1. Integers

Lesson

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 the United States 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. Let's say we visit Seeley, California, which is about $15$15 m below sea level. Then our altitude at this location would be $-15$−15 m. Later we might hike up to the peak of Red Butte, which is about $2228$2228 m above sea level. On the summit, our altitude would be $2228$2228 m.

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).

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$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$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$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
- $0$0°C, the freezing point of water

With the zero selected, we can then orient the number line by choosing a positive direction. Again, we are free to work with any orientation we like, but quite often the words we use to describe the physical situation suggest the most sensible orientation. Here are some physical directions that are common in situations with integers:

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 meters, so in that case, it would make sense to let $1$1 unit on the number line represent $1$1 m of altitude in the real world. Here are some quantities we could identify with $1$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

Let's look at an example to see how to set up a number line for a particular situation.

Fred uses a straight length of highway to drive to his office, which is $9$9 km east of his home and to drive to the mountains, which are $5$5 km west of his home. Use a number line to describe the location of the office and the mountains, relative to Fred's home.

**Think**: We want to describe each location with respect to Fred's home, so we can use that as the zero. Next, we can choose east to be the positive direction, and let $1$1 km be $1$1 unit on the number line.

**Do**: In the image below we have shown all the given information about the places of interest along the highway, and we have drawn a number line alongside this that we will use to describe the location of each place.

Using this setup, with $0$0 at Fred's home and the positive direction to the east, we can see that Fred's office corresponds to the integer $9$9, and the mountains correspond to the integer $-5$−5.

**Reflect**: If we had chosen the positive direction to be to the west, then the mountains would be represented by the integer $5$5, and the office by the integer $-9$−9.

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? Let's look at an example.

A high rise apartment building has several basement levels for car parking and storage. The ground floor is floor $0$0. Amy lives on the $7$7th floor and needs to take the elevator down $11$11 stories to get to her car. How many stories below ground is Amy's car?

**Think**: Let's set up a number line with $0$0 at the ground floor, the positive direction going upwards, and $1$1 unit on the number line representing one story in the building. Notice that the answer needs to be a positive number, even though we may need to use negative integers in our solution.

**Do**: We want to evaluate the subtraction $7-11$7−11. On the number line below, we start at $7$7 and move $11$11 units in the negative direction. This gets us to the integer $-4$−4.

Is $-4$−4 the answer? Not quite. This integer contains all the information we need to answer the question, but we were asked to describe the location of Amy's car in terms of the number of stories below the ground floor. So we need to interpret the integer $-4$−4 in this context.

The sign of $-4$−4 is negative, which tells us that Amy's car is below ground, as we expected. The magnitude (or size) of $-4$−4 is $4$4, and this tells us the number of stories below the ground. So we can conclude by saying that Amy's car is $4$4 stories below ground.

**Reflect**: 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.

In our example, we had the result $-4$−4, and we reinterpreted the meaning of this integer to be "$4$4 stories below ground". It would be unnecessarily messy to say "$-4$−4 stories from the ground" or "$-4$−4 stories above ground".

Careful!

Answers that are integers can be positive or negative:

Question: What is the balance of your account? |

Answer: $\text{Balance }=-\$31$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$$31. |

The image below shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line.

What integer represents $3$3 m above the surface?

What integer represents $4$4 m below the surface?

If Nadia is initially $2$2 m above the surface, and descends $6$6 m in the elevator, what integer represents her end point?

If Nadia is at a location represented by the integer $-4$−4, and ascends $3$3 m, which option describes her new location?

$7$7 m below the surface

A$1$1 m below the surface

B$3$3 m above the surface

C$7$7 m above the surface

D$7$7 m below the surface

A$1$1 m below the surface

B$3$3 m above the surface

C$7$7 m above the surface

D

Tara is waiting for the next flight to Los Angeles, which was scheduled to be in $36$36 minutes, but there is a $48$48 minute delay. She takes a nap, and wakes up $24$24 minutes later. How much longer does Tara have to wait before the plane departs?

Luigi enters an elevator at the $7$7^{th} floor (the ground floor being floor $0$0). The elevator goes down $3$3 floors, then up $9$9 floors and finally it goes down $2$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?

Solve real-world and mathematical problems involving the four operations with integers and other rational numbers.