Negative Integers

What Are Negative Integers?

A negative integer is a whole number that is less than zero. Integers themselves are all the whole numbers (like $0,1,2,3\dots$) and their opposites (like $-1,-2,-3\dots$). Negative integers are simply the ones with a minus sign ($-$) in front of them. They represent the opposite of positive numbers.

Think about a number line. If positive numbers are steps you take to the right of zero, negative numbers are the steps you take to the left. For every positive integer, there is a corresponding negative integer that is the same distance from zero, just in the opposite direction. These pairs are called opposites.

• The opposite of $5$ is $-5$.
• The opposite of $-12$ is $12$.

We use negative integers all the time in the real world, often without even thinking about it:

• Temperature: When the weather gets very cold, the temperature can drop below zero. A temperature of $-{10}^{\circ }$ Fahrenheit is $10$ degrees colder than $0^{\circ }$ F.
• Money & Debt: If you have $\mathrm{\$}50$ in your bank account, you have a positive balance. But if you owe someone $\mathrm{\$}50$, you could think of your balance as $-\mathrm{\$}50$.
• Elevation: The height of land is measured relative to sea level. Mount Everest has an elevation of $29,032$ feet above sea level. The Dead Sea shore, however, is about $1,412$ feet below sea level, which we can represent as $-1,412$ feet.
• Games: In some card games or video games, you can lose points and have a negative score.

Understanding negative integers is the first step to mastering more advanced algebra concepts, so getting comfortable with them now is key!

How Do We Visualize and Compare Negative Integers?

The best tool for understanding negative integers is the number line. A number line is a straight line with numbers placed at equal intervals along its length. It helps us see the relationship between numbers.

Here’s how it works:

1. Zero is the Center: Zero is neither positive nor negative. It's the starting point that separates the two sides.
2. Positives on the Right: All positive integers ($1,2,3,\dots$) are located to the right of zero. Numbers increase in value as you move to the right. For example, $8$ is greater than $3$.
3. Negatives on the Left: All negative integers ($-1,-2,-3,\dots$) are located to the left of zero. Numbers decrease in value as you move to the left.

This last point is very important! It can be confusing at first. On the left side of zero, the further away a number is, the smaller its value. This means $-10$ is actually smaller than $-2$. Think about temperature: $-{10}^{\circ }$ is much colder (and therefore a lower temperature) than $-2^{\circ }$.

To compare any two integers on a number line, just remember this simple rule: The number on the right is always greater.

• Comparing $-5$ and $-1$: Find both on the number line. $-1$ is to the right of $-5$, so $-1>-5$.
• Comparing $-7$ and $2$: Find both on the number line. $2$ is to the right of $-7$, so $2>-7$.

What Is Absolute Value?

Absolute value is a fundamental concept related to integers. The absolute value of a number is its distance from zero on the number line. Since distance can't be negative, the absolute value of a number is always positive or zero.

We show absolute value by putting two vertical bars around the number, like this: $|x|$. This is read as "the absolute value of $x$".

Let's look at some examples:

• To find $|5|$, we ask: "How far is $5$ from zero?" It is $5$ units away. So, $|5|=5$.
• To find $|-5|$, we ask: "How far is $-5$ from zero?" It is also $5$ units away. So, $|-5|=5$.

Notice that opposites have the same absolute value. The number's sign tells you the direction from zero (left or right), while the absolute value tells you the distance from zero. The absolute value of zero itself is just zero, because it has no distance from itself: $|0|=0$.

How Do You Add Negative Integers?

Adding with negative integers can be thought of as movement on the number line. A positive number means you move to the right, and a negative number means you move to the left. There are two main scenarios you'll encounter.

Case 1: Adding Two Negative Integers

When you add two negative integers, you are combining two 'leftward' movements. This means you will move further to the left, resulting in a more negative number.

Rule: To add two negative numbers, add their absolute values and then make the answer negative.

For example, let's solve $-4+(-3)$.
The absolute value of $-4$ is $4$.
The absolute value of $-3$ is $3$.
Add them: $4+3=7$.
Since both original numbers were negative, the answer is negative: $-7$.

Case 2: Adding a Positive and a Negative Integer

When you add a positive and a negative integer, you are moving in opposite directions. The result depends on which number has the larger 'pull' or absolute value.

Rule: To add a positive and a negative number, find the difference between their absolute values. The answer takes the sign of the number with the larger absolute value.

For example, let's solve $8+(-5)$.
The absolute value of $8$ is $8$.
The absolute value of $-5$ is $5$.
Find the difference: $8-5=3$.
Since $8$ has the larger absolute value and it is positive, the answer is positive: $3$.

Now let's solve $-9+2$.
The absolute value of $-9$ is $9$.
The absolute value of $2$ is $2$.
Find the difference: $9-2=7$.
Since $-9$ has the larger absolute value and it is negative, the answer is negative: $-7$.

How Do You Subtract Negative Integers?

Subtracting negative integers is the place where most students get confused, but there is one simple, powerful rule that makes it easy. The rule is: to subtract an integer, you add its opposite. This is often called "Keep-Change-Change" or "Add the Opposite".

Here's how it works:

1. Keep the first number the same.
2. Change the subtraction sign to an addition sign.
3. Change the sign of the second number to its opposite.

After you do this, you just have a simple addition problem that you already know how to solve!

Let's see this in action.

Scenario 1: Subtracting a Positive Number

Let's solve $-5-3$.
1. Keep the $-5$.
2. Change the $-$ to a $+$.
3. Change the $3$ to its opposite, $-3$.
The problem becomes $-5+(-3)$. Now we just add two negatives, which gives us $-8$.

Scenario 2: Subtracting a Negative Number

This is the most important case. Let's solve $4-(-6)$.
1. Keep the $4$.
2. Change the $-$ to a $+$.
3. Change the $-6$ to its opposite, $6$.
The problem becomes $4+6$, which is simply $10$. Subtracting a negative number made the result bigger!

Think of it this way: taking away a debt (a negative) is the same as giving you money (a positive).

How Do You Multiply and Divide Negative Integers?

Compared to addition and subtraction, the rules for multiplying and dividing integers are very straightforward. It all comes down to the signs of the numbers you are working with. The rules are the same for both multiplication and division.

The Sign Rules

There are only two rules you need to memorize:

1. If the signs are the same (both positive or both negative), the answer is always positive.
2. If the signs are different (one positive and one negative), the answer is always negative.

That's it! You perform the multiplication or division as you normally would with positive numbers, and then you apply the correct sign rule to the answer.

Here is a table to help you remember:

A simple way to remember: Same signs, positive answer. Different signs, negative answer.

What Are Common Mistakes When Working with Negative Integers?

Working with negative integers can be tricky, and a few common errors often trip students up. Being aware of these mistakes is the best way to avoid making them!

• Confusing Value and Absolute Value: A common mistake is thinking that $-10$ is a larger number than $-2$. Remember, on the number line, numbers to the left are always smaller. While $-10$ has a larger absolute value ($|-10|=10$), its actual value is less than $-2$.
• Mixing Up Addition and Multiplication Rules: Students sometimes apply the multiplication sign rules to addition. For example, they might see $-4+(-2)$ and think "two negatives make a positive," leading to an incorrect answer of $6$. The correct answer is $-6$. The sign rules for multiplication/division are different from the rules for addition.
• Forgetting to "Add the Opposite" when Subtracting: The most frequent error in subtraction is forgetting the Keep-Change-Change rule. For a problem like $8-(-5)$, many will incorrectly calculate $8-5=3$. You must change the problem to $8+5$ to get the correct answer, $13$.
• Sign Errors with Zero: Remember that zero is neutral—it's neither positive nor negative. Any number plus its opposite equals zero (e.g., $-7+7=0$). Any number multiplied by zero is zero (e.g., $-7\times 0=0$).
• Order of Operations with Negatives: When dealing with exponents, be careful. $(-3{)}^2$ means $(-3)\times (-3)$, which is $9$. However, $-3^2$ means $-(3\times 3)$, which is $-9$. The parentheses are very important!

Quick Summary and Reference

Here is a quick reference guide to the key rules for operating with negative integers. Keep this handy when you're doing homework!

Comparing Integers

• On a number line, the number to the right is always greater.
• Example: $-3>-9$

Absolute Value

• A number's distance from zero. It is always positive or zero.
• Example: $|-6|=6$

Addition Rules

• Same Signs: Add the absolute values and keep the original sign. (e.g., $-5+(-2)=-7$)
• Different Signs: Subtract the smaller absolute value from the larger one. The answer takes the sign of the number with the larger absolute value. (e.g., $10+(-4)=6$)

Subtraction Rule

• Add the Opposite (Keep-Change-Change): To subtract an integer, add its opposite.
• Example: $5-(-9)=5+9=14$
• Example: $-2-8=-2+(-8)=-10$

Multiplication and Division Rules

• Same Signs: The result is always positive. (e.g., $-4\times (-5)=20$)
• Different Signs: The result is always negative. (e.g., $18÷(-3)=-6$)