Back to the number line

The 'infinite lift' we saw above looked very much like a number line, didn't it? In fact, if we rotate it by 90°, it basically becomes a line. It also tells us how to complete the number ray to a number line, answering the question that we had asked at the beginning of the chapter. To the left of 0 are the negative numbers – 1, – 2, – 3, …

Usually we drop the '+' sign on positive numbers and simply write them as 1, 2, 3, …

Instead of traveling along the number line using a lift, we can simply imagine walking on it. To the right is the positive(forward) direction, and to the left is the negative (backward) direction.

Smaller numbers are now to the left of bigger numbers and bigger numbers are to the right of smaller numbers.

So, 2 <> 5; –3_ <> 2; and –5 <> –3.

If, from 5 you wish to go over to 9, how far must you travel long the number line?

Instructions

You must travel steps. That is why 5+4=. (Remember: Starting number + Movement = Target number)

The corresponding subtraction statement is 9−5=. (Remember: Target number – Starting number = Movement needed)

Now, from 9, if you wish to go to 3, how much must you travel along the number line?

You must move steps backward, i.e., you must move –6. Hence, we write 9+−6=. (Remember again: Starting number + Movement = Target number)

The corresponding subtraction statement is 3−9=. (Remember again: Target number – Starting number = Movement needed)

Now, from 3, if you wish to go to –2, how far must you travel?

You must travel –5 steps, i.e., steps backward. Thus, 3+−5=. The corresponding subtraction statement is: −2−3=.

1. Mark 3 positive numbers and 3 negative numbers on the number line.

Answer:

2. Write down the above 3 marked negative numbers in the following boxes: [[]], [[]], [[]]

1. Is 2 > –3 ? Why? Is –2 < 3? Why?

2. What are:

a. −5+0=

b. 7+−7=

c. −10+20=

d. 10−20=

e. 7−−7=

f. −8−−10=

Using the unmarked number line to add and subtract

Just as you can do additions, subtractions and comparisons with small numbers using the number line above, you can also do them with large numbers by imagining an 'infinite number line' or drawing an 'unmarked number line' as follows:

This line shows only the position of zero. Other numbers are not marked. It can be convenient to use this unmarked number line to add and subtract integers. You can show, or simply imagine, the scale of the number line and the positions of numbers on it.

For example, this unmarked number line (UNL) shows the addition problem:

85+−60=?

We then can visualise that 85+−60=

The following UNL shows a subtraction problem which can also be written as a missing addend problem:

−100−+250=? or 250+?=−100.

We can then visualise that ? = in this problem.

In this way, you can carry out addition and subtraction problems, with positive and negative numbers, on paper or in your head using an unmarked number line.

Use unmarked number lines to evaluate these expressions:

a. −125+−30=

b. +105-−55=

c. +80-−150=

d. −99-−200=

Converting subtraction to addition and addition to subtraction

Recall that Target Floor – Starting Floor = Movement needed

or

Target Floor = Starting Floor + Movement needed

If we start at 2 and wish to go to –3, what is the movement needed?

First method: Looking at the number line, we see we need to move –5 (i.e., 5 in the backward direction).

Therefore, −3−2=−5.

The movement needed is –5.

Second method: Break the journey from 2 to –3 into two parts.

a. From 2 to 0, the movement is 0−2= .

b. From 0 to –3, the movement is −3−0= .

The total movement is the sum of the two movements: −3+−2= .

Look at the two coloured expressions. There is no subtraction in the second one!

Examples:

a. +7−+5=+7+

b. −3−+8=−3+

c. +8−−2=+8+

d. +6−−9=+6+