Adding and Subtracting Integers (Positive and Negative Numbers)

Adding and subtracting integers is a critical skill for advanced math and algebra  -- and it's a skill that we find many students are not very good at!

It's worth investing time and practice in learning how to add and subtract negative numbers quickly and accurately. 

There are many ways to add and subtract integers as well as a number of tricks.  See our alternate methods for the most common "trick" that students learn.  But, ideally you should learn how to add and subtract positive and negative numbers conceptually.  That way -- you don't have to remember a crazy set of rules, you just have to understand what you're doing!

First of all, adding and subtracting integers includes adding and subtracting positive numbers.  It might be helpful to think of all numbers as numbers with a positive (+) or negative (-) sign from now on.  Remember any number that does not have a sign in front of it is positive (we just often drop the positive sign).  So,

$$\eqalign{3+ 4&=\\^+3 + ^+4&=}$$

Likewise:

$$\eqalign{8 - 5&=\\^+8 - ^+5&=}$$

You know the answers to both of these problems: 

$$\eqalign{3 + 4&=7\\^+3 + ^+4&=7}$$

And,

$$\eqalign{8- 5&=3\\^+8 - ^+5&=3}$$

Once you start to work with integers, one of the fun things you get to do is forget the rule that says you can't subtract a bigger number from a small number.  Of course you can, it just means that you end up with a negative number!

What is a negative number, exactly?

There are lots of ways to think of a negative number.   Some people like to picture a ladder standing in a hole in the ground.  When you are standing on the ladder at ground level, you are at 0.  When you climb up a rung, you are at positive 1 (+1) and the higher you go, the higher your positive number.  But, if you climb below ground, you go into negative numbers, first negative 1 (-1) then negative 2 (-2) and so on.

We prefer to think of positive and negative numbers in terms of money.  Postive numbers represent money that you have.  A positive 5 (+5) is five dollars in your pocket.  A negative 5 (-5) is five dollars that you owe to someone else.  0 is when you have no money at all.  If you have 5 dollars and someone gives you another dollar, you just add, 5 + 1 = 6 and you have six dollars.  But if you owe money, you have to pay off your debt before you can start having money in your pocket.  So, if you owe 5 dollars (-5) and someone gives you a dollar, you lessen your debt, but you still owe money: -5 + 1 = -4.   Now, you only owe 4 dollars (-4).

Let's try a problem and let's try to think about it logically:

$$\eqalign{-10+ 5&= &&\text{You owe 10 but someone gives you five}}$$

Logically, if you combine your debt of 10 and your new 5 dollars, do you have money or do you still owe money?

You still owe, correct, so you know that your answer will be negative.  So, if you owe 10 and pay off 5 of it, how much do you still owe?

$$\eqalign{^-10+ ^+5&= &&\text{You owe 10 but someone gives you five}\\^-10 + ^+5&=^-5&&\text{If you owe 10 and pay off five, you owe five more}}$$

You can also think of this type of problem as a number line.  Start at -10, then hop 5 spaces to the right (adding means move to the right), and you land on -5.

$$\eqalign{&^-10\Large{\text{ }\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}\normalsize{^-5}\\&\large{\overline{\mathbf{^-10} \text{  }^-9\text{  }^-8 \text{  }^-7\text{  } ^-6\text{  } \mathbf{^-5}\text{  } ^-4\text{  } ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

Based on this logic, there are basically four types of operations:

You have some money (+) and someone gives you more money (+):

$$\eqalign{^+3+ ^+5&= &&\text{You have 3 and someone gives you five}\\^+3 + ^+5&=^+8&&\text{If you have 3 and someone gives you 5 more, you have 8}}$$

Here's the number line.  Start at +3, then hop 5 spaces to the right (adding means move to the right), and you land on +8.

$$\eqalign{&^+3\Large{\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}\normalsize{^+8}\\\large{\overline{^-10 \text{  }^-9\text{  }^-8 \text{  }^-7\text{  } ^-6\text{  } ^-5\text{  } ^-4\text{  } ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  }}} &\large{\overline{^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

You have a debt (-) and someone gives you money (+):

$$\eqalign{^-3+ ^+5&= &&\text{You owe 3 but someone gives you five}\\^-3 + ^+5&=^+2&&\text{If you owe 3, and if you get 5, you can pay that off and have 2 left}}$$

Here's the number line.  Start at -3, then hop 5 spaces to the right (adding means move to the right), and you land on +2.

$$\eqalign{&^-3\Large{\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}\normalsize{^+2}\\\large{\overline{\mathbf{^-10} \text{  }^-9\text{  }^-8 \text{  }^-7\text{  } ^-6\text{  } \mathbf{^-5}\text{  } ^-4\text{  }}}&\large{\overline{ ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

You have some money (+) and takes money away (-) or gives you a debt (-):

Once you start working with positive and negative numbers, it's almost always easier to think about combining positive and negative numbers, rather than adding or subtracting.  When you think of it that way, someone taking away your money (-) and someone giving you a debt (-) are the same: they are combining your money with a negative. 

Essentially, whenever you see a minus sign, just attach it to the number it's in front of.  So, if you see $10-5$ just stick the minus sign on the five and you are combining $10$ and $-5$

$$\eqalign{&^+2-5= &&\text{You have 2 and someone takes away 5} &&&^+2 + ^-5= &&&&\text{Or you have 2 and someone gives you a debt of 5}\\&^+2 - 5=^-3&&\text{You owe 3} &&&^+2 + ^-5=^-3 &&&&\text{You owe 3}}$$

No matter how you think of this problem (as subtracting a positive, or adding a negative), you are subtracting, which means your number is going down (to the left on the number line).

Here's the number line.  Start at +2, then hop 5 spaces to the left (subtracting means move to the left), and you land on -3.

$$\eqalign{&^-3\Large{\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}\normalsize{^+2}\\\large{\overline{^-10\text{  }^-9\text{  }^-8 \text{  }^-7\text{  } ^-6\text{  } ^-5\text{  } ^-4\text{  }}}&\large{\overline{ ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

You have a debt (-) and someone takes money away (-) or gives you more debt (-):

Here is another instance where you can think of subtracting, or think of adding a negative.  Think about it in whichever way makes more sense to you.

$$\eqalign{&^-1-5= &&\text{You owe 1 and someone takes away 5} &&&^-1 + ^-5= &&&&\text{Or you owe 1 and someone gives you a debt of 5}\\&^-1 - 5=^-6&&\text{You owe 6} &&&^-1 + ^-5=^-6 &&&&\text{You owe 6}}$$

No matter how you think of this problem (as subtracting a positive, or adding a negative), you are going farther into debt, into the negative, so you will move to the left on the number line.

$$\eqalign{&^-6\Large{\text{ }\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}\normalsize{^-1}\\\large{\overline{^-10 \text{  }^-9\text{  }^-8 \text{  }^-7\text{  }}}&\large{\overline{^-6\text{  } ^-5\text{  } ^-4\text{  } ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

You have a debt (-) and someone takes away a debt (- - = +):

What? This is what happens when you subtract a negative. Think of it this way, if someone takes away a debt, it's like adding.  You owe someone 5, then that person says "nevermind, I don't want the money back" and you just went up 5 bucks.  It's like someone took away a debt.  And, in math, that means, you add.  So, whenever you see a minus sign and a negative sign touching, you can turn them into a plus sign).

$$\eqalign{^-3-^-3&= &&\text{You owe 3 and someone takes away the debt of 3} \\^-3 + 3&=&&\text{Taking away a debt is like adding!}\\^-3 + 3&=0 &&\text{If you owe 3, but take away that debt of 3, you go back up to 0!}}$$

Here's the number line:

$$\eqalign{&^-3\Large{\curvearrowright \curvearrowright \curvearrowright}\normalsize{0}\\\large{\overline{^-10 \text{  }^-9\text{  }^-8 \text{  }^-7\text{  }^-6\text{  } ^-5\text{  } ^-4\text{  }}}&\large{\overline{ ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

You have a money (+) and someone takes away a debt (- - = +):

This also works when you have money.  Let's say you have 5 dollars.  Well, actually you have 8 dollars, but you know that you owe your sister 3 bucks, you so just think of it as 5 dollars.  Then, your sister tells you, you know what, you don't have to pay me back that 3 dollars you owe me.  That's her taking away a debt of 3 dollars.  And now, you actually have 8 bucks!  By her "taking away the debt," she added to your money.  Here's what it looks like in math:

$$\eqalign{^+5-^-3&= &&\text{You 5 and your sister takes away the debt of 3} \\^+5 + 3&=&&\text{Taking away a debt is like adding!}\\^+5 + 3&=8 &&\text{If you have 5, and someone takes away that debt of 3, you go back up to 8!}}$$

Here's the number line:

$$\eqalign{&^+5\Large{\curvearrowright \curvearrowright \curvearrowright}\normalsize{8}\\\large{\overline{^-10 \text{  }^-9\text{  }^-8 \text{  }^-7\text{  }^-6\text{  } ^-5\text{  } ^-4\text{  } ^-3\text{  } ^-2 \text{  }^-1 \text{  }0 \text{  }^+1\text{  } ^+2\text{  } ^+3\text{  } ^+4 \text{  }}}&\large{\overline{^+5\text{  } ^+6\text{ } ^+7\text{  }^+8\text{  } ^+9 \text{  }^+10}}}$$

Overall, adding and subtracting with integers takes practice -- often pages and pages of practice! -- but we find that once students do several hundred problems (sorry!) they start doing them fast and correct!

Try our strategy of thinking about money.  Every + is money you have.  Every - is money you owe.  Every problem is just you combining the money you have with the money you owe and seeing what you come out with.  Here are our mental steps:

1. Am I starting with money?  Or in debt?  How much?
2. Am I going up (adding money?) or going down (subtracting money or adding debt)?
3. Will I end up with money in my pocket (+) or owing money (-)?
4. Then, do the math and see if it makes sense!

Try these steps on the practice problems below and see if you can start getting all of your answers correct!

(And read the alternate method too -- even though we don't recommend it as the only method you use, it can be helpful as you learn).