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.