Adding and Subtracting Negative Numbers (page 2 of 4)

Sections: Introduction, Adding and subtracting, Multiplying and dividing, Negatives and exponents

How do you deal with adding and subtracting negatives? The process works similarly to adding and subtracting positive numbers. If you are adding a negative, this is pretty much the same as subtracting a positive, if you view "adding a negative" as adding to the left.

Let's return to the first example from the previous page: "9 – 5" can also be written as "9 + (–5)". Graphically, it would be drawn as "an arrow from zero to nine, and then a 'negative' arrow five units long":

...and you get "9 + (–5) = 4".

Now look back at that subtraction you couldn't do: 5 – 9. Because you now have negative numbers off to the left of zero, you also now have the "space" to complete this subtraction. View the subtraction as adding a negative 9; that is, draw an arrow from zero to five, and then a "negative" arrow nine units long:

...or, which is the same thing:

Then 5 – 9 = 5 + (–9) = –4.

Of course, this method of counting off your answer on a number line won't work so well if you're dealing with larger numbers. For instance, think about doing "465 – 739". You certainly don't want to use a number line for this. You know that the answer to "465 – 739" has to be negative, (because "minus 739" will take you somewhere to the left of zero), but how do you figure out which negative number is the answer?   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

Look again at "5 – 9". You know now that the answer will be negative, because you're subtracting a bigger number than you started with (nine is bigger than five). The easiest way of dealing with this is to do the subtraction "normally" (with the smaller number being subtracted from the larger number), and then put a "minus" sign on the answer: 9 – 5 = 4, so 5 – 9 = –4. This works the same way for bigger numbers (and is much simpler than trying to draw the picture): since 739 – 465 = 274, then 465 – 739 = –274.

Adding two negative numbers is easy: you're just adding two "negative" arrows, so it's just like "regular" addition, but in the opposite direction. For instance, 4 + 6 = 10, and –4 – 6 = –4 + (–6) = –10. But what about when you have lots of both positive and negative numbers?

• Simplify 18 – (–16) – 3 – (–5) + 2

Probably the simplest thing to do is convert everything to addition, group the positives together and the negatives together, combine, and simplify. It looks like this:

18 – (–16) – 3 – (–5) + 2
= 18 + 16 – 3 + 5 + 2
= 18 + 16 + (–3) + 5 + 2
= 18 + 16 + 5 + 2 + (–3)
= 41 + (–3)
= 41 – 3
= 38

"Whoa! Wait a minute!" you say. "How do you go from ' – (–16)' to ' + 16' in your first step?" This is actually a fairly important concept, and, if you're asking, I'm assuming that your teacher's explanation didn't make much sense to you. So I won't give you a "proper" mathematical explanation of this "the minus of a minus is a plus" rule. Instead, here's a mental picture that I ran across in an algebra newsgroup:

Imagine that you're cooking some kind of stew, but not on a stove. You control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number), the temperature goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.

Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, –2(–3) = + 6.

There's another analogy that I've been seeing recently. Letting "good" be "positive" and "bad" be "negative", you could say:

good things happening to good people: a good thing

The above isn't a technical explanation or proof, but I hope it makes the "minus of a minus is a plus" and "minus times minus is plus" rules seem a bit more reasonable. Let's look at a few more examples:

• Simplify –43 – (–19) – 21 + 25.
• –43 – (–19) – 21 + 25
= –43 + 19 – 21 + 25
= (–43) + 19 + (–21) + 25
= (–43) + (–21) + 19 + 25
= (–64) + 44
= 44 + (–64)

Technically, I can only move the numbers around as I did in the steps above after I have converted everything to addition. I cannot reverse a subtraction, only an addition. In practical terms, this means that I can only move the numbers around if I move their signs with them. If I move only the numbers and not their signs, I will have changed the value and will end up with the wrong answer. Continuing...

44 + (–64) = 44 – 64

Since 64 – 44 = 20, then 44 – 64 = –20.

• Simplify 84 + (–99) + 44 – (–18) – 43.
• 84 + (–99) + 44 – (–18) – 43
= 84 + (–99) + 44 + 18 + (–43)

= 84 + 44 + 18 + (–99) + (–43)

= 146 + (–142)

= 146 – 142

= 4

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 Cite this article as: Stapel, Elizabeth. "Adding and Subtracting Negative Numbers." Purplemath. Available from     http://www.purplemath.com/modules/negative2.htm. Accessed [Date] [Month] 2016

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