Why it happens: ‘– (– 55)’ asks how far above – 55 is + 105. To climb out of a level that is 55 below the ground you must first come up those 55 levels — so the 55 gets added to the journey, not removed from it. In short, a – (– b) = a + b.
Q2.
Take a look at the ‘infinite lift’ above. Does it remind you of a number line? In what ways?
Answer
Yes — the infinite lift is a number line, only standing up instead of lying down.
Level 0 of the lift is the 0 of the number line.
Levels above 0 are the positive numbers; levels below 0 are the negative numbers.
The levels are equally spaced, just like the marks on a number line.
Pressing ‘+’ is moving forward; pressing ‘–’ is moving backward.
A higher level means a greater number, exactly as a mark further right means a greater number.
Turn the infinite lift through 90° and it becomes … – 3, – 2, – 1, 0, 1, 2, 3 … — a complete number line.
Math Talk: This is exactly the answer to the question asked at the very start of the chapter — the number ray starting at 0 becomes a full number line once the negative numbers are put to the left of 0.
Q3.
If, from 5 you wish to go over to 9, how far must you travel along the number line?
Answer
You must travel 4 steps forward (to the right).
Four steps forward take you from 5 to 9.
Starting Number + Movement = Target Number → 5 + 4 = 9
Target Number – Starting Number = Movement needed → 9 – 5 = 4
Q4.
Now, from 9, if you wish to go to 3, how much must you travel along the number line?
Answer
You must move 6 steps backward, that is a movement of – 6.
Six steps backward take you from 9 to 3, so the movement is – 6.
Starting Number + Movement = Target Number → 9 + (– 6) = 3
Target Number – Starting Number = Movement needed → 3 – 9 = – 6
Why the answer is negative: the sign shows the direction. Forward (right) is +, backward (left) is –. The 6 tells you how far, the minus tells you which way.
Q5.
Now, from 3, if you wish to go to – 2, how far must you travel?
Answer
You must travel – 5, that is 5 steps backward. Three steps bring you to 0, and two more take you to – 2.
From 3 to – 2 is five steps backward — three to reach 0 and two more beyond it.