Introduction to Negative Numbers How to Add and Subtract

Positive and Negative Numbers

When you first learned your numbers, way back in elementary school, you started with the

counting numbers: 1, 2, 3, 4, 5, 6, and so on. Your number line looked something like this:

Addition, multiplication, and division always made sense — as long as you didn't try to

divide by zero — but sometimes subtraction didn't work. If you had "9 – 5", you got 4:

...but what if you had "5 – 9"? You just couldn't do this subtraction, because there wasn't

enough "space" on the number line to go back nine units:

You can solve this "space" problem by using negative numbers. The "whole" numbers

start at zero and count off to the right; these are the positive integers. The negative

integers start at zero and count off to the left:

Numbers Can be Positive or Negative:

Negative Numbers (-) Positive Numbers (+)

"-" is the negative sign. "+" is the positive

Note the arrowhead on the far right end of the number line above. That arrow tells you the

direction in which the numbers are getting bigger. In particular, that arrow also tells you

that the negatives are getting smaller as they move off to the left. That is, –5 is smaller

than –4.

This might seem a bit weird at first, but that's okay; negatives take some getting used to.

Opposites. Pairs of numbers −1 and 1,−2 and 2, -3 and 3 etc. are called the opposites.

They lie at the same distance from zero on the number line, but in the opposite directions.

For any number x (whether positive or negative), we will denote by −𝑥 the opposite of 𝑥.

For example, −  (−2) is the opposite of negative 2, which is equal to 2.

Absolute value. The distance of a number from zero on the number line is called absolute

value. So, we can say that the opposites have the same absolute value. The symbol for

absolute value is | |. For example, |4|=|−4|=4.

Comparing negative numbers. When comparing negative numbers, remember that the

smaller number is the one to the left. For example, −2 < −1.

Addition and subtraction. If we add a positive number to any number, we move to the

right along the number line. If we add a negative number to any number, we move to the

left along the number line. So, adding (−5) is moving 5 units to the left on the number line

— which is the same as subtracting 5. This rule holds in general:

𝑎   +  (−𝑏)  =  𝑎   −  𝑏  

More generally,

𝒂 − −𝒃 = 𝒂 + 𝒃

Now if we recall that – 𝑎 = −1 ∗ 𝑎 and the distributive law :

   𝑐× 𝑎 + 𝑏 = 𝑐×𝑎 + 𝑐𝑏

Then, combining them we get another rule:

−(𝒂 + 𝒃) = −𝒂 + (−𝒃)

Indeed, − 𝑎 + 𝑏 = −1× 𝑎 + 𝑏 = −1×𝑎 + −1 ×𝑏 = −𝑎 + (−𝑏)

Equations:

− −2𝑥 + 8 = 4

Switching the signs

2𝑥 + 8 = −4

Moving 8 to the other side

−2𝑥 = −4 − 8  

−2𝑥 = −12

Switching signs again:

2𝑥 = 12  

𝑥 = 6

What about 𝑥 = 5?

The fact that the absolute value of x equals 5 means that x is either equal to 5 or -5.

Thus the equation has two solutions:

𝑥! = 5,        𝑥! = −5.

And, finally, let’s solve

𝑥 + 2 + 3 = 10

|𝑥 + 2| = 7

 

That means that either 𝑥 + 2 = 7 or 𝑥 + 2 = −7. Solving these two cases separately, we

get

𝑥 = 7 − 2    𝑎𝑛𝑑  𝑥 = −7 − 2

Or

𝑥! = 5      𝑥! = −9.

Multiplication and division of negative numbers:

Rules:

𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒×𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒: −1 ×3 = −3

𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒×𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒: −1 × −3 = 3

𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ÷ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒: −1 ÷ 3 = 13

𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ÷ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒: −1 ÷ −3 = 3