes) and we get 12. Notice that another way to think of this picture is that we took a rectangle with one side of length 4 and one side of length 3, and then found the area of the rectangle. But that way's not as much fun, because then we don't get to overuse the word "things."
Even if the numbers aren't all whole numbers, a similar diagram will still work.
3.5 × 6.2
Simply draw a rectangle with side lengths 3.5 and 6.2, and 3.5 × 6.2 is the area of that rectangle. If we're thinking about it in a real-life scenario, imagine that we have 3.5 of something and 6.2 of something else. Let's just hope they're not living things, or this could get messy.
If we switch the order of the numbers being multiplied, our rectangular box gets turned onto one end (apparently the multiplication symbol didn't notice the "FRAGILE" warning written on the side of it), but the final answer is still the same.
4 × 3 = 12
This means that multiplication is commutative: the order in which we write the numbers doesn't matter. They can "commute" past each other without changing the final answer.
a × b = b × a
Man, a and b sure are lucky they're at the beginning of the alphabet. They get to be in everything.
Multiplication is also associative: we don't care how the numbers associate with each other. In symbols, for real numbers a, b, and c:
(a × b) × c = a × (b × c)
At least c got a little love that time. We still can't help but feel bad for w though.
5 × 2 × 3
With multiplication, we can start from either end. Either way we slice it, we'll still get the same answer.
(5 × 2) × 3 = 10 × 3 = 30
5 × (2 × 3) = 5 × 6 = 30
If one or more of the numbers we're multiplying together has a negative sign, we first multiply the absolute values of the numbers together. Then, for each negative sign, we reflect our answer to the other side of the number line. If we come across enough negative signs, we might find ourselves doing more reflecting than Lindsay Lohan during her next jail sentence.
2 × (-3)
First multiply 2 by 3 to get 6. Since we have just one negative sign, we reflect 6 to the other side of the number line once to get our final answer: -6. Kind of hard to draw the rectangle for that one, so we recommend that you don't even try.
(-6) × (-4)
First multiply 6 by 4 to get 24. Since we have two negative signs, we reflect 24 across 0 and back again. Our final answer is positive 24.
There are several different symbols used for multiplication. We can write "a times b" as:
a × b
Be Careful: 4 × - 3 doesn't mean anything. It might have some sentimental value to you, but it certainly doesn't mean anything mathematically: "4 times subtract 3?" What in the world is that? Let's put parentheses around the second term to keep track of the negative sign:
4 × (-3)
Also keep in mind that it's really easy to multiply by the number 1. We say 1 is the multiplicative identity, because multiplying by 1 allows numbers to keep their identities. Even if some of them are a little embarrassed by their identities and would prefer to trade them in. Like 37, for example. Terrible self-image.
1 × 8 = 8
9 × 1 = 9
1 × 1 = 1
-π × 1 = -π
In symbols, we say that if n is any real number, then:
1 × n = n × 1 = n
Any real number n, except for 0 (sorry, 0, you'll get 'em next time), has a multiplicative inverse or reciprocal, written 1/n. This is the number that, if we multiply it by n, we get back to 1.
What's the multiplicative inverse of 2?
The multiplicative inverse of 2 is , because . Got two bagel halves? We bet they'll make a full bagel if you press them together.
So why doesn't 0 have a multiplicative inverse? Can't a number get a break around here? The number 0 is special in multiplication, because 0 "kills" everything. Wow. Real nice, 0. That's the last time we ever stick up for you.
For any real number n:
0 × n = n × 0 = 0
This means there can't be any real number n where 0 × n = 1. Zero kills everything because if we take n groups of zero things, or zero groups of n things, we don't have any things at all. Or at least that's the defense zero's lawyer is using at trial.
This can be very useful. If you're multiplying a lot of numbers together and one of the numbers happens to be 0, you don't have to do any work. Since anything multiplied by 0 is 0, all you have to do is write down 0 as your answer! Or, if you're taking the SAT, fill in that 0-shaped bubble next to the 0.
789,234 × 67,623,746,374 × 23,432,432 × 0 × 234,872,384,723 = 0
That lone 0 in the mix turns the entire product into 0, no matter how gnarly the other terms are.