## The Meaning of Negative Numbers

Negative numbers are numbers
that are below 0. Now what kind of sense does it make to talk about
numbers that are below 0? Well, you have heard of temperatures being
below zero, but even that is kind of artificial because if you used
the Kelvin scale, you wouldn't have to deal with them anyway. There
are other things that kind of correspond to negative numbers like
below sea level, but it's not like you're gong to deal with
temperature or elevation all of the time. Perhaps a better reason to
have negative numbers is the purely mathematical one, that it makes
our number system more complete and easier to work with.
Particularly when you are doing algebra and are working with unknown
quantities, it is nicer if most if not all of the operations you can
write down are do-able, and negative numbers allow you to subtract
bigger numbers from smaller numbers. This comes up in the real world
too, but it isn't a very pleasant subject. If you have 5 dollars and
you spend 7 dollars then you are in debt 2 dollars, which could be
represented as having -2 dollars.

We can represent negative numbers on the number line by writing
them to the left of 0, since on the number line smaller numbers are
written to the left of larger numbers. When you are comparing
negative numbers with each other you must be careful, because the
ones that look bigger are really smaller. If you are 7 dollars in
debt you have less money than if you are 5 dollars in debt.

## Opposites

Another good reason for having negative numbers is
that it allows addition to have something that is called an inverse,
which is also helpful when you study algebra. Every number has a
number that can be added to it to get zero, and this number is
called its opposite. To find the opposite of a number you simply
change its sign, so the opposite of a positive number is a negative
number, and the opposite of a negative number is a positive number.
The symbol for opposite is the same as that for negative so this
could be a little confusing. But it doesn't really matter that much
to determine which meaning of the minus sign is appropriate, because
they sort of are the same thing anyway. If you wanted to, you could
simply think of all of the minuses as meaning opposite, because when
it is being used to indicate a negative number, that number is the
opposite of the positive number that the minus sign is on anyway, so
it comes to the same thing. But all you really have to do is
interpret the minus sign as indication of negativeness whenever that
makes sense, and otherwise take it as meaning opposite. This
basically means that the first minus sign on a number means negative
and any further ones mean opposite.

## Addition

What adding signed numbers is really about is
combining adding and subtraction into one operation just as
multiplying of fractions is combining multiplication and division
into one operation. Adding positive numbers is adding, and adding
negative numbers is subtracting. If you want to think of it on the
number line you start from 0 and when you add a positive number you
go that much to the right, and when you add a negative number you go
that much to the left.

### Rules for Adding

If you work this out case by case, you can
come up with the rules for adding plus and minus numbers. Looking at
the possibilities for combinations of signs, you can see that there
are 4 possibilities.

#### + +

Nothing new here, just add as usual.

#### + -

You are going to the right and then to the left,
so they are fighting with each other, and you don't get very far.
Ignore the signs and subtract the bigger one minus the smaller one.
The bigger one wins out as far as which direction you are going, so
it determines the sign of your answer.

#### - +

You are going to the left and then to the right, so
again they are fighting, so just like in the last case, you ignore
the signs and subtract the bigger one minus the smaller one and the
bigger one wins and determines the sign of your answer.

#### - -

Now you are going to the left and then going to the left
again, so to determine how far you are going you ignore the signs
and add, and since you will end up to the left of 0, the sign of
your answer will be negative.

Actually you can collapse these rules into two cases, and this
gives you a simpler statement of the rules.

- Like Signs - Add and the sign is the common sign.
- Different Signs - Subtract and use the sign of the bigger one.

## Absolute Value

You might have noticed that I referred a
couple of times to ignoring the signs in the above explanation. To
avoid having to talk about ignoring the signs in such explanations
and other times when it is called for, it is convenient to have a
name for the number stripped of its sign, and the name given for
this is absolute value. The absolute value of a number is simply the
number without its sign. This means that if the number is positive,
the absolute value of it is just itself, but if it is negative it is
the number you get when you strip it of its minus sign, that is, the
corresponding positive number. On the number line you can think of
absolute value also as the distance from 0. The notation for the
absolute value of a number x is |x|. So for example |7|=7, but
|-4|=4. The answer to an absolute value evaluation is always
positive.

## Subtraction

With signed numbers we really don't need
subtraction, but sometimes real world or intuitive reasons lead us
to thinking about a problem in terms of subtraction, so we need a
definition that would fit this for subtraction of signed numbers.
Our original subtraction of positive numbers like 7-5 can be take
care of by adding a negative, since 7+-5 give the same answer as
7-5, so we can generalize this to a definition for subtracting any
two signed numbers. So to subtract two signed numbers you change it
into an addition problem by changing the second number to its
opposite. If you know about division of fractions, you might notice
that this is very much like the case there where you divide my
multiplying by the reciprocal. To subtract, change the sign of the
second number and add. And that's really all there is to it. You
just have to remember to do it, and that takes practice.

One thing that I have noticed can be confusing with subtraction
is that the negative sign is the same symbol as the subtraction
sign, so you have to make sure you don't get them confused or make
one do double duty. One way to keep this straight is to draw a
circle the two number and a different colored circle around the
subtraction sign. To do this, first circle the first number. Then
the next minus sign you see is the subtraction sign, so circle that
in a different color. The any further minus sign must be a negative
sign attached to a number, so circle whatever is left with the first
color. Then to change the subtraction to addition, do this. First
copy the first number, the one in the first circle. Then write an
addition sign in place of the original subtraction sign, the one in
the different colored circle. Then write down the opposite of the
next number, the one that is circled in the same color as the first
number.

### Example 1:

3-7
### Solution:

### Explanation:

The two numbers are circles in red and the
operation sign in blue. Write down the first number, the 2. Then
change the operation to addition. Then change the sign of the second
number. It was a positive 7, so now it becomes a -7. Now for the
addition problem we have different signs, so they are fighting with
each other, one number telling us to go forward and the other number
telling us to go backward, so we subtract and get 4, but since the
negative was the bigger one, it is a -4.

### Example 2

-4-9
### Solution:

### Explanation:

This time the first number is -4 and the second
number is 9, circled in gold with the minus sign circled in green.
First write down the -4. Then change the operation to addition and
write the addition plus sign down. Then change the sign of the
second number the 9, which will make it a -9. Now for the addition
problem we have two negative numbers to add. Since they have the
same sign, they both want us to go in the same direction, so we add
them to get 13. But since they are both negative numbers, it is a
negative 13.

### Example 3:

6-(-7)
### Solution:

### Explanation:

This time the first number is 6, and the second
number is -7. The parentheses aren't really necessary here. They are
just here to make it easier to read by keeping the minus signs from
running together. Then we change the problem to an addition problem.
First write down the 7. Then change the subtraction to addition, so
write down a plus sign for that. That change the sign of the second
number from negative to positive to make it a positive 7 instead of
a negative 7. Now for the addition, it is just adding to positive
numbers, which we already knew how to do before learning about
negative numbers.

### Example 4:

-5-(-6)
### Solution:

### Explanation:

Again the parentheses aren't really necessary.
The two numbers are -5 and -6, so to change it to an addition
problem we are adding -5 and 6. For that addition problem we have
different signs, but this time the positive one is bigger, so we
subtract and get a positive answer.

## Multiplication

Now lets look at the various combinations of
plus and minus for multiplication of signed numbers.

#### + +

No negative numbers here, so nothing new.

#### + -

Multiplying by positive numbers means repeated addition,
so the same thing should be true when you multiply it by a negative
number. The repeated addition of a negative number gives a negative
number and the absolute value, that is the size without the minus
sign, is simply the product of the two absolute values. This means
when you multiply a positive number times a negative number, you
multiply the two numbers ignoring the signs, and the sign of
the answer is negative.

#### - +

We want multiplication to be commutative, so this should
be done the same way as +-.

#### - -

This one is slightly trickier to understand, and I've
never seen a convincing physical interpretation of it. Mainly you
have to accept this on the basis that it is the only mathematically
consistent way to define it given the other definitions. There are a
number of ways to think about it. If -+ is - then -- somehow has to
be something different, so it must be +. Or you can think of it as,
since -+ is - then multiplying by a minus must change the sign, so
in a minus times a minus the first minus must change the sign of the
second one to plus. An interesting more formal way of seeing it is
to use the distributive property. Take an example with numbers to
make it friendlier.

-5(6+-6)=(-5)(0), but also by the distributive
property

-5(6+-6)=(-5)(6)+(-5)(-6)=-30+(-5)(-6).

So whatever (-5)(-6) is, when you add it to -30 you have to get
0, and the only thing you can add to -30 and get 0 is 30. Anyway,
however you see it, it seems that the only possible thing for the
product of two negatives to be is a positive. I know two wrongs
don't make a right, but strange as it seems, in mathematics the
product of two negatives is indeed a positive.

I think one important thing to think about if you get bothered by
the idea of negative times negative being positive is that
multiplication is really a much more complicated operation than
addition, and it is definitively not the same as addition. If you
are thinking this can't be true because two wrongs don't make a
right, you need to realize that combining two wrongs is adding them,
not multiplying. Multiplying two negative numbers is something
different. If multiplication is repeated addition, what does it mean
to repeatedly add something a negative number of times? That doesn't
really literally make a lot of sense. Before you can decide what a
negative times a negative should be, you have to first decide what
is meant by multiplying by a negative number. To some extent we
define negative times negative without really thinking about this,
and just defining it the only way it would make sense given the
above considerations, but if we were to give some thought to what
multiplication by a negative means possibly the best way to think
about it would be as repeated subtraction. Since multiplying by a
positive number is repeated addition, it would make sense to think
of multiplication by a negative number as repeated subtraction, and
that indeed would make the product of two negatives a positive,
since subtracting a negative is the same as adding a positive.

Extra for Experts: If you are a more advanced student or
instructor or parent, who has learned about quadratic equations, and
you would like to learn about another reason that minus times minus
is plus, read my article A Geometrical Approach to Completing the
Square.

Again just like with addition we can make this easier to remember
by collapsing it down to just two cases, and here it is really much
simpler that with addition, because with multiplication you always
multiply, so all you have to worry about is what the sign of the
answer will be.

- Always multiply
- Like signs, sign is +
- Different signs, sign is -

### Big Products

For multiplying it is also interesting to see
what happens when you multiply several different numbers. What
happens then is every time you have two minus signs the get together
and make a plus. so for each minus sign the answer flip flops
between - and +, so to determine the sign of your answer you just
need to count up the minuses and see whether it is even or odd. If
it is even the answer is +, and if it is odd the answer is -.

### Multiplication and Addition

It is interesting to compare the
rules for multiplication with those for additions so that you don't
get them confused.

- For multiplication you always multiply, but for addition
sometimes you add and sometimes you subtract.
- For both multiplication and addition you do different things
depending on whether the signs are like or different.
- For multiplication like signs mean the answer is +, and
for addition like signs mean you add and the sign is the
common sign.
- For multiplication different signs mean the answer is -, and
for addition different signs means you subtract and the sign
is the sign of the larger.
- Multiplication ++=+, Addition ++=+.
- Multiplication +-=-, Addition +-=the sign of the larger.
- Multiplication -+=-, Addition -+=the sign of the larger.
- Multiplication --=+, Addition --=-.

### Powers

Powers mean repeated multiplication, so from the
rules for multiplying you should be able to raise numbers to powers.
When you raise negative numbers to powers you can also use our rule
about counting the minus signs. When you raise a negative number to
an even power, then you have an even number of minus signs, so the
answer is +. when you raise a negative number to an odd power there
are an odd number of minus signs, so the answer is -.

### A little Notational Matter

There is a little thing you have
to be careful with in the notation of negative numbers raised to
powers. When you wish to denote a negative number raised to a power
you always enclose it in parentheses. So when you wish to write -2
raised to the 5th power, you write it

the reason for this is that because
multiplying by -1 puts a minus sign on a number we think of the
minus signs on numbers as equal to multiplying in the order of
operations. That means that powers get done before minus signs get
attached to numbers, so if you write

this doesn't mean -2 is being raised to
the 5th power, instead because the power gets done first, it means
that 2 is getting raised to the 5th, and then the minus sign is
attached to that number.

## Division

The same rules about signs hold for division as for
multiplication. Also if you have a problem that is all
multiplication and division, you can just ignore the signs and then
figure out what the sign of the answer is by counting up the minuses
and if it is even the answer is plus and if it is odd the answer is
minus. But be careful, you can only do this if the problem is all
multiplication and division.

## More Examples and Practice

The MathHelp collection of
problem sets

*Integers* will give you some more examples
and practice with addition, subtraction, multiplication, and
division of integers.

### Links

Smith, D. E. History of Mathematics Volume I and Volume
II

This is one of the few books on the history of mathematics with
an emphasis on elementary mathematics. It is also the only book I
have found that has much information about the history of negative
numbers. Volume II has a chapter called Artificial Numbers that is
quite interesting.