I count, therefore I count. Well, I’m paraphrasing Descartes a bit there.
For as long as there have been human beings, there have been mathematicians.
Think about it.
No one really knows where math began, but counting seems like a pretty reasonable start.
And we humans probably counted things almost from the get-go
-- the number of fingers on a hand, the number of mammoths grazing on a hill, how many siblings
we had...we could count on and on.
Which has led us to tough questions, like “how do you organize your number system
to deal with things you can’t count on your fingers?”
or “is there even such a thing as a negative number?”
Different cultures have come up with different answers.
The Mayan calendar system is well-known for its impressive accuracy,
while the Incas could record vast numerical information
-- like accurate census data -- using knotted string quipus.
And negative numbers first showed up in China during the Han dynasty as a way to solve multiple
equations at once.
In India, they were used to represent debts.
And in Europe negative numbers were called “absurd numbers” and, up until the 18th
century, dismissed as meaningless.
Basically, arithmetic -- the study of numbers and how we manipulate them -- has changed
a lot throughout history.
So today we’ll take a look at the current consensus and decide how all this relates
to algebra.
G’Day! I’m James Tanton, and this is Study Hall Algebra presented
by Arizona State University and Crash Course.
To unlock the secrets of arithmetic, we’re going to use 10 minutes to build a few of
the same realizations that human beings have had over millennia.
If establishing counting numbers, which are positive whole numbers like 1 or 2 or 77,
was the very beginning,
maybe the next leap would be combining groups of things.
So let me draw a picture.
But rather than fingers, mammoths, or siblings, which are fun but somewhat distracting, let’s
just draw dots.
Let’s start with five dots, which we can also think of as two dots followed by three
dots. 2 plus 3 equals 5.
I just read the picture from left to right, because I’m used to reading in English.
But if we read the picture from right to left, like if we were reading Arabic or Hebrew,
we see 3 + 2.
Of course, the five dots stayed the same, so 3 + 2 also equals 5.
We see this same pattern in multiplication: reading left-to-right
or right-to left doesn’t matter.
We get the same answer.
With counting numbers, “4 times 5” means “four groups of five.”
So let’s be organized and draw our “four groups of five” in rows.
And if we shift our perspective and look at the columns --
we also have five groups of four.
This is called the commutative property, or the idea that you can add and multiply numbers
in any order and get the same answer. Right now our pictures have only told us the
commutative property works with counting numbers.
So in algebra, we have to decide whether x + 2 = 2 + x, or 5*x = x*5 works no matter
what type of number x is.
(Though spoiler alert! It will.)
Of course, drawing dots has its limits.
Think about how awful it would be to have to draw picture of 37 times 34 in dots.
Even using numbers, we humans really aren’t good at picturing quantities larger than we
can count on our fingers.
But we’ve come up with ways to get by.
For instance, we can think of this multiplication problem as finding the area of a rectangle,
and then rewrite 37 as 30 + 7 and 34 as 30 + 4.
These smaller rectangles are friendlier.
30 times 30 is 900, 30 times 7 is 210, 30 times 4 is 120, and 7 times 4 is 28.
We can add all those together and get 1258, exactly as if we’d drawn 37 groups of 34
dots and counted them.
This rectangle-chopping is the heart of the distributive property,
which says if you multiply two things by a number and then add the results,
you get the same answer as adding two things up and then multiplying the result.
And we’ll see the distributive property also holds in all of algebra as well, including
for numbers like fractions and negatives.
We can use it to solve a classic algebra problem like this and figure out how to rewrite it
without parentheses.
Rectangles are again the answer! We’ll multiply everything out and then add the results.
x times x is x^2. x times 2 is 2x. x times 5 is 5x.
And 2 times 5 is positive 10. Adding the like terms together, the whole expression becomes
x^2 + 7x + 10.
It’s just like we saw when multiplying 37 and 34, only with some unknowns because we’re
dealing with algebra. We’ll revisit expressions like this later,
especially when we discuss polynomials.
But for now, remember that you can always divide and conquer.
So we’ve come up with different ways to count up. What about counting down?
Not long after people began to work with counting numbers, we developed the concept of negative
numbers:
losses, debts, or days in the past.
If I owe my sister a sheep, and then I acquire two sheep, at the end of the day I’m left
with just one lonely sheep.
This is him. His name is Dale.
Today we call positive and negative whole numbers, like 5, 0, and -47, the integers.
In modern mathematics, to keep track of our debts, we represent a negative number by attaching
a dash to its left.
But if we keep stacking up negatives, that gets a little confusing.
What’s “negative negative one” or “negative negative negative one”?
To think about this, we can go back to dots for a moment and represent negative numbers
as anti-dots that cancel out dots.
So “negative one” appears as the opposite of a dot: an anti-dot.
Negative negative one would be the opposite of the opposite of a dot: a dot.
And negative negative negative one is the opposite of the opposite of the opposite of
a dot: back to an anti-dot.
Using dots, let’s look at 5 + (-3). We see some dots and antidots cancel out and we’re
left with two dots.
And if we have something like five minus negative one, that looks like 5 plus the opposite of
negative one.
So this is 5 plus the opposite of an anti-dot, or six dots.
The idea that subtracting a negative is the same as adding a positive can get tricky when
it intersects with the distributive property,
especially when you have variables in play.
Let’s look at this through the Math Goggles!
Step right up, here it is folks, the fastest hands in the west! You won’t believe your
eyes!
Here we have black cards and we have red cards, 10 of each!
Shuffle ‘em up and we get two piles -- count ‘em, 1! 2! with 10 cards in each.
Now watch closely here -- there’s nothing up my sleeve! -- I say... the number of black
cards on the left is the same as the number of red cards on the right.
GASP! Someone get that poor lady some smelling salts, she was overwhelmed by the wonder…
of math!
Can you see the trick?
If we want to represent this with algebra, let’s say that B is the number of black
cards in the left-hand stack.
Then the remaining 10 – B black cards are in the right-hand stack.
Because there are 10 cards in the right-hand stack and 10 –
B of them are black, the number of red cards in the right-hand stack must be 10 – (10
– B).
Let’s turn our subtraction into addition, like before.
This is really 10 plus the opposite of 10-B, or 10 plus negative 10 plus B.
We just take the opposite of each term.
Ah! The 10s must cancel and we’re just left with B.
This makes sense for two reasons.
First, we know we have to have positive numbers of red cards and black cards on each side;
a negative B means nothing here. Second, we’ve already seen in our demonstration
that the number of black cards in one stack is equal to the number of red cards in the
other stack,
and the left-hand stack has B black cards. But now we have proved with the power of algebra
that this will always be the case!
Okay, algebra can be pretty tricky...but we can handle it.
So we can add and subtract negative numbers.
And multiplying them doesn’t seem so bad as long as only one of them is negative.
If I’m given –5 * 4, it feels a little weird to think of making –5 groups of four,
or adding four –5 times. But 4 * –5 is easier.
Four groups of five anti-dots is twenty anti-dots. Thanks, commutative property.
But when you multiply a negative by a negative, we’ve all been told you get a positive.
That seems kind of arbitrary.
Let’s look at an example to try and explain why this must be true.
We can set up 17 times 18 like we did earlier7, where we multiply each bit separately and
then just collect them together at the end.
10 times 10 is 100, 10 times 7 is 70, 10 times 8 is 80, and 7 times 8 is 56. Add all those
up, and you get 306.
But, what if we wrote 17 and 18 as (20 – 3) and (20 – 2)?
The sides of the rectangle are still the same: 17 and 18. We just changed the way we wrote them.
20 times 20 is 400, 20 times –3 is –60, 20 times –2 is –40, and …
let’s leave –2 times –3 a mystery for now. So 400 plus –60 plus –40 plus our
mystery M must equal 306, or 300 plus M equals 306.
So the product of –2 and –3 has to be positive 6 because we never changed the area,
just the way we wrote our numbers. It’s weird!
There’s no denying that’s weird.
But there is just no way that it can be anything else.
A negative times a negative has to be a positive for all the rest of mathematics to hold together.
Who knew negative numbers were so important?
When I first learned these ideas I took them at face value and thought they’d existed
since the dawn of time.
But our ideas about math are constantly changing as we learn more about the world.
And in algebra, we need to pay attention to these patterns that emerge since our variables
can be any type of number.
Next time we’ll look at another system of numbers where the rules may seem strange:
the world of fractions. Until then... cheers!
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the Crash Course team at Complexly.
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