Adding and Subtracting Signed Numbers

In the last lesson, you learned about positive and negative numbers. In this lesson, we will learn how to add and subtract them. At first we’ll use a number line, then try some without it after we get some rules. I hope you remember the number line from the last lesson. Here is a picture of part of it:

Adding numbers of the same sign

Everybody knows 2+3 = 5. Let’s see what that looks like on a number line.

How about adding two negatives? We’ll look at a picture of -4+(–6).

If your checking account is overdrawn $4 and you write a check for $6, your checking account is then overdrawn by $10. When you add a negative number to a negative number, you add the numbers and keep the – sign, so –4+(–6)= –10. Another way you can think of this is to think of playing in a pile of dirt. Negative numbers are holes, and positive nymbers are piles. If we dig a 6 foot hole in the bottom of a 4 foot hole, we would have a 10 foot hole, so –4+(–6)= –10. If we put a 3 foot pile on top of a 2 foot pile, we have a 5 foot pile, so 2+3 = 5. Test yourself on these examples:

1) 4+5= | |

2) –3+(–4)= |

Easy, isn’t it?

Adding numbers whose signs differ

This isn’t so very hard either. On a number line, let’s add 6 and –4.

We can see that 6+(–4)=2. Do you think –4+6 should be the same? Let’s look at it on a number line.

Using the "playing in the dirt" idea, if we have a 4 foot hole and fill it with a 6 foot pile of dirt, we would have a 2 foot pile of dirt, so –4+6=2 as the picture shows.

If we have a 6 foot dirt pile and dig a 4 foot hole in it, we are 2 feet above ground level, so 6+(–4)=2.

Similarly, if we have a checking account that is *overdrawn* by $4 and
make a $6 deposit, we would have a $2 balance. If we had a $6 balance and
wrote a $4 check, we would have a $2 balance.

I think it’s time for you to try these!

3) 7+(-3)= | |

4) -9+2= | |

5) -5+1= | |

6) 1+(-5)= | |

7) -15+7= | |

8) 8+(-20)= |

If you got these right, then you must have the idea. Now you are ready to learn subtraction.

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Subtraction of Signed Numbers

Picture this on a number line: 7–3 =4

How about this one: -4–5=-9

You might note from these two examples that we get the same result for 7–3
as we do for 7+(-3) and similarly

-4–5 is the same as -4+(-5). The fact of the matter is, we really
don’t need a separate set of rules for subtraction. We just need to
remember to **add** the *opposite* of the subtracted number. This
isn’t so far fetched.

Consider the hole-dirt-pile way of thinking. When you add a hole (a negative), it is just like subtracting a dirt-pile (a positive). If you subtract a dirt-pile (a positive), it’s just like adding a hole (a negative).

In a checking account, deposits are positive numbers and checks are negative
numbers. If we want to *remove *a deposit of 5$ (subtract a positive)
it has the same effect as adding a check for $5 (add a negative). On the
other hand if we wish to *remove* a $10 check from our records (subtract
a negative), we wind up doing the same thing as if we added a $10 deposit
(add a positive).

Review:

subtract a *positive *èadd a
*negative*

subtract a *negative*èadd a
*positive*

**subtract a number èadd its opposite**

The rule for subtraction of signed numbers is that simple: we don’t
subtract, we **add** the *opposite*.

Therefore 7–(-3)=7+3=10

and -9–7=-9+(-7)=-16.

You should try these!

9) -6–(-10) = -6 + | = | ||

10) 4–(-8)= 4 + | = | ||

11) -3–8 = -3+ | = | ||

12) 5–22 = 5+ | = |

Now here is a list of mixed addition and subtraction problems you can practice on:

13) -7+5= | |

14) 7+(-11)= | |

15) -6– (-17)= | |

16) 22–(-12)= | |

17) -14–5= |

18) On January 15, the high temperature in Honolulu, Hawaii was 79 degrees while the high in Nome, Alaska was -18 degrees. How many degrees warmer was it in Honolulu?

19) The highest land point in California is Mount Whitney, with a height
of 14494 feet *above* sea level. The lowest point in California is Death
Valley with a height of 287 feet *below* sea level. What is the difference
in their heights? (Don't use a comma!)

20) If you have *overdrawn* your checking account by $75, make a $50
deposit and write a check for $15, what is your new balance?