# Negative numbers

Here is everything you need to know about negative numbers, including how to add, subtract, multiply, and divide negative numbers and how to order negative numbers.

Students will first learn about negative numbers as a part of the number system in 6 th grade.

## What are negative numbers?

Negative numbers are any numbers less than zero. They have a negative or minus sign (-) in front of them.

Numbers greater than zero are referred to as positive numbers. If there is no sign in front of a number, the number is positive.

On the number line below, you can see some positive and negative integers (whole numbers):

The numbers in orange are negative values and the blue numbers are positive values. Zero is neither positive nor negative.

Just like you can add, subtract, multiply and divide positive numbers, you can do the same with negative numbers whether they are integers, decimals or fractions.

### Adding and subtracting negative numbers

When adding and subtracting negative numbers, use a number line.

If you are adding, move to the right of the number line, or the positive direction.

If you are subtracting, move to the left of the number line, or the negative direction.

Sometimes a question might have two operations next to each other.

For example, 4+(-2).

If the signs are the same, replace them with a positive sign.

If the signs are different, replace them with a negative sign.

The chart below summarizes this:

As a rule of thumb: Same signs add, different signs subtract.

For example,

Solve: -4+13.

-4+13=9

Solve 4+-7.

There are two different signs written next to each other. These become negative.

So, 4+-7 = 4-7

4-7=-3

### Multiplying and dividing negative numbers

Similar rules apply for multiplying and dividing negative numbers.

If the signs are the same, the answer is positive.

If the signs are different, the answer is negative.

When multiplying negative numbers:

The same rules apply for dividing negative numbers:

For example,

Solve: -4 \times-3.

4 \times 3=12

Both numbers have the same sign, so the answer will be positive.

-4 \times-3=12

Solve: 12 \div-2.

12 \div 2=6

The numbers have different signs, so the answer will be negative.

12 \div-2=-6

### Ordering integers

Integers (whole numbers) are ordered on a number line based on positive integers and negative integers. The center of the number line is marked as 0.

The integers that are greater than 0 are positive integers. The integers that are less than 0 are negative integers.

The more an integer is negative, the smaller the value is. The more an integer is positive, the bigger the value is.

For example,

Put these numbers in ascending order.

4, \; -2, \; 0, \; 3, \; -3, \; -5, \; 1, \; 2

Remember: ascending means smallest to biggest.

-5 is the most negative number, so this is the smallest value.

4 is the most positive number, so this is the biggest value.

So, putting them in ascending order you get, -5, \; -3, \; -2, \; 0, \; 1, \; 2, \; 3, \; 4

For example,

Put these numbers in descending order.

-1, \; -6, \; -12, \; -4, \; -3, \; -8

Remember, descending means biggest to smallest.

-1 is the least negative number, so this is the biggest value.

-12 is the most negative number, so this is the smallest value.

So, putting them in descending order you get, -1, \; -3, \; -4, \; -6, \; -8, \; -12

## Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6: The Number System (6.NS.C.5)
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

• Grade 6: The Number System (6.NS.C.6)
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

• Grade 6: The Number System (6.NS.C.7)
Understand the ordering and absolute value of rational numbers.

## How to add and subtract negative numbers

In order to add and subtract negative numbers:

1. If you have two signs next to each other, change them to a single sign.
2. Circle the first number on the number line.

## Negative numbers examples

### Example 1: adding a positive number

Solve: -5+8.

1. If you have two signs next to each other, change them to a single sign.

• If the signs are the same in the middle of the calculation, replace them with a positive sign (+).
• If the signs are different, replace them with a negative sign (-).

In this case you do not have two signs next to each other.

2Circle the first number on the number line.

The first number in the question is (-5).

• If you are adding, move to the right of the number in step 2 \; (\rightarrow).
• If you are subtracting, move to the left of the number in step 2 \; (\leftarrow).

In this case you are adding the 8 so move 8 spaces right from the (-5) on the number line:

-5+8=3

### Example 2: subtracting a negative number (two operations)

Solve: -7-(-5).

If you have two signs next to each other, change them to a single sign.

Circle the first number on the number line.

## How to multiply and divide negative numbers

In order to multiply and divide negative numbers:

1. Multiply or divide the numbers normally.
2. Change the sign using the rules of multiplying and dividing negative numbers.

### Example 3: multiplying negative numbers

Solve: -6 \times (-7).

Multiply or divide the numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers.

### Example 4: dividing negative numbers

Solve: -25 \div 5.

Multiply or divide the numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers.

### Example 5: real life word problem

The table below shows the temperatures recorded in Chicago at different times of the day.

a) What is the product of the highest and lowest temperatures?

b) What is the difference between the temperatures at 2 am and 1 pm?

Part \textbf{a)} What is the product of the highest and lowest temperatures?

The highest temperature is 3^{\circ} \mathrm{F}.

The lowest temperature is -8^{\circ} \mathrm{F}.

Multiply or divide the numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers.

\textbf{b)} What is the difference between the temperatures at \bf{1} pm and \bf{2} am?

The temperature at 2 am was -8^{\circ}\mathrm{F} and the temperature at 1 pm was 3^{\circ}\mathrm{F}.

To find the difference we must work out -8 - 3.

If you have two signs next to each other, change them to a single sign.

Circle the first number on the number line.

## How to order negative numbers

In order to order negative numbers:

1. Plot the given numbers on a number line.
2. List the numbers in ascending or descending order.

### Example 6: ascending order

Put these numbers in ascending order.

5, \; -2, \; 0, \; -4, \; -8, \; 3, \; 1

Plot the given numbers on a number line.

List the numbers in ascending or descending order.

### Example 7: descending order

Put these numbers in descending order.

-5, \; 3, \; -3, \; -7, \; 1, \; 5

Plot the given numbers on a number line.

List the numbers in ascending or descending order.

### Teaching tips for negative numbers

• Use real-world examples when discussing negative numbers. Use a bank account as an example and explain when you spend money, the amount of money spent is usually shown as a negative, and the amount in your account decreases.

• Teach students about opposites, explaining that every negative number has a corresponding positive number with the same magnitude.

• Present students with common misconceptions about negative numbers and have students work through why the example presented is incorrect.

### Easy mistakes to make

• Thinking a greater negative means a larger number
Students sometimes assume that the larger a negative number the greater it is. For example, students might incorrectly assume -3 is greater than 2 because 3 is a larger number.

• Raising a negative number to a power greater than one
Remember, when raising a negative number to a power greater than 1 , the resulting answer could be positive or negative. When you raise a negative number to an odd exponent, the resulting answer is negative; when you raise a negative number to an even exponent, the resulting answer is positive.

• Confusing when to change the signs
When adding or subtracting with negative numbers, the signs change only if they are next to each other in the middle of the calculation and are different.

### Practice negative numbers questions

1. Solve: -8+10.

-2

2

-18

18

Find the first number (–8) on the number line and move to the right the number of places as the second number (10).

2. Solve: -11-(-8).

-3

3

-19

19

There are two negative signs (-) together, so change it to a positive (+).

-11+8

3. Solve: -8 \times (-9).

-72

72

17

-17
8 \times 9 = 72

The signs are the same so the answer is positive.

-8 \times (-9)=72

4. Solve: -144 \div 3.

147

-147

-48

48
144 \div 3 = 48

The signs are different so the answer will be negative.

-144 \div 3= -48

5. The table shows the temperature in Buffalo, NY at different times of the day in degrees Fahrenheit.

What is the difference between the highest and lowest temperatures?

1^{\circ} \mathrm{F}

7^{\circ} \mathrm{F}

10^{\circ} \mathrm{F}

17^{\circ} \mathrm{F}

The calculation you need to solve here is 9-\,-8.  There are two negative signs (-) together so this becomes a positive (+).

The signs are different so the answer will be negative.

9-\,-8=9+8=17

6. Place the following numbers in ascending order:

-2, \; 5, \; -6, \; 0, \; 4, \; -3

5, \; 4, \; 0, \; -2, \; -3, \; -6

-6, \; -3, \; -2, \; 0, \; 4, \; 5

0, \; -2, \; -3, \; 4, \; 5, \; -6

-6, \; 5, \; 4, \; -3, \; -2, \; 0

Plot the numbers on a number line.

Ascending order means from least to greatest, so find the smallest number (left side of the number line) and list numbers moving from left to right on the number line.

## Negative numbers FAQs

Are negative numbers considered natural numbers?

Negative numbers are not natural numbers. Natural numbers are all positive integers from 1 to infinity, often referred to as counting numbers.

How do you compare negative numbers?

When you compare negative numbers, you compare the magnitude of each number. The farther left a number is on the number line, the smaller it is.

Can negative numbers be rational or irrational?

Negative numbers can be rational or irrational depending on whether it is expressed as fractions or not. For example, –3 is rational because it can be expressed as \cfrac{-3}{1}, while the square root of –2 is irrational.

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