Introduction
There are many situations that require us to compare two or more amounts where the second amount can be above or below the first. The integers are the perfect tool to use in these cases.
For example, if we are on a road trip through Australia we might be interested in our altitude at different parts of our journey. Altitude is a quantity that tells us how far we are from sea level, and in which direction.
In both cases, the vertical distance from sea level is given by the magnitude of the altitude (the size of the integer), and the direction from sea level is given by the sign of the altitude (whether the integer is positive or negative).
Set up of a number line
In general, there are three things we need to keep in mind when applying a number line to a particular real world situation:
- What point in the real world shall be represented by the integer 0 on the number line?
- What direction in the real world shall be represented by the positive direction on the number line?
- What length in the real world shall be represented by 1 unit on the number line?
Once we have covered these three things, we can use our knowledge of addition and subtraction on the number line to describe how the real world quantities change.
We are free to choose any point we like as the number 0, but there are some common choices for certain situations. Here are some physical points that are often chosen:
- Ground level
- Sea level
- A starting location in space
- A starting event in time
- An account balance of \$0
- 0\degree \text{C}, the freezing point of water
| Positive | Negative |
|---|---|
| Right | Left |
| Up | Down |
| Above | Below |
| North | South |
| East | West |
| After | Before |
| Credit | Debit |
| Hotter | Colder |
Finally, we need to relate the magnitude of a real world quantity with the distance between each integer on the number line. We can think of this as scaling the number line to suit the physical problem. In our earlier example, we were measuring altitude in metres, so in that case, it would make sense to let 1 unit on the number line represent 1\text{ m} of altitude in the real world. Here are some quantities we could identify with 1 unit on the number line:
- Length - cm, m, km, etc.
- Time - seconds, minutes, hours, days, years, etc.
- Temperature - degrees Celsius (or Centigrade), degrees Fahrenheit, Kelvin, etc.
- Money - cents, dollars, etc.
Examples
Example 1
The image below shows how the location of a miner travelling up and down a mine shaft relates to an integer on the number line.
What integer represents 3\text{ m} above the surface?
What integer represents 4\text{ m} below the surface?
In setting up the number line, note these when applying a number line to a particular real world situation:
The point in the real world shall be represented by the integer 0 on the number line, such as:
Ground level
Sea level
A starting location in space
A starting event in time
An account balance of \$0
0\degreeC, the freezing point of water
The direction in the real world shall be represented by the positive direction on the number line, such as:
Positive: Right, Up, Above, North, East, After, Credit, Hotter
Negative: Left, Down, Below, South, West, Before, Debit, Colder
The length in the real world shall be represented by 1 unit on the number line, such as:
Length - cm, m, km, etc.
Time - seconds, minutes, hours, days, years, etc.
Temperature - degrees Celsius (or Centigrade), degrees Fahrenheit, Kelvin, etc.
Money - cents, dollars, etc.
Movement on the number line
After setting up a number line, we can talk about changes in the quantity we are representing using integer arithmetic. Given a starting temperature, and some change in a certain direction, what is the final temperature? Given a starting balance and an ending balance of money in an account, what has been the magnitude and sign of the change?
Answers that are integers can be positive or negative:
Question: What is the balance of your account?
Answer: \text{Balance }=-\$31
When describing this situation in words it is more natural to combine a positive number with a directional word:
Question: How much do you owe the bank?
Answer: I owe the bank \$31.
Often we will start to think about a problem using directional words like "above/below" or "east/west" or "credit/debit". Then we will set up a number line and use integers to make calculations. It is tempting to combine the integer that is the result of this calculation with the directional word we first used to describe the situation, but this can lead to confusion.
Examples
Example 2
Tara is waiting for the next flight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34 minute delay. She takes a nap, and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?
Example 3
Luigi enters an elevator at the 7th floor (the ground floor being floor 0). The elevator goes down 3 floors, then up 9 floors and finally it goes down 2 floors, where Luigi gets out.
On which floor does Luigi end up?
When Luigi gets off the elevator, how many floors from his starting point is he?
Answers that are integers can be positive or negative. The sign of the integer determines the location of a thing or person, or whether we have a profit or loss, or savings or debt.