Absolute Differences
Subtraction has two meanings. When I write $5-3=2$5−3=2 it can refer to two things.
1) You have $5$5 of something. Minus or take away $3$3. We are left with $2$2.
2) You have $5$5 of something. You have $3$3 of something else. What is the difference between the amounts? The two amounts are different by $2$2.
This is why the terms 'subtraction', 'minus', 'take away' and 'difference' are all used to mean the same thing.
Difference of Two Integers
We're going to now look at subtraction of any two integers as a difference, by seeing how many places you have to move on the number line to get from one integer to the other.
Let's think about $5-3=2$5−3=2 again.
Look at where number $5$5 is. Look at where number $3$3 is. How many places to the left do you have to move to get from $5$5 to $3$3? You have to move $2$2 places to the left.
Finding the difference between two integers works in the exact same way as finding the difference between two positive whole numbers.
However, the difference depends on the order we subtract the integers in.
Hence, we can have the positive difference resulting from $5-3=2$5−3=2, and we can also have the negative difference from $3-5=-2$3−5=−2.
You might ask, 'But how can we have a negative difference?' The reason for the difference $3-5=-2$3−5=−2 being negative is that the subtraction $3-5$3−5 asks us, 'If we start at $3$3 and want to get to $5$5, how many places to the left do we have to move?'
As you can clearly see, we in fact want to move in the opposite direction by $2$2 places, hence the difference of $-2$−2.
Differences therefore represent movements between integers to the left or the right.
The difference of two integers is how many places you have to move from one to the other, AND in which direction you have to do it.
The difference $a-b$a−b is positive if the first number is greater, and negative if the second number is greater.
Absolute Difference (Distance Between Integers)
Let's say we want to know the distance between two integers, say, $9$9 and $3$3. We can clearly see on the number line that they are a distance of $6$6 places apart.
We have the difference $9-3=6$9−3=6, because we have to move $6$6 places to the left to get from one number to the other.
However, $3-9=-6$3−9=−6 because we have to move $6$6 places to the right to get from one number to the other.
But what if we don't care about the direction we have to move? What if we just want the distance? How would we do this? The answer: Absolute Values.
Distance is always positive, so the absolute value will tell us how far we've moved, without the direction. Hence,
| $\left|9-3\right|$|9−3| | $=$= | $\left|6\right|$|6| |
| $=$= | $6$6 |
OR
| $\left|3-9\right|$|3−9| | $=$= | $\left|-6\right|$|−6| |
| $=$= | $6$6 |
This is called the absolute difference, and it means the same thing as the distance between two integers. As you can see, when we find the absolute difference between two integers, it doesn't matter what order we subtract them in, as the final absolute value result will be positive either way!
The absolute difference (distance) between two integers can be found by taking the absolute value of their difference. The order won't matter!
$\text{Absolute Difference}=\left|a-b\right|$Absolute Difference=|a−b|$=$=$\left|b-a\right|$|b−a|
Examples
Question 1
To find the distance between $-10$−10 and $5$5, Neil explained and performed the following steps:
Step 1: First I find the difference between $-10$−10 and $5$5: $-10-5=-15$−10−5=−15
Step 2: Next I take the absolute value of this difference, since distance is positive: $\left|-15\right|=15$|−15|=15
Which of the other calculations below could be used to find the distance between $-10$−10 and $5$5? Select all the correct options.
$\left|5-\left(-10\right)\right|$|5−(−10)|
A$5+10$5+10
B$\left|5\right|-\left|10\right|$|5|−|10|
C
Question 2
Given two numbers $5$5 and $2$2:
Write "the distance between $5$5 and $2$2" as a mathematical expression.
Now, evaluate the distance between $5$5 and $2$2.