integers

A number that is less than zero is known as a negative number and a number that is greater than zero is known as a positive number. Zero is neither positive nor negative.

Integers are a subset of the real numbers that consist of all the positive and negative whole numbers including zero. Integers can be represented as {...,-3,-2,-1,0,1,2,3,...} and are used in a variety of mathematical operations and applications such as counting, measuring, and describing quantities.

A negative number is written by putting a minus sign “-“ in front of a positive number. For example, -2 is a negative number and is read as “minus two” or “negative two” and it means the opposite of 2. 

Negative numbers are left of zero on the number line. A number and its opposite (or negative) are always the same distance from zero. The negative number -2 is just as far to the left of zero as 2 to the right of zero.

Let's take a real-world example where negative numbers are easily noticed. We measure temperature to know how hot or cold something is. Let's consider a scenario where the weather forecast predicts that tomorrow it will be 4 degrees colder than today. Today's temperature is 3 degrees celsius. What would the temperature be tomorrow? Let's find out using the number line below.

To find out what would be tomorrow's temperature, start at 3 and go back 4 steps.

Here we reach the temperature of -1. Notice that -1 is colder than 3.

As we go to the right of the number line the positive value increases, similarly as we go left to the number line negative value increases. But remember -100 is much smaller than -1. Numbers appearing farther to the right on the number line are larger or greater, while numbers appearing farther to the left are smaller or fewer.

Another situation where you notice negative numbers is in a bank statement. Let's consider a scenario where we deposited $300 last month and we receive a bank statement.

Last Month's Balance = 300
Gas station = -20
Departmental store = -50
Total Balance = 230

The negative numbers -20 and -50 represent expenditures, while the positive numbers represent a credit or deposit to the account.

 

Example 1:  2 − 5 = ?
Let's solve this with a number line

Move 5 times to the left of 2

2 - 5 = -3

To subtract, move backward, or to the left, on the number line.

Example 2:  -2 + 2 = ?
Start with negative 2 and move right 2 times on the number line.

We reach 0, so -2 + 2 = 0.

To add, move forward, or to the right, on the number line.

 

Example 3:  -2 − 3 = ?

Start with negative 2 and move left 3 times in the number line.

-2 − 3 = -5

Example 4:  -3 + 2 = ?

Here we are adding 2 to negative 3. Start with -3 and more 2 times right on the number line to reach -1.

-3 + 2 = -1

No sign means positive

If a number has no sign it usually means that it is a positive number. For example, 5 is +5

 

Operations on positive and negative numbers

Addition

When adding a positive to a positive or a negative to a negative, add them together and give them the same sign. For instance, 5 + 5 is equal to 10, while -5 + -7 is -12. 

When adding a positive number and a negative number together, use subtraction by taking the absolute value — the numbers without their signs — and subtract the smaller from the larger. Then give the answer the sign of the larger number. For instance, -7 + 4 means you take 7, subtract 4 and give the answer a negative sign since the absolute value of -7 is greater than 4.

Subtraction

Subtracting a negative number from something is the same as adding a positive number to it. For example, to subtract the negative number −8 from the number 6 is the same as adding the number 6 and the number 8. In symbols:

6 − (-8) = 6 + 8 = 14

When subtracting a negative from a negative, such as -6 and -4, switch -4 to positive 4 and add the values together to have -6 + 4, giving -2 following the addition rules.

-6 − (-4) = -6 + 4 = -2

To subtract a positive and negative number, 12 - (-9), switch the -9 to 9 and add the values to get 21.

12 − (-9) = 12 + 9 = 21

 

Multiplication

A positive number multiplied by a negative number gives you a negative number. For example, to multiply the positive number 3 by the negative number -2 is the same as multiplying the number 3 by the number 2 and the result will have a negative sign.

(3) × (-2) = -6

A negative number multiplied by another negative number gives you a positive number. For example, to multiply the negative number -3 by the negative number -2 is the same as multiplying the number 3 by the number 2 but the answer is positive.

(-3) × (-2) = 6

 

Division

In division, rules vary slightly from multiplication.

A positive number divided by a positive is always positive. For example, when you divide the positive number 15 by the positive number 3, you will get 5.

15 ÷ 3 = 5

A negative number divided by a positive or a positive number divided by a negative will always be negative. For example, when you divide the negative number -15 by the positive number 3, you will get -5. Also, when you divide the positive number 15 by the negative number -3, you will get -5.

15 ÷ (-3) = (-5)

When dividing a negative number by a negative number, you divide the absolute value by each other and get a positive number. For example, when you divide the negative number -15 by the negative number -3, you will get 5.

(-15) ÷ (-3) = 5