Whole Number Vs Integer

What Exactly Are Whole Numbers?

Whole numbers are the set of positive numbers including zero, without any fractions or decimals. Think of them as the most basic numbers you use for counting things. When you count the number of pencils on your desk, the number of friends in your class, or the number of cookies in a jar, you are using whole numbers.

The list of whole numbers is easy to remember. It starts at zero and goes on forever. It looks like this:

The three dots (called an ellipsis) mean that the list continues infinitely. There is no largest whole number! However, there is a smallest whole number: $0$. A key rule for whole numbers is that they cannot be negative, and they cannot have a fractional or decimal part. So, numbers like $-5$, $8.25$, or $\frac{1}{2}$ are not whole numbers.

On a number line, whole numbers start at $0$ and are marked at equal intervals as you move to the right into the positive territory.

So, What Are Integers?

Integers are the set of all whole numbers and their negative opposites, without any fractions or decimals. Integers expand our number world by introducing negative values. They answer questions like, "What's the temperature when it's colder than zero?" or "How can we represent a debt?"

The set of integers includes all the whole numbers $(0,1,2,3,...)$, but it also includes the negative versions of all the counting numbers $(-1,-2,-3,...)$. The full set of integers stretches infinitely in both the positive and negative directions.

Why the letter $Z$? It comes from the German word Zahlen, which means "numbers." Just like whole numbers, integers cannot have fractional or decimal parts. This means that $-4.5$ and $9\frac{3}{4}$ are not integers. The number $0$ is also an integer, and it's special because it is neither positive nor negative.

How Do Whole Numbers and Integers Relate?

The relationship between whole numbers and integers is a fundamental concept. The simplest way to think about it is this: all whole numbers are also integers, but not all integers are whole numbers.

Imagine you have a big box labeled "Integers." Inside that box, you have a smaller box labeled "Whole Numbers." Any number you find in the "Whole Numbers" box (like $0,1,2,...$) is automatically inside the bigger "Integers" box. However, there are many numbers in the "Integers" box that are outside the smaller "Whole Numbers" box, such as $-1,-2,-100$, and so on. These are the negative integers.

So, if someone asks you if the number $5$ is an integer, the answer is yes! Since $5$ is a whole number, it must also be an integer. But if they ask if $-5$ is a whole number, the answer is no, because whole numbers cannot be negative.

This table breaks down the key differences and similarities:

How Do You Identify Which Number is Which?

Categorizing numbers can be easy if you ask yourself a few simple questions. When you see a number, follow these steps to determine if it's a whole number, an integer, or neither.

1. Does the number have a fraction or a decimal part? If the answer is yes (like $6.7$ or $\frac{1}{4}$), it is neither a whole number nor an integer. Stop here.
2. If it has no fraction or decimal, is the number negative? If the answer is yes (like $-8$ or $-50$), it is an integer. It is not a whole number.
3. If it has no fraction or decimal and is not negative, is it $0$ or a positive counting number? If the answer is yes (like $0$, $19$, or $300$), it is both a whole number and an integer!

How Can a Number Line Help Us See the Difference?

The number line is one of the best tools for understanding integers. It gives us a picture of how numbers relate to each other, especially when negatives are involved.

An integer number line has zero in the center. As you move to the right, the numbers become positive and increase in value ($1,2,3,...$). As you move to the left of zero, the numbers become negative and decrease in value ($-1,-2,-3,...$). A number to the right is always greater than a number to its left.

This can sometimes feel tricky. For example, $-2$ is greater than $-8$. Why? Because on a number line, $-2$ is located to the right of $-8$. Think about temperature: a day that is $-2$ degrees is warmer (a greater temperature) than a day that is $-8$ degrees.

Where Do We Use These Numbers in Real Life?

You use whole numbers and integers all the time, probably without even thinking about it! They help us describe and measure the world around us.

Whole Number Examples:

• Counting Objects: The number of students in your class (e.g., $28$ students). You can't have a negative or fractional student!
• Age: You might be $10$ or $11$ years old.
• Scores in some sports: A basketball team can score $98$ points.

Integer Examples:

• Temperature: A cold winter day might be $-{10}^{\circ }$ Fahrenheit. A hot summer day could be ${95}^{\circ }$ Fahrenheit.
• Money & Finance: If you have $\mathrm{\$}50$ in your account, that's $+50$. If you owe your friend $\mathrm{\$}5$, you can think of your balance with them as $-5$.
• Elevation: The peak of a mountain might be $14,000$ feet above sea level ($+14000$), while a valley like Death Valley is $282$ feet below sea level ($-282$).
• Games: In a video game, you might gain $500$ points ($+500$) for finding treasure but lose $200$ points ($-200$) for hitting an obstacle.

What Are Some Common Mistakes to Avoid?

When learning about whole numbers and integers, a few common mix-ups can happen. Being aware of them is the best way to avoid them!

• Forgetting Zero: Many people remember the positive counting numbers but forget that $0$ is a crucial member of the whole number set. Remember, the whole numbers start at $0$, not $1$. Zero is also an integer.
• Confusing Integers with All Numbers: A common error is to think that any number with a negative sign is an integer. But numbers like $-3.14$ or $-1/2$ are not integers because they have decimal or fractional parts. Integers must be 'whole' units.
• Thinking Negative Numbers are Whole Numbers: This is the most direct confusion between the two sets. If a number is negative (other than zero), it cannot be a whole number. Whole numbers are exclusively $0$ and positive.
• Mixing up Value with Negatives: It's easy to think that $-10$ is a "bigger" or "greater" number than $-3$ because $10$ is bigger than $3$. But when comparing numbers, always think of a number line. Since $-3$ is to the right of $-10$, it is the greater number. $-3>-10$.

A Quick Summary for Your Notes

Here is a simple breakdown you can use as a quick reference guide when you're studying or doing homework.

Whole Numbers

• What are they? The basic counting numbers, plus zero.
• The Set: $\{0,1,2,3,4,...\}$
• Key Features: No negatives, no fractions, no decimals.
• Smallest Value: $0$

Integers

• What are they? All the whole numbers plus their negative opposites.
• The Set: $\{...,-3,-2,-1,0,1,2,3,...\}$
• Key Features: Includes positives, negatives, and zero. Still no fractions or decimals.
• Smallest Value: None! It goes on forever in the negative direction.

The Golden Rule: Remember, every whole number is an integer. But only the positive integers and zero are whole numbers.