Hey it’s Professor Dave; let’s take a look at negative numbers.
We learned about simple subtraction, whereby if you have ten apples, and someone buys three
of them from you, you have seven left.
Ten minus three equals seven.
But what happens if someone wants more apples than you have?
Let’s say someone wants fifteen apples.
You could promise them fifteen apples, saying you’ll get five more tomorrow, so in a sense
you could say that after the transaction you have negative five apples, which is five fewer
apples than no apples.
This is because once you get more, you immediately owe five of them, so it’s like you will
need five apples just to get to zero apples.
To make sense of this, let’s look at a number line.
Until now we’ve been looking at the positive side, which continues climbing all the way
to infinity.
But on the other side of zero, if we continue subtracting, we start with negative one and
we can go all the way to negative infinity.
This helps us understand the magnitude of negative numbers, because numbers to the right
on the number line are always greater than numbers to the left.
This means that negative two is greater than negative five, which can be counterintuitive
because five is greater than two.
But if you picture the number line in your head it will always make sense.
Think of your bank account.
If you overdraft, your balance will be in the negative, and if you owe the bank ten
dollars, you have more money than if you owe twenty dollars, because you will need less
money to get back to zero.
Now let’s try some operations with negative numbers.
If we are doing addition, adding a negative number is the same as subtracting the positive
version of that number.
Five plus negative two is the same as five minus two, which gives us three.
If we are doing subtraction, subtracting a negative number is the same as adding the
positive version of that number, because the minus signs cancel each other out.
This cancellation isn’t magic, it actually makes perfect sense, because taking away a
negative amount is like adding.
If your bank account is overdrawn by ten dollars, and the bank were to forgive or subtract that
overdraft, you’d suddenly be at zero, which is ten more than negative ten, so it’s like
the bank gave you ten dollars.
So five minus negative two is the same as five plus two, which gives us seven.
We can even verify this on the number line, as there are indeed seven units between negative
two and five.
Incorrectly subtracting negative numbers is a very common source of error for math students,
so make sure you don’t make these kinds of mistakes.
With multiplication and division, if one of the numbers is negative, the answer will be
negative.
Five times negative two is negative ten, which makes sense, since adding negative two five
times will give us negative ten.
Similarly, fifteen over negative three is negative five.
However, if both numbers are negative, again the minus signs cancel each other out and
the result will be positive.
Negative five times negative two is positive ten.
Negative fifteen divided by negative three is positive five.
This makes sense if we think of each number as negative one times the positive version
of the number.
Negative one times fifteen divided by negative one times three is clearly positive five,
because the negative ones cancel out, since anything over itself is one.
Little tricks like this make it easier to remember all kinds of arithmetic identities
and help us go beyond simple memorization, which by itself doesn’t always lead to a
deep appreciation of mathematics.
Now that we know how to perform operations with negative numbers, let’s check comprehension.