Explainer: Integers

In this explainer, we will learn how to read and write integers including describing quantities having opposite directions or values.

We will model wins and losses, temperature increases and decreases, elevation, and more. We will also learn to model opposite quantities and learn about the opposite (or additive inverse) of integers.

We will start with a discussion of what the integer numbers are.

You already know that the whole numbers, which are the counting numbers 1, 2, 3, 4, 5, and so on together with zero, can be drawn on a number line. When you do this, you can think of each number having a distance from zero. For example, 5 is 5 units to the right from zero, or a distance of 5 from zero.

The numbers 1,2,3,, count from zero to the right. We call these numbers the positive numbers. We can also think of counting from zero in the other direction. When we do this we get the negative numbers. Negative numbers are always written with a negative sign (). We can write positive numbers with the positive sign (+) but we do not have to.

For example, 5 units to the left of zero we find the number negative 5, which we write 5. The positive number 5 and the negative number 5 are the same distance from zero in opposite directions.

When we put these positive and negative numbers together with zero we get a set of numbers called the integers.

Definition: The Integers

The integers are the positive counting numbers 1,2,3,4,5,, together with their additive inverses, the negative numbers, 1,2,3,4,5,, and 0, which is neither positive nor negative.

When we plot these numbers on a number line, 0 is in the centre, the positive numbers count up to the right of zero, and the negative numbers count down to the left of zero.

When we measure from zero, there are always two numbers that are the same distance away but in opposite directions. We call these numbers opposites. For example,

  • 1 and 1 are opposites,
  • 2 and 2 are opposites, and
  • 53 and 53 are opposites.

The positive integers are more than zero and the negative integers are less than zero. Therefore, integers can be used to describe real-world situations, which represent quantities that are more or less than zero. This is what we will discuss here.

Suppose that Hannah has a bank account. If the bank account contains $50, then Hannah has $50 to spend. We can represent this amount with the positive integer 50 because she has an amount that is more than zero.

If, however, her account is in debt by $50, then she owes $50. To represent the balance in her account, we would use the negative integer 50.

In this situation, a positive integer represents money she has or money she has gained, and a negative number represents money she has lost or money she has to pay. When we talk about the balance in a bank account, the amount can either be more than zero (having money to spend) or less than zero (owing money).

We can model this on a number line.

If the bank account is empty, the balance can be represented by 0. If she deposits $50 into the account when the balance is 0, this represents a gain of $50 that we represent with the positive integer 50. If instead she withdraws $50 from the account when the balance is zero, this represents a debt, or a loss, of $50 that we can represent with the negative integer 50.

A deposit (gain) of $50 and a withdrawal (loss) of $50 are represented by integers that are the same distance from zero but in opposite directions. This is because they represent the same amount of money each time but we use either a positive or negative number to indicate whether the amount is being gained or lost.

The numbers 50 and 50 are called opposites, or additive inverses, because they sum to zero. To see this, observe that a deposit of $50 and then a withdrawal of $50 takes the balance in the account back to zero. When we model this addition on a number line, we see that 50+(50)=0.

Equally, if she withdrew $50 and then deposited $50 the account balance would return to zero, so 50+50=0.

Above, we saw an example of representing bank deposits and withdrawals with integers; there are many other situations that can be modeled by positive and negative numbers. We will look at some more examples. First, let us look at representing elevation with integers.

Example 1: Using Integers to Represent Elevation

Answer the following.

  1. How can the elevations of points 𝐴 and 𝐵 with respect to sea level be represented?
  2. What does it mean if a location has an elevation of zero?

Answer

  1. We can use positive numbers to represent elevations above sea level and negative numbers to represent elevations below sea level.
    The point 𝐴 is 60 m above sea level; we represent it with the positive number 60.
    The point 𝐵 is 100 m below sea level; we represent it with the negative number 100.
  2. If something has an elevation of zero, then it means that it is at sea level.

Next, we will represent temperature with integers.

Example 2: Using Integers to Represent Temperature

  1. Isabella saw on the weather report that Alaska’s temperature was 17 degrees below zero. Represent the temperature with an integer.
  2. The temperature in Cape Town can be represented by the opposite of this integer. What is the temperature, in Fahrenheit, in Cape Town?

Answer

  1. If the temperature was 17 degrees above zero (or 17F or just 17 degrees), we would represent it with the positive integer 17. But we are asked to represent a temperature of 17 degrees below zero. To do this we use negative integers. A tempertaure of 17 degrees below zero is represented by 17 or written 17F.
  2. To find the opposite (or additive inverse) of 17, we have to find the number that is the same distance from zero on a number line in the opposite direction. This is also the number that when added to 17 gives an answer of 0. This number is 17, or positive 17.
    Hence, the temperature in Cape Town is 17F.

So far, we have looked at examples where we can use integers to represent real-world scenarios that compare an amount to zero. For example, positive numbers represent elevation above sea level (zero), or temperatures above zero, or having money in a bank account, whereas negative numbers represent elevation below sea level, or temperatures below zero, or owing money.

We can also use integers to represent change.

Suppose that the temperature at midday is 7 degrees above zero but at midnight the temperature is 2 degrees below zero. We would like to use an integer to represent the change in temperature. First, let us model this on a number line. A temperature of 7 degrees above zero is represented by positive 7 and a temperature of 2 degrees below zero is represented by negative 2 or (2).

From the number line, we see that between midday and midnight the temperature decreased by 9 degrees.

We want to represent this with an integer. Although the difference in temperature is 9, we also want to record whether it is an increase or decrease. If it was an increase, we would use a positive number but, since it is a decrease, we will use a negative number. Hence, the change in temperature from midnight to midday is 9.

Example 3: Using Integers to Represent Change

Jacob is playing a game. He starts with 30 points.

  1. If, after his first turn, the change in his score is 20, what is his score after the first turn?
  2. If instead, after his first turn, the change in his score is +20, what is his score after the first turn?

Answer

  1. Since the change in his score is a negative number, this represents a loss of 20 points, so his score is decreased by 20 points, which will be 10.
  2. Since the change is a positive number, it represents a gain of 20 points, so his score increased by 20 points to be 50.

We can also model these two situations on a number line. In both cases, you start with 30 points. When the change is 20, you decrease the score, and when the change is 20, you increase the score.

Let us end with one more example.

Example 4: Using Integers to Represent Temperature

A certain stock lost 6 points in one day and gained 9 points the next day. Write integers to represent the stock’s losses and gains for the two days.

  1. 6,9
  2. 6,9
  3. 6,9
  4. 6,9
  5. 6,6

Answer

A loss of 6 points can be modelled with the negative integer 6.

A gain of 9 points cents can be modelled with the positive integer 9.

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