I wanted to quickly comment on a couple of mathy things

that have been floating around the internet, just

so you know I'm still alive.

So there's this video that's been floating around

about how to multiply visually like this.

Pick two numbers, let's say, 12 times 3.

And then you draw these lines.

12, 31.

Then you start counting the intersections-- 1, 2,

3 on the left; 1, 2, 3, 4, 5, 6, 7 in the middle;

1, 2 on the right, put them together, 3, 7, 2.

There's your answer.

Magic, right?

But one of the delightful things about mathematics

is that there's often more than one way to solve a problem.

And sometimes these methods look entirely different,

but because they do the same thing,

they must be connected somehow.

And in this case, they're not so different at all.

Let me demonstrate this visual method again.

This time, let's do 97 times 86.

So we draw our nine lines and seven lines

time eight lines and six lines.

Now, all we have to do is count the intersections-- 1,2, 3, 4,

5, 6, 7, 8, 9, 10.

OK, wait.

This is boring.

How about instead of counting all the dots,

we just figure out how many intersections there are.

Let's see, there's seven going one way and six

going the other.

Hey, let's do 6 times 7, which is-- huh.

Forget everything I ever said about learning

a certain amount of memorization in mathematics being useful,

at least at an elementary school level.

Because apparently, I've been faking my way

through being a mathematician without having

memorized 6 times 7.

And now I'm going to have to figure out 5 times 7, which

is half of 10 times 7, which is 70, so that's 35,

and then add the sixth 7 to get 42.

Wow, I really should have known that one.

OK, but the point is that this method breaks down

the two-digit multiplication problem

into four one-digit multiplication problems.

And if you do have your multiplication table memorized,

you can easily figure out the answers.

And just like these three numbers

became the ones, tens, and hundreds place of the answer,

these do, too-- ones, tens, hundreds-- and you

add them up and voila!

Which is exactly the same kind of breaking down

into single-digit multiplication and adding

that you do during the old boring method.

The whole point is just to multiply every pair of digits,

make sure you've got the proper number of zeroes on the end,

and add them all up.

But of course, seeing that what you're actually doing

is multiplying every possible pair

is not something your teachers want you to realize,

or else you might remember the every combination concept

when you get to multiplying binomials,

and it might make it too easy.

In the end, all of these methods of multiplication

distract from what multiplication really

is, which for 12 times 31 is this.

All the rest is just breaking it down

into well-organized chunks, saying, well, 10 times 30

is this, 10 times 1 this, 30 times 2 is that, and 2 times 1

is that.

Add them all up, and you get the total area.

Don't let notation get in the way of your understanding.

Speaking of notation, this infuriating bit of nonsense

has been circulating around recently.

And that there has been so much discussion of it is sign

that we've been trained to care about notation way too much.

Do you multiply here first or divide here first?

The answer is that this is a badly formed sentence.

It's like saying, I would like some juice or water with ice.

Do you mean you'd like either juice

with no ice or water with ice?

Or do you mean that you'd like either juice with ice or water

with ice?

You can make claims about conventions

and what's right and wrong, but really the burden

is on the author of the sentence to put in some commas

and make things clear.

Mathematicians do this by adding parenthesis and avoiding

this divided by sign.

Math is not marks on a page.

The mathematics is in what those marks represent.

You can make up any rules you want about stuff

as long as you're consistent with them.

The end.