One of the tricky things you encounter in Algebra is **negative numbers**, like
-1 and -24.

What makes negative numbers particularly tricky, is that *we don't use them in normal
conversation*, and on the few occasions when we do use them (like for *temperature*, which I'll
explain in more detail below), it's for something where
the difference between negative and positive is somewhat arbitrary.

To give an example, think of a young man spending a day by the seaside, who has just met an attractive young lady who is lying on the beach, somewhat curiously dressed in a very long dress, so long that her feet aren't even showing. After engaging her in conversation, the young man discovers that she lives in a house with her Dad, "Keeng" Neptunious, in a large house with a wonderful view. He suggests that he would like to go to her house and meet her Dad, but she replies that it might not be a good idea because he would have difficulty breathing. Thinking of thin air and mountain sickness, the young man asks "How high is your house above sea level?"

And the young lady replies, her house isn't *above* sea level, it's 223 metres *below*
sea level.

You will notice that the young lady hasn't answered the young man's question – How high is the house
above sea level? – instead she has answered a different question: How high is the house
*below* sea level?

But, if the young lady had been more mathematical, and willing to answer questions in Mathematics-speak,
she could have answered the original question by stating that her house is -223 metres *above* sea level.

A man goes to the bank with his wife, to investigate the state of their finances. Unbeknown to his wife, he has been spending too much money. The wife asks the bank clerk: "How much money is in our account?", and the bank clerk replies "Well, actually, you have an overdraft of $256.32".

But, if the bank clerk had been able to talk Mathematics-speak, he could have answered the original question directly by saying "There is $-256.32 in your account". (Actually people in banks don't talk Mathematics-speak, they talk Accounting-speak, which has its own peculiarities involving the words "credit" and "debit", but we won't go into that here.)

There's a general pattern in these examples: when we want to measure something in one direction, but actually it goes in the opposite direction, we can still measure it in the original direction, if we are willing to use negative numbers to state our measurement.

That's not something that we do in normal conversation, which as I said earlier, explains why
negative numbers can be tricky to understand. To get the hang of them, we have to be willing
to sound a bit silly, and use negative numbers, *instead* of reversing the direction and
continuing to use positive numbers.

One case where we *do* use negative numbers in normal conversation is temperature, like
"Today it's -40^{o}C".

The curious thing about temperature is that the location of zero is somewhat arbitrary. For
example, in the **Celsius** scale, 0^{o} is the freezing point of water and
100^{o} is the boiling point of water. In the **Fahrenheit** scale, the freezing
point of water is 32^{o} and the boiling point is 212^{o}, and the location
of 0^{o} is 32 degrees below the freezing point.

Eventually scientists discovered that there is a proper zero temperature called **absolute zero**,
which corresponds to -273.15^{o}C. So they invented the **Kelvin** scale, which has the
same size degrees as the Celsius scale, but 0^{o}K is absolute zero, and the freezing point
of water is 273.15^{o} and the boiling point of water is 373.15^{o}, and there aren't
any negative temperatures at all. Well actually you *can* have negative temperatures, but only
in certain special situations, and they are actually hotter than the positive temperatures, and we
are getting into very esoteric phyics if we discuss them any further so I won't.

The upshot is, that although temperatures are the one thing that we talk about using negative
numbers, they aren't really a very good example to help you *understand* negative numbers.

A good example for understanding negative numbers is walking backward and forward, and it's an even better example if we imagine someone taking steps that are always the same length.

For example, if the person steps *backward* 3 steps, we can say that they walked *forward*
-3 steps. And if they step *forward* 5 steps, we can say that they stepped *backward* -5 steps.

If I walk 3 steps forward and then 5 steps forward, it's the same as if I walked 8 steps forward, because:

\[3 + 5 = 8\]

If I walk 3 steps forward and then 5 steps *backward*, I'll end up 2 steps behind where I started,
i.e., as if I walked 2 steps backward. To describe this example as an addition, I need to describe
all my walking in terms of steps *forward*. So I can say that if I walk 3 steps forward and then
-5 steps forward, it's the same as if I had walked -2 steps forward, because:

\[3 + (-5) = -2\]

If I walk 9 steps forward and then 6 steps backward, it's the same as if I had walked 3 steps forward, because:

\[9 - 6 = 3\]

If I walk 9 steps forward and then 6 steps forward, it's the same as if I has walked 15 steps forward. To turn that into a subtraction problem, I have to describe the 6 steps forward as -6 steps backward. That is, if I walk 9 steps forward and then -6 steps backward, it's the same as if I had walked 15 steps forward, because:

\[9 - (-6) = 15\]

If I walk 3 steps forward 5 times, then in total I will have walked 15 steps forward. This corresponds to the multiplication:

\[5 \times 3 = 15\]

If I walk 3 steps *backward* 5 times, then in total I will have walked 15 steps backward. Restating
this in terms of steps *forward*, I can say that if I walk -3 steps forward 5 times, then in total
I will have walked -15 steps forward, corresponding to the multiplication:

\[5 \times -3 = -15\]

We could also say, since 3 steps forward is the opposite of 3 steps backward, that
walking 3 steps backward 5 times is the same as walking 3 steps *forward* -5 times, corresponding
to the multiplication:

\[-5 \times 3 = -15\]

The holy grail of multiplying negative numbers is to work out what happens when you multiply
two numbers together and they are *both* negative. And to solve that problem, we can return
to the first example in this section, i.e. walking 3 steps forward 5 times. Because 3 steps forward
is the opposite of 3 steps backward, we can say that this is the same as walking 3 steps backward
-5 times. But 3 steps backward is the same as -3 steps forward. So we can say that I walked
-3 steps forward -5 times, and the total number of steps I walked forward is 15 steps (remember,
I'm not changing anything about the example, I'm only changing how it's *described*).
And this corresponds to the multiplication:

\[-5 \times -3 = 15\]