Absolute Value and addition

Absolute values can be very useful when adding together positive and negative numbers. To refresh your memory on adding integers, click here. 

Remember that the absolute value of any number $x$x is written as $\left|x\right|$|x| and means the distance of $x$x from zero on the number line. For instance, $\left|7\right|=7$|7|=7 and $\left|-21\right|=21$|−21|=21. Notice also that the absolute value is never negative.

We know so far that numbers exist on the number line. Positive numbers are to the right of zero, and negative numbers are to the left of zero. 

All the numbers on the number line are ordered in increasing size from left to right. So, for example, $-8$−8 is less than $-4$−4, and $5$5 is greater than $-7$−7.

When we add, we grow in size and move to the right, and when we subtract, we shrink in size and move to the left.

Take for example the sum $-8+13$−8+13. We start on the number line at $-8$−8, and we move to the right by $13$13 units.

What about the sum $-2+\left(-5\right)$−2+(−5)? We remember our rules for adjacent signs and realise that $-2+\left(-5\right)$−2+(−5) means $-2-5$−2−5. This time we start on the number line at $-2$−2, and we move to the left by $5$5 units.

 

The absolute value of a number tells us how far we are going to have to move when adding, regardless of the direction. It is good in general when adding positive and negative numbers to first think about the absolute value of the number being added so you know how far you're moving. Then you can figure out the direction.

Question 1

Question 2

You can even think of any of number line movements in these sums as starting from zero! In the sum $-13+18$−13+18 you can think about starting at $-13$−13, or you can think about starting first at zero, then moving to $-13$−13 before you actually do the sum. Just like before, this follows our idea of absolute values representing the distance you move.

So, if I told you I had a number with an absolute value of $22$22, then we performed a sum that involved a movement in the opposite direction of $23$23, what is the absolute value of the sum?

Without even knowing whether the numbers I've used are positive or negative, you know that the absolute value of the sum is $1$1! Both of the following could have been the example referred to.

$-22+23=1$−22+23=1
$22+\left(-23\right)=-1$22+(−23)=−1

Question 3

Question 4