The System of Rational Numbers and Integers

Positive and Negative Numbers in Context

In everyday life, numbers do more than count how many things we have. Sometimes numbers show direction, change, or whether something is above or below a starting point.

That is why we use positive numbers and negative numbers. These numbers help us describe real situations like temperature, money, height, and movement.

In this lesson, you will learn what positive and negative numbers mean, how to read and write them, and how to use them in real-world contexts.

1. What are positive and negative numbers?

A positive number is greater than zero. A negative number is less than zero.

We write a positive number with a plus sign or with no sign, like \(+5\) or just \(5\). We write a negative number with a minus sign, like \(-5\).

The number \(0\) is neither positive nor negative. It is the point in the middle.

You can think of numbers on a number line:

Numbers to the right of \(0\) are positive. Numbers to the left of \(0\) are negative.

2. Why do we use signed numbers?

Signed numbers are numbers with a positive or negative sign. They are useful when a situation has opposites.

Here are some common examples of opposites:

• above zero and below zero
• up and down
• gain and loss
• deposit and withdrawal
• forward and backward

Positive and negative numbers help us describe these opposite ideas clearly.

3. Contexts where we use positive and negative numbers

Temperature: Temperatures above \(0^\circ\) are positive. Temperatures below \(0^\circ\) are negative.

• \(8^\circ\) means 8 degrees above zero
• \(-3^\circ\) means 3 degrees below zero

Elevation: Elevation tells how high or low something is compared to sea level.

• \(+200\) meters means 200 meters above sea level
• \(-50\) meters means 50 meters below sea level

Money: A positive number can show money you have or money added to an account. A negative number can show money owed or money taken away.

• \(+\$25\) could mean a deposit of 25 dollars
• \(-\$10\) could mean a withdrawal of 10 dollars

Movement: Positive and negative numbers can show direction from a starting point.

• \(+4\) steps could mean 4 steps forward
• \(-4\) steps could mean 4 steps backward

4. Understanding zero in context

Zero is the starting point or reference point. It is not positive and it is not negative.

In different situations, zero can mean different things:

• \(0^\circ\) means zero degrees
• \(0\) meters elevation means sea level
• \(\$0\) means no money gained or owed
• position \(0\) means the starting point

5. Comparing positive and negative numbers

When comparing numbers, remember:

• Any positive number is greater than any negative number.
• Any negative number is less than any positive number.
• Among negative numbers, the number farther left on the number line is smaller.

For example:

• \(4 > -2\)
• \(-1 > -5\) because \(-1\) is to the right of \(-5\)
• \(-7 < 0\)

This can feel tricky at first. A number like \(-10\) looks big because 10 is big, but \(-10\) is actually less than \(-2\) because it is farther below zero.

6. The sign tells the direction or type of value

The number part tells how much. The sign tells which side of zero or which direction.

For example, in \(-6\):

• the \(6\) tells the amount is 6 units
• the negative sign tells it is below zero, backward, or a loss, depending on the situation

In \(+6\):

• the \(6\) still means 6 units
• the positive sign tells it is above zero, forward, or a gain

7. Opposites

Numbers that are the same distance from zero but on opposite sides are called opposites.

Examples of opposites:

• \(5\) and \(-5\)
• \(12\) and \(-12\)
• \(\frac{1}{2}\) and \(-\frac{1}{2}\)

Opposites have the same distance from zero, but they mean opposite directions or opposite situations.

8. Absolute value: distance from zero

Absolute value means the distance a number is from zero on the number line.

Distance is always positive or zero.

We write absolute value with bars:

This means \(-4\) and \(4\) are both 4 units from zero.

Absolute value helps us focus on how far, not which direction.

9. Worked examples

Example 1: Temperature

In the morning, the temperature is \(-2^\circ\). In the afternoon, it is \(5^\circ\).

Question: Which temperature is warmer?

Step 1: Put the numbers on a number line in your mind. \(5\) is to the right of \(-2\).

Step 2: The number farther right is greater.

Answer: \(5^\circ\) is warmer than \(-2^\circ\).

Example 2: Elevation

A diver is at \(-18\) meters. A boat is at \(0\) meters because it is on sea level.

Question: How do the signed numbers describe their positions?

Step 1: \(0\) is the reference point, which is sea level.

Step 2: The diver's elevation is negative, so the diver is below sea level.

Answer: \(-18\) meters means the diver is 18 meters below sea level. \(0\) meters means the boat is at sea level.

Example 3: Bank account

A bank account shows \(-\$7\).

Question: What does this mean in context?

Step 1: A negative amount in a bank account means the amount is below zero.

Step 2: This often means money is owed.

Answer: \(-\$7\) means the account is 7 dollars below zero, so 7 dollars are owed.

Example 4: Comparing movement

Player A moves \(+6\) spaces. Player B moves \(-3\) spaces.

Question: Who moved in the positive direction, and whose move was farther from the starting point?

Step 1: \(+6\) means 6 spaces in the positive direction. \(-3\) means 3 spaces in the negative direction.

Step 2: Compare distances from zero using absolute value:

Answer: Player A moved in the positive direction. Player A also moved farther from the starting point because 6 spaces is more than 3 spaces.

10. How to decide whether a number should be positive or negative

Ask yourself these questions:

1. What is the reference point? Is it zero, sea level, the starting point, or no money owed?
2. Is the value above or below that point?
3. Does the situation describe a gain or a loss, up or down, forward or backward?

If the value is above the reference point or in the positive direction, use a positive number.

If the value is below the reference point or in the opposite direction, use a negative number.

11. Common mistakes to avoid

• Forgetting that zero is neither positive nor negative.
• Thinking negative numbers are always larger because the digit is larger. For example, \(-9\) is less than \(-2\).
• Ignoring the context. The same number can mean different things in different situations. For example, \(-4\) could mean 4 degrees below zero, 4 meters below sea level, or a loss of 4 dollars.
• Mixing up amount and direction. The sign shows direction or type, while the number shows how much.

12. Quick practice ideas

Try reading each signed number in words:

• \(-6^\circ\)
• \(+12\) meters
• \(-\$15\)
• \(0\)

Then ask:

• Is it above, below, or at the reference point?
• What does the sign tell me?
• What does the number tell me?

Summary

Positive and negative numbers help describe real-world situations with opposite meanings, such as above and below zero, gains and losses, or forward and backward movement.

Zero is the reference point. Positive numbers are greater than zero, and negative numbers are less than zero.

The sign tells the direction or type of value, and the number tells the amount. In context, understanding the reference point helps you choose whether a number should be positive, negative, or zero.

Opposites and Absolute Value

In this lesson, you will learn about opposites and absolute value. These ideas help us understand positive and negative numbers on a number line.

Negative numbers are numbers less than zero, like \(-3\) and \(-10\). Positive numbers are numbers greater than zero, like \(4\) and \(12\). The number \(0\) is neither positive nor negative.

A number line helps show where numbers are placed. Numbers to the right of \(0\) are positive. Numbers to the left of \(0\) are negative.

1. Opposites

Two numbers are opposites if they are the same distance from \(0\) but on different sides of \(0\) on the number line.

For example, \(5\) and \(-5\) are opposites. They are both 5 units away from \(0\), but one is to the right and one is to the left.

Opposites always add up to zero.

$$5 + (-5) = 0$$

Here are some pairs of opposites:

• \(1\) and \(-1\)
• \(7\) and \(-7\)
• \(12\) and \(-12\)

The opposite of a positive number is negative. The opposite of a negative number is positive.

What about zero? The opposite of \(0\) is \(0\), because

$$0 + 0 = 0$$

2. Absolute Value

The absolute value of a number is its distance from \(0\) on the number line.

Distance is never negative, so absolute value is always zero or positive.

We use vertical bars to show absolute value. For example:

$$|4| = 4$$

This means the distance from \(4\) to \(0\) is 4 units.

Also,

$$|-4| = 4$$

This means the distance from \(-4\) to \(0\) is also 4 units.

Even though \(4\) and \(-4\) are different numbers, they have the same absolute value because they are the same distance from zero.

Here are more examples:

• \(|9| = 9\)
• \(|-9| = 9\)
• \(|0| = 0\)

Important idea: Absolute value tells how far a number is from zero, not which side of zero it is on.

3. Opposites and Absolute Value Together

Opposites have the same absolute value.

For example, \(8\) and \(-8\) are opposites, and

$$|8| = 8 \quad \text{and} \quad |-8| = 8$$

Both numbers are 8 units from zero.

This is true for any pair of opposites. They are on opposite sides of zero, but they are the same distance from zero.

4. How to Find an Opposite

To find the opposite of a number:

• Change a positive number to negative.
• Change a negative number to positive.
• Leave zero as zero.

Examples:

• The opposite of \(6\) is \(-6\).
• The opposite of \(-2\) is \(2\).
• The opposite of \(0\) is \(0\).

5. How to Find Absolute Value

To find absolute value, ask: How far is the number from zero?

• If the number is positive, its absolute value is the same number.
• If the number is negative, its absolute value is the positive version of that number.
• If the number is zero, its absolute value is zero.

Examples:

• \(|3| = 3\)
• \(|-3| = 3\)
• \(|0| = 0\)

Worked Example 1

Question: What is the opposite of \(9\)?

Step 1: \(9\) is positive.

Step 2: The opposite of a positive number is negative.

Answer: The opposite of \(9\) is \(-9\).

Worked Example 2

Question: What is \(|-6|\)?

Step 1: Absolute value means distance from zero.

Step 2: \(-6\) is 6 units from zero.

Answer:

$$|-6| = 6$$

Worked Example 3

Question: Are \(11\) and \(-11\) opposites? What is their sum?

Step 1: Check whether they are the same distance from zero.

Both are 11 units from zero.

Step 2: Check whether they are on different sides of zero.

Yes. \(11\) is positive and \(-11\) is negative.

Step 3: Add them.

$$11 + (-11) = 0$$

Answer: Yes, they are opposites, and their sum is zero.

Worked Example 4

Question: Compare \(|-7|\) and \(|5|\).

Step 1: Find each absolute value.

$$|-7| = 7$$

$$|5| = 5$$

Step 2: Compare 7 and 5.

$$7 > 5$$

Answer:

$$|-7| > |5|$$

Common Mistakes to Avoid

• Mistake: Thinking absolute value can be negative.
  Absolute value is a distance, so it cannot be negative.
• Mistake: Thinking a number and its opposite are the same number.
  For example, \(4\) and \(-4\) are different numbers, even though they have the same absolute value.
• Mistake: Forgetting that zero is its own opposite.
  The opposite of \(0\) is still \(0\).

Quick Check

1. What is the opposite of \(-8\)?
2. What is \(|12|\)?
3. What is \(|-12|\)?
4. Do \(3\) and \(-3\) add to zero?

Answers:

1. \(8\)
2. \(12\)
3. \(12\)
4. Yes, because \(3 + (-3) = 0\).

Summary

Opposites are two numbers the same distance from zero but on opposite sides of zero. Opposites always add to zero.

Absolute value is the distance a number is from zero. It is always zero or positive. Opposite numbers have the same absolute value.

Comparing and Ordering Integers

Integers are whole numbers and their opposites, including zero. This means integers include numbers like \(-5\), \(-2\), \(0\), \(4\), and \(9\).

When we compare integers, we decide which number is greater, which is less, or whether they are equal. When we order integers, we put them in a sequence from least to greatest or greatest to least.

This is an important skill because negative numbers appear in real life. For example, temperature, elevation, and money owed can all be shown with negative integers.

1. Understanding integers on a number line

A number line helps us compare integers easily. On a number line:

• Numbers to the right are greater.
• Numbers to the left are smaller.
• Zero is in the middle between positive and negative numbers.

Here is a simple idea to remember:

$$\text{left} \quad \longrightarrow \quad -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \quad \longrightarrow \quad \text{right}$$

As you move farther left, the values get smaller. As you move farther right, the values get greater.

2. Comparing integers

We use these symbols to compare numbers:

• \(>\) means greater than
• \(<\) means less than
• \(=\) means equal to

For example:

• \(3 > -2\) because \(3\) is to the right of \(-2\) on the number line.
• \(-5 < -1\) because \(-5\) is to the left of \(-1\).
• \(0 > -4\) because zero is to the right of \(-4\).

A very important idea is this: among negative numbers, the one with the greater absolute value is actually smaller.

For example, \(-8\) and \(-3\) are both negative. Since \(-8\) is farther left on the number line, \(-8 < -3\).

3. Using absolute value to think about distance from zero

The absolute value of a number is its distance from zero. Distance is always positive or zero.

We write absolute value with bars:

$$|-4| = 4 \qquad |6| = 6 \qquad |0| = 0$$

Absolute value tells how far a number is from zero, but it does not tell whether the number is greater or less than another number by itself. You still need the number line idea.

For example:

• \(|-7| = 7\) and \(|2| = 2\)
• Even though \(7 > 2\), the number \(-7\) is still less than \(2\).

4. Ordering integers from least to greatest

To order integers from least to greatest:

1. Find the numbers that are farthest left on the number line.
2. Put negative numbers first, starting with the most negative.
3. Then place zero, if it is included.
4. Finally place positive numbers from smallest to largest.

Example pattern:

$$-10, -6, -2, 0, 3, 8$$

5. Ordering integers from greatest to least

To order integers from greatest to least, do the reverse:

1. Start with the numbers farthest right on the number line.
2. List positive numbers first from largest to smallest.
3. Then zero, if it is included.
4. Then negative numbers, ending with the most negative.

Example pattern:

$$8, 3, 0, -2, -6, -10$$

6. Comparing rational numbers written as decimals and fractions

Rational numbers include integers, fractions, and decimals. They can also be placed on a number line and compared in the same way.

For example, \(-1.5\) is between \(-2\) and \(-1\). Since it is to the left of \(-1\), we know:

$$-1.5 < -1$$

Also, \(\frac{1}{2}\) is the same as \(0.5\), which is to the right of zero. So:

$$\frac{1}{2} > 0$$

When comparing fractions and decimals, it often helps to think about where they belong on the number line.

Worked Example 1: Compare two integers

Compare \(-4\) and \(2\).

Step 1: Locate both numbers on the number line.

\(-4\) is left of zero, and \(2\) is right of zero.

Step 2: The number farther right is greater.

So, $$-4 < 2$$

Answer: \(-4\) is less than \(2\).

Worked Example 2: Compare two negative integers

Compare \(-9\) and \(-6\).

Step 1: Both numbers are negative, so look for which one is farther left.

Step 2: \(-9\) is farther left than \(-6\).

So, $$-9 < -6$$

Answer: \(-9\) is less than \(-6\).

Worked Example 3: Order integers from least to greatest

Order these numbers from least to greatest:

$$3, -1, 0, -5, 2$$

Step 1: Find the most negative number. That is \(-5\).

Step 2: Next is \(-1\).

Step 3: Then comes \(0\).

Step 4: Then list the positive numbers from smallest to largest: \(2, 3\).

So the order is:

$$-5, -1, 0, 2, 3$$

Answer: From least to greatest, the numbers are \(-5, -1, 0, 2, 3\).

Worked Example 4: Order rational numbers from greatest to least

Order these numbers from greatest to least:

$$-2, \frac{1}{2}, -1.5, 3$$

Step 1: Identify the largest number. \(3\) is greatest.

Step 2: \(\frac{1}{2} = 0.5\), so it comes next.

Step 3: Compare \(-1.5\) and \(-2\). Since \(-1.5\) is to the right of \(-2\), it is greater.

So the order is:

$$3, \frac{1}{2}, -1.5, -2$$

Answer: From greatest to least, the numbers are \(3, \frac{1}{2}, -1.5, -2\).

7. Helpful tips

• Think of a number line every time you compare integers.
• Right means greater; left means less.
• Negative numbers are always less than zero.
• Among negative numbers, the one with the bigger distance from zero is actually smaller.
• When ordering, move across the number line in the direction asked.

8. Common mistakes to avoid

• Thinking \(-8\) is greater than \(-3\) because \(8 > 3\). This is not correct. On the number line, \(-8\) is farther left, so \(-8 < -3\).
• Forgetting that zero is greater than any negative number.
• Mixing up least to greatest with greatest to least.
• Ignoring where decimals or fractions belong on the number line.

Summary

To compare and order integers, use the number line. Numbers farther right are greater, and numbers farther left are smaller. Negative numbers are less than zero, and among negative numbers, the one farther from zero is the smaller number. This same idea also works for rational numbers like decimals and fractions.

Adding Integers means finding the sum of whole numbers that can be positive, negative, or zero.

Integers are numbers like \,\(-5\), \(-2\), \(0\), \(3\), and \(9\). Positive integers are greater than zero, and negative integers are less than zero.

When we add integers, we can use two helpful models:

• Zero pairs
• A number line

These models help us understand why the rules work, not just memorize them.

1. Understanding positive and negative numbers

You can think of positive and negative numbers as opposites.

• A positive number can mean moving right on a number line.
• A negative number can mean moving left on a number line.

For example:

• \(+4\) means 4 steps to the right.
• \(-4\) means 4 steps to the left.

Another way to think about them is with counters:

• A positive counter represents \(+1\).
• A negative counter represents \(-1\).

One positive counter and one negative counter make a zero pair because

$$1 + (-1) = 0$$

2. Using zero pairs

Zero pairs are useful when adding a positive integer and a negative integer.

If a positive counter and a negative counter are together, they cancel out. Each pair has a value of zero.

For example, to find \(3 + (-2)\):

• Start with 3 positive counters.
• Add 2 negative counters.
• Match one positive counter with one negative counter to make a zero pair.

After making zero pairs, 1 positive counter is left.

So,

$$3 + (-2) = 1$$

3. Using a number line

A number line shows integers in order, with negative numbers to the left of 0 and positive numbers to the right of 0.

To add integers on a number line:

1. Start at the first number.
2. Move right if the second number is positive.
3. Move left if the second number is negative.

For example, to find \(-3 + 5\):

• Start at \(-3\).
• Since \(5\) is positive, move 5 steps right.
• You land on \(2\).

So,

$$-3 + 5 = 2$$

4. Rules for adding integers

After using models, we can see patterns that lead to rules.

Rule A: If the integers have the same sign, add their absolute values.

The absolute value of a number is its distance from 0. It is always positive.

Examples:

• The absolute value of \(-6\) is \(6\).
• The absolute value of \(6\) is also \(6\).

If both integers are positive, the answer is positive.

If both integers are negative, the answer is negative.

Example:

$$(-4) + (-3) = -(4+3) = -7$$

Rule B: If the integers have different signs, subtract their absolute values.

Then keep the sign of the integer with the greater absolute value.

Example:

$$(-8) + 3$$

The absolute values are 8 and 3. Subtract:

$$8 - 3 = 5$$

Since \(-8\) has the greater absolute value, the answer is negative:

$$(-8) + 3 = -5$$

5. Worked examples

Example 1: Add two positive integers

Find \(4 + 6\).

Both integers are positive, so add normally:

$$4 + 6 = 10$$

Answer: \(10\)

Example 2: Add two negative integers

Find \((-5) + (-7)\).

Both integers are negative, so add the absolute values:

$$5 + 7 = 12$$

Keep the negative sign:

$$(-5) + (-7) = -12$$

Answer: \(-12\)

Example 3: Add integers with different signs

Find \(9 + (-4)\).

The signs are different, so subtract the absolute values:

$$9 - 4 = 5$$

The greater absolute value is 9, which is positive, so the answer is positive:

$$9 + (-4) = 5$$

Answer: \(5\)

Example 4: Add integers with different signs

Find \((-11) + 6\).

The signs are different, so subtract the absolute values:

$$11 - 6 = 5$$

The greater absolute value is 11, and it belongs to \(-11\), so the answer is negative:

$$(-11) + 6 = -5$$

Answer: \(-5\)

6. Helpful steps to remember

• Look at the signs of the integers.
• If the signs are the same, add and keep the sign.
• If the signs are different, subtract and keep the sign of the number farther from 0.

7. Common mistakes to avoid

• Mistake: Always adding the numbers no matter what the signs are.
  Instead, check whether the signs are the same or different first.
• Mistake: Forgetting the sign of the answer.
  After subtracting, make sure to use the sign of the integer with the greater absolute value.
• Mistake: Confusing addition with subtraction.
  For example, \(5 + (-2)\) means start with 5 and add negative 2, which is the same as moving 2 left on the number line.

8. Quick practice ideas

Try these on your own:

• \(7 + (-3)\)
• \((-6) + (-2)\)
• \((-9) + 12\)
• \(8 + (-8)\)

You can solve them with zero pairs, a number line, or the rules.

Summary

Adding integers becomes easier when you understand what positive and negative numbers mean.

Use zero pairs to show how positive and negative counters cancel out. Use a number line to show movement left and right.

Remember:

• Same signs: add and keep the sign.
• Different signs: subtract and keep the sign of the number with the greater absolute value.

With practice, you will be able to add integers quickly and correctly.

Subtracting Integers as Adding the Opposite

Sometimes subtracting integers can look tricky, especially when negative numbers are involved. The good news is that there is one simple rule that makes integer subtraction much easier.

Rule: To subtract an integer, add its opposite.

In math symbols, this means:

$$a-b=a+(-b)$$

This rule works for all integers: positive numbers, negative numbers, and zero.

What is the opposite? The opposite of a number is the same distance from 0 on the number line, but on the other side.

• The opposite of \(5\) is \(-5\).
• The opposite of \(-3\) is \(3\).
• The opposite of \(0\) is \(0\).

So when you see subtraction, you can change it to addition by changing the second number to its opposite.

For example:

$$7-2=7+(-2)$$

And:

$$7-(-2)=7+2$$

This is called adding the opposite.

Why does this work? Subtraction asks, “What happens when we take away a number?” Adding the opposite gives the same result.

On a number line, subtracting a positive number moves left. Adding a negative number also moves left. That is why \(6-4\) and \(6+(-4)\) both end at the same place.

Also, subtracting a negative number moves right. Adding a positive number also moves right. That is why \(3-(-5)\) and \(3+5\) have the same answer.

Steps for subtracting integers

1. Keep the first number the same.
2. Change subtraction to addition.
3. Change the second number to its opposite.
4. Add.

Here is the pattern:

$$\text{keep }a,\quad -\,b\;\text{ becomes }\;+(-b)$$

Worked Example 1: Subtracting a positive integer

Solve \(8-3\).

Step 1: Keep the first number: \(8\)

Step 2: Change subtraction to addition.

Step 3: Change \(3\) to its opposite, which is \(-3\).

$$8-3=8+(-3)$$

Now add: starting at 8, adding \(-3\) means move 3 units left.

$$8+(-3)=5$$

Answer: \(8-3=5\)

Worked Example 2: Subtracting a negative integer

Solve \(6-(-4)\).

Keep the first number: \(6\)

Change subtraction to addition.

Change \(-4\) to its opposite, which is \(4\).

$$6-(-4)=6+4$$

Now add:

$$6+4=10$$

Answer: \(6-(-4)=10\)

Notice something important: subtracting a negative becomes adding a positive.

Worked Example 3: Starting with a negative number

Solve \((-5)-2\).

Keep the first number: \(-5\)

Change subtraction to addition.

Change \(2\) to its opposite, which is \(-2\).

$$(-5)-2=(-5)+(-2)$$

Now add two negative numbers. When both numbers are negative, add their distances from 0 and keep the negative sign.

$$(-5)+(-2)=-7$$

Answer: \((-5)-2=-7\)

Worked Example 4: Negative minus negative

Solve \((-9)-(-6)\).

Keep the first number: \(-9\)

Change subtraction to addition.

Change \(-6\) to its opposite, which is \(6\).

$$(-9)-(-6)=(-9)+6$$

Now add. Starting at \(-9\), moving 6 units right lands on \(-3\).

$$(-9)+6=-3$$

Answer: \((-9)-(-6)=-3\)

Helpful patterns to remember

• Subtracting a positive number means add a negative.
• Subtracting a negative number means add a positive.
• After rewriting the problem as addition, use your integer addition rules.

Here are some quick examples:

• \(4-7=4+(-7)=-3\)
• \(4-(-7)=4+7=11\)
• \((-2)-3=(-2)+(-3)=-5\)
• \((-2)-(-3)=(-2)+3=1\)

Common mistake to avoid

Do not just change the subtraction sign to an addition sign and leave the second number the same. You must also change the second number to its opposite.

For example, this is wrong:

$$7-2=7+2$$

This is correct:

$$7-2=7+(-2)$$

How to check your answer

You can check by thinking about the number line.

• If you subtract a positive number, your answer should usually be less than where you started.
• If you subtract a negative number, your answer should be greater than where you started.

For example, \(5-(-2)\) should be greater than 5, because subtracting a negative means moving right. And it is:

$$5-(-2)=5+2=7$$

Summary

Subtracting integers becomes much easier when you use one rule: subtracting is the same as adding the opposite.

Whenever you see a subtraction problem with integers:

1. Keep the first number.
2. Change subtraction to addition.
3. Change the second number to its opposite.
4. Add.

If you practice this rule, integer subtraction will feel much simpler.

Multiplying and Dividing Integers

Integers are whole numbers and their opposites, including zero. That means integers include numbers like \(-5\), \(-2\), \(0\), \(4\), and \(9\).

When we multiply and divide integers, we need to think about two things:

• the sign of the answer: positive or negative
• the basic fact: the multiplication or division part without the signs

This lesson will help you learn the sign rules and use them correctly.

1. What happens when we multiply integers?

Start by remembering what multiplication means. For example, \(3 \times 4\) means 3 groups of 4, which equals 12.

With integers, the sign matters. There are simple rules you can follow.

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative
• Negative \(\times\) Positive = Negative

You can remember this by thinking:

• same signs \(\rightarrow\) positive answer
• different signs \(\rightarrow\) negative answer

Here is the sign pattern:

Why does a negative times a negative equal a positive?

A helpful way to see this is by looking at a pattern.

Each time the first number goes down by 1, the answer goes down by 2. That pattern helps show why multiplying by a negative changes the sign.

Now look at another pattern:

The answers keep increasing by 2. This pattern shows that a negative times a negative gives a positive.

2. What happens when we divide integers?

Division uses the same sign rules as multiplication.

• Positive \(\div\) Positive = Positive
• Negative \(\div\) Negative = Positive
• Positive \(\div\) Negative = Negative
• Negative \(\div\) Positive = Negative

Again, you can remember:

• same signs \(\rightarrow\) positive answer
• different signs \(\rightarrow\) negative answer

Examples:

3. Use the basic fact first

A good strategy is to ignore the signs for a moment and solve the basic multiplication or division fact. Then decide whether the answer should be positive or negative.

For example, in \((-6) \times 7\):

• basic fact: \(6 \times 7 = 42\)
• signs are different, so the answer is negative

So,

4. What about zero?

Zero is an important integer.

• Any integer multiplied by 0 equals 0.
• 0 divided by any nonzero integer equals 0.
• You cannot divide by 0.

Examples:

But this is not allowed:

Division by 0 does not make sense, so it is undefined.

5. Multiplication and division are related

Multiplication and division are inverse operations. That means they undo each other.

For example, since

we also know:

and

This connection can help you check your answers.

Worked Examples

Example 1: Multiply a positive and a negative integer

Find \(5 \times (-3)\).

Step 1: Find the basic fact.

Step 2: The signs are different, so the answer is negative.

Answer: \(-15\)

Example 2: Multiply two negative integers

Find \((-6) \times (-4)\).

Step 1: Find the basic fact.

Step 2: The signs are the same, so the answer is positive.

Answer: \(24\)

Example 3: Divide integers with different signs

Find \((-28) \div 7\).

Step 1: Find the basic fact.

Step 2: The signs are different, so the answer is negative.

Answer: \(-4\)

Example 4: Divide two negative integers

Find \((-45) \div (-9)\).

Step 1: Find the basic fact.

Step 2: The signs are the same, so the answer is positive.

Answer: \(5\)

6. Common mistakes to avoid

• Do not add or subtract the signs. First solve the basic fact, then decide the sign.
• Do not forget: two negatives multiply or divide to make a positive.
• Do not divide by 0.
• Watch carefully for different signs. Different signs give a negative answer.

7. Quick sign guide

For both multiplying and dividing integers:

• Same signs: positive
• Different signs: negative

You can think of it like this:

Brief Summary

To multiply or divide integers, first solve the basic fact without the signs. Then use the sign rules: same signs give a positive answer, and different signs give a negative answer.

Also remember that any number times 0 is 0, and you can never divide by 0. If you use these rules carefully, you can solve integer multiplication and division problems with confidence.

Operations with Negative Rational Numbers

In this lesson, you will learn how to work with negative rational numbers. Rational numbers include integers, fractions, and decimals that can be written as a fraction.

Examples of rational numbers are:

• (5\)
• \(-3\)
• \(\frac{1}{2}\)
• \(-\frac{7}{4}\)
• \(0.25\)
• \(-1.8\)

A negative rational number is just a rational number that is less than zero, like \(-2\), \(-\frac{3}{5}\), or \(-0.7\).

When we do operations with negative rational numbers, we use the same ideas as with integers, fractions, and decimals. The most important part is paying attention to the sign: positive or negative.

1. Adding negative rational numbers

When adding rational numbers, first look at the signs.

• If the signs are the same, add the numbers and keep the sign.
• If the signs are different, subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value.

Absolute value means the distance from zero. For example, the absolute value of \(-4\) is \(4\).

Examples:

• \(-2.5 + (-1.3) = -3.8\)
• \(-\frac{3}{4} + \frac{1}{2}\)

For the fraction example, rewrite with a common denominator:

The signs are different, so we subtract \(2\) fourths from \(3\) fourths and keep the negative sign.

2. Subtracting negative rational numbers

Subtraction can be changed into addition:

This means subtracting a number is the same as adding its opposite.

Examples:

• \(3 - (-2) = 3 + 2 = 5\)
• \(-1.5 - 0.4 = -1.9\)
• \(-\frac{2}{3} - \left(-\frac{1}{6}\right) = -\frac{2}{3} + \frac{1}{6}\)

Now use a common denominator:

3. Multiplying negative rational numbers

To multiply signed numbers, multiply the number parts first. Then decide the sign.

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative
• Negative \(\times\) Positive = Negative

Examples:

• \((-3)(-2) = 6\)
• \((-4)(0.5) = -2\)
• \(\left(-\frac{2}{3}\right)\left(\frac{9}{10}\right)\)

Multiply the fractions:

4. Dividing negative rational numbers

Division follows the same sign rules as multiplication.

• Positive \(\div\) Positive = Positive
• Negative \(\div\) Negative = Positive
• Positive \(\div\) Negative = Negative
• Negative \(\div\) Positive = Negative

Examples:

• \((-12) \div 3 = -4\)
• \((-6) \div (-2) = 3\)
• \(\left(-\frac{3}{4}\right) \div \left(\frac{1}{2}\right)\)

To divide fractions, multiply by the reciprocal:

5. Order of operations with negative rational numbers

When an expression has more than one operation, use the order of operations:

1. Parentheses
2. Multiplication and division from left to right
3. Addition and subtraction from left to right

This is very important when negative numbers are involved.

For example:

Multiply first:

Then add:

Worked Example 1: Adding decimals

Find \(-2.7 + 5.1\).

The signs are different, so subtract the absolute values:

Since \(5.1\) has the larger absolute value and it is positive, the answer is:

Worked Example 2: Subtracting fractions

Find \(\frac{1}{3} - \left(-\frac{5}{6}\right)\).

Change subtraction to addition:

Use a common denominator of \(6\):

Now add:

Worked Example 3: Multiplying and dividing signed numbers

Find $$\left(-1.2\right) \times \left(-\frac{5}{2}\right)$$

First decide the sign. A negative times a negative is positive.

Rewrite \(-1.2\) as a fraction:

Now multiply:

So the answer is:

Worked Example 4: A full expression

Find:

Use order of operations. Multiply first:

Now add:

Rewrite \(-2\) as halves:

Now add:

Helpful tips

• Always check the sign before you calculate.
• For addition and subtraction with fractions, use a common denominator.
• For division with fractions, multiply by the reciprocal.
• Follow the order of operations carefully.
• If the signs are different when adding, subtract the absolute values.

Common mistakes to avoid

• Forgetting to change subtraction into addition of the opposite.
• Mixing up the sign rules for multiplication and division.
• Adding denominators when adding fractions.
• Ignoring parentheses in an expression.

Summary

Negative rational numbers include negative integers, fractions, and decimals. To add and subtract them, pay close attention to the signs and use common denominators when needed.

To multiply and divide, use the sign rules: same signs give a positive answer, and different signs give a negative answer. When solving longer expressions, always use the order of operations.