Integer

What Are Integers?

Integers are the set of all whole numbers, their opposites (which are the negative numbers), and zero. Think of them as the solid, whole steps you can take on a number line, moving both forward and backward from your starting point. A key rule to remember is that integers never include fractions or decimals.

The complete set of integers is often represented by the symbol $\mathbb{Z}$ in higher math. It looks like an infinite list stretching in both positive and negative directions:

We can break this set down into three parts:

• Positive Integers: These are the integers greater than zero ($1,2,3,4,\dots$). They are the numbers we first learned to count with.
• Negative Integers: These are the integers less than zero ($\dots ,-4,-3,-2,-1$). They are the opposites of the positive integers.
• Zero: The number $0$ is an integer, but it is special because it is neither positive nor negative. It acts as the central point on the number line.

A number line is the best way to visualize integers. It's a line with zero in the middle, positive numbers increasing to the right, and negative numbers decreasing to the left.

<--|----|----|----|----|----|----|----|----|----|-->
...-5  -4  -3  -2  -1   0   1   2   3   4  ...

Where Do We See Integers in Real Life?

Integers aren't just abstract numbers for math class; they are all around us! Understanding them helps us make sense of the world. Here are some common examples:

• Temperature: Weather reports use integers to describe how hot or cold it is. A temperature of ${25}^{\circ }$ Celsius is a warm day, while $-{10}^{\circ }$ Celsius is very cold, meaning it's $10$ degrees below zero.
• Bank Accounts: When you deposit money, your balance increases (a positive change). When you withdraw money or pay a bill, your balance decreases (a negative change). A balance of $-\mathrm{\$}50$ means you owe the bank money.
• Elevation: Geographers use integers to describe height relative to sea level. Mount Everest has an elevation of $+8,848$ meters (above sea level), while the Dead Sea is at $-430$ meters (below sea level).
• Sports: In American football, a team gaining $10$ yards is a positive integer ($+10$), while a penalty or a sack might result in a loss of $5$ yards, which is a negative integer ($-5$).
• Timelines: Historians use negative numbers to refer to years before year 1 (B.C.E.). The Roman Empire was founded in the year $-753$ (or 753 B.C.E.).

How Do You Compare Integers?

Comparing integers means figuring out which one is larger or smaller. The number line is your best friend for this. The rule is simple: any number to the right on the number line is greater than any number to its left.

We use the greater than ($>$) and less than ($<$) symbols to compare them.

• $5>2$ ($5$ is to the right of $2$ on the number line)
• $-1>-4$ (This can be tricky! Find $-1$ and $-4$ on the number line. $-1$ is to the right of $-4$, so it is greater.)
• $3>-3$ (Any positive integer is always greater than any negative integer.)
• $-6<0$ ($-6$ is to the left of $0$, so it is less than zero.)

A helpful tip for negative numbers: Think about temperature. Is it colder at $-{10}^{\circ }$ or $-2^{\circ }$? It's colder at $-{10}^{\circ }$, so $-10$ is the smaller number. Therefore, $-10<-2$.

What Is Absolute Value?

The absolute value of an integer 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: $|x|$.

To find the absolute value, you just ask yourself, "How many steps does it take to get from this number to zero?"

• The absolute value of $6$ is $6$, because it takes $6$ steps to get from $6$ to $0$. We write this as $|6|=6$.
• The absolute value of $-6$ is also $6$, because it also takes $6$ steps to get from $-6$ to $0$. We write this as $|-6|=6$.

That second part looks strange, but it works! If $x=-9$, then $|-9|=-(-9)=9$.

How Do You Add and Subtract Integers?

Adding and subtracting integers can be visualized by moving along the number line. Start at the first number. Adding a positive number means moving to the right. Adding a negative number (or subtracting a positive number) means moving to the left.

Rules for Adding Integers

1. Same Signs: If the integers have the same sign (both positive or both negative), add their absolute values and keep the common sign.
   Example: $-3+(-5)$. Add $3+5=8$. Since both are negative, the answer is $-8$.
2. Different Signs: If the integers have different signs, subtract the smaller absolute value from the larger absolute value. The answer takes the sign of the number with the larger absolute value.
   Example: $7+(-2)$. Subtract $7-2=5$. Since $|7|>|-2|$ and $7$ is positive, the answer is $+5$.
   Example: $-9+4$. Subtract $9-4=5$. Since $|-9|>|4|$ and $9$ is negative, the answer is $-5$.

Rules for Subtracting Integers

The easiest way to subtract integers is to use the "Keep-Change-Change" method. You change the subtraction problem into an addition problem.

For example, to solve $8-12$:

• Keep the $8$.
• Change the subtraction sign to an addition sign.
• Change the $12$ to its opposite, $-12$.

So, $8-12$ becomes $8+(-12)$. Now use the addition rules: subtract the absolute values ($12-8=4$) and take the sign of the number with the larger absolute value ($-12$). The answer is $-4$.

How Do You Multiply and Divide Integers?

Multiplying and dividing integers is actually easier than adding and subtracting! You just need to remember two simple rules for the signs. First, multiply or divide the numbers as you normally would, then apply the sign rule.

The Sign Rules for Multiplication and Division

1. If the signs of the two integers are the same, the result is always positive.
2. If the signs of the two integers are different, the result is always negative.

Here is a table to help you remember:

Why does a negative times a negative equal a positive? Think of it as 'removing a debt'. If someone removes 3 debts of $\mathrm{\$}7$ from you ($-3\times -7$), your net worth has gone up by $\mathrm{\$}21$.

Common Mistakes to Avoid with Integers

Working with integers can be tricky at first. Here are some common pitfalls to watch out for:

• Confusing Value and Absolute Value: A common mistake is thinking that $-10$ is greater than $-2$ because $10$ is greater than $2$. Always use a number line to check! The number further to the right is always greater ($-2>-10$).
• Subtracting Negatives: The expression $5-(-3)$ often causes confusion. Many students incorrectly get $2$. Remember to use Keep-Change-Change: $5-(-3)$ becomes $5+3$, which equals $8$. Taking away a negative is a positive!
• Mixing Up Rules: Students sometimes apply the multiplication/division sign rules to addition/subtraction. For example, they might see $-2+(-3)$ and think the answer is $+5$ because 'two negatives make a positive'. This rule is only for multiplication and division. For addition, $-2+(-3)=-5$.
• Absolute Value Errors: The absolute value of a number can never be negative. It is a distance. So, $|-8|$ is $8$, not $-8$. However, an expression like $-|-8|$ is different. This means "the opposite of the absolute value of -8", which would be $-(8)=-8$.

Quick Reference: Integer Rules Summary

Here's a quick summary of the key rules for operating with integers. Use this as a study guide!

Addition

• Same Signs: Add the numbers, keep the sign. (e.g., $-5+(-2)=-7$)
• Different Signs: Subtract the smaller absolute value from the larger, keep the sign of the number with the larger absolute value. (e.g., $-8+3=-5$)

Subtraction

• Keep-Change-Change: Change the problem to addition and add the opposite. (e.g., $4-(-9)$ becomes $4+9=13$)

Multiplication and Division

• Same Signs: The answer is POSITIVE. (e.g., $(-6)\times (-3)=18$)
• Different Signs: The answer is NEGATIVE. (e.g., $12÷(-4)=-3$)

Absolute Value

• The distance from zero. It's always non-negative. (e.g., $|-15|=15$)