Let's write down some of the things we have discovered about addition and multiplying on the number line.

Absolute value

The absolute value of a number is a measure of its size or magnitude. On the number line the absolute value of a number is its distance from the the number $0$0. Distances have no sign attached to them, and neither do absolute values. 

We use small vertical lines to surround a number to indicate its absolute value, so that the absolute value of minus three is written as $|-3|$|−3|. Its value is simply $3$3. The value of $|3|$|3| is also $3$3, because $3$3 and $-3$−3 are an equal distance away from $0$0 on the number line.

The meaning of the two absolute values $|-3|$|−3| and $|2|$|2| are illustrated here on this number line:

Operating with negative numbers

We will use the numbers $6$6, $-6$−6, $2$2, and $-3$−3 to illustrate what happens when we add, subtract or multiply using signed numbers in mathematical sentences.

The same rules apply for any number on the number line.   

 

Addition and subtraction

The sentence $3+(-3)$3+(−3) changes to $3-3$3−3 and so the answer is $0$0

The sentence $-3+(-3)$−3+(−3) changes to $-3-3$−3−3 and so the answer is $-6$−6

The sentence $6-(-6)$6−(−6) changes to $6+6$6+6 and so the answer is $12$12

The sentence $-6-(-6)$−6−(−6) changes to $-6+6$−6+6 and so the answer is $0$0

 

Multiplication

The sentence $6\times(-3)$6×(−3) becomes $-18$−18, because the product of a $+$+ and a $-$− is always a $-$−.

The sentence $-6\times(-3)$−6×(−3) becomes $18$18, because the product of a $-$− and a $-$− is always a $+$+. 

The sentence $-6\times3$−6×3 becomes $-18$−18, because the product of a $+$+ and a $-$− is always a $-$−.

The sentence $6\times3$6×3 becomes $18$18, because, well, thats always been the answer!

 

Division

Division works in the same way as multiplication. The division of a positive number by a negative number, or a negative number by a positive number is always negative.

so $6\div(-2)$6÷(−2) is $-3$−3 and $-6\div2$−6÷2 is $-3$−3. 

Division of any negative number by a negative number is always positive, so $-6\div(-2)$−6÷(−2) becomes $3$3.

And, as usual, division of a positive number by a positive number is positive! 

Of course, dividing two integers can lead to numbers that are not integers. So $-3\div-6$−3÷−6 becomes $\frac{1}{2}$12​. 

 

The order of operations

Mathematicians use an important convention when simplifying number sentences.

A convention is an agreed way of interpreting things so that we don't confuse meaning.

For example, we might think that a sentence like $2+(-3)\times6-(-2)$2+(−3)×6−(−2) is worked out from left to right so that we might simplify it as:

$2+(-3)\times6-(-2)$2+(−3)×6−(−2)	$=$=	$-1\times6-(-2)$−1×6−(−2)
	$=$=	$-6-(-2)$−6−(−2)
	$=$=	$-6+2$−6+2
	$=$=	$-4$−4

We tend to think this is correct, because we are used to reading words in a sentence from left to right.

But in mathematics, ... "convention not is this the".  

In fact we have a strict and agreed order to reading certain sentences.

This includes working within brackets first, then simplifying divisions and multiplications from left to write, and then finally doing the additions and subtractions, also from left to right.

The sentence above,  $2+(-3)\times6-(-2)$2+(−3)×6−(−2) is correctly worked out as:

$2+(-3)\times6-(-2)$2+(−3)×6−(−2)	$=$=	$2+(-18)-(-2)$2+(−18)−(−2)
	$=$=	$2-18+2$2−18+2
	$=$=	$-16+2$−16+2
	$=$=	$-14$−14

Here is another interesting example involving a fraction where the line in the fraction (called a vinculum) acts as a bracket, so that the numbers in the numerator are simplified first:

$\frac{6-(-4)}{-5}$6−(−4)−5​	$=$=	$\frac{6+4}{-5}$6+4−5​
	$=$=	$\frac{10}{-5}$10−5​
	$=$=	$-2$−2

Worked Examples

Question 1

Question 2

Question 3