Negative Numbers

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.

A Conversation by the Seaside

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 Trip to the Bank

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.)

Negative Numbers Go in the Opposite Direction

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.

Temperatures

One case where we do use negative numbers in normal conversation is temperature, like "Today it's -40oC".

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

Eventually scientists discovered that there is a proper zero temperature called absolute zero, which corresponds to -273.15oC. So they invented the Kelvin scale, which has the same size degrees as the Celsius scale, but 0oK is absolute zero, and the freezing point of water is 273.15o and the boiling point of water is 373.15o, 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.

Walking Backward and Forward

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:

$3 + (-5) = -2$

Subtracting Negative Numbers

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:

$9 - (-6) = 15$

Multiplying Negative Numbers

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:

$-5 \times -3 = 15$