Whole Number
Introduction
Whole numbers are essentially all the numbers you can count on your fingers, starting from `0` and going up to infinity. They include natural numbers, which are the counting numbers like `1, 2, 3,` and so on, along with the number `0`. Now, think of integers; they're like whole numbers but also include negative versions of the counting numbers, like `-1, -2, -3,` and so forth. Finally, real numbers are like the big umbrella that covers all these types of numbers: natural numbers, whole numbers, integers, and even fractions. So, while all whole numbers are real numbers, not all real numbers are whole numbers. Examples of whole numbers could be `0, 7, 15, 28,` and so on.
What Are Whole Numbers?
Whole numbers are basically what you get when you take into account all the natural numbers and throw in the number `0` as well in the list. So, they start from `0` and go on forever, just like natural numbers do. Think of it this way: if natural numbers are a set of numbers starting from `1`, then whole numbers are like the same set but this time it starts from `0`. The key difference between natural and whole numbers is that whole numbers include zero, while natural numbers don't. Some whole numbers examples could be `0, 10, 24, 37,` and so forth.
Symbol for Whole Numbers
To represent whole numbers, we use a special symbol: the capital letter '`W`'. It's like giving a name tag to all the numbers in the set. So, when you see '`W`', you know we're talking about whole numbers. Picture it like this: `W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}`. This means that the whole numbers list starts from `0` and goes on till infinity.
Natural Numbers and Whole Numbers
A natural number encompasses all positive whole numbers excluding zero. Additionally, it's important to note that every natural number is also categorized as a whole number. Hence, the set of natural numbers is a component of the set of whole numbers.
Difference Between Whole Numbers and Natural Numbers
Properties of Whole Numbers
`1`. Closure Property
`2`. Commutative Property
`3`. Associative Property
`5`. Additive Identity
`6`. Multiplicative Identity
`7`. Multiplication by Zero
`8`. Division by Zero
Closure Property
The closure property of whole numbers states that when you add or multiply two whole numbers, the result will always be a whole number.
Example:
Let's take two whole numbers, `6` and `9`.
Solution:
`1`. Adding them: `6 + 9 = 15` (a whole number)
`2`. Multiplying them: `6 × 9 = 54` (a whole number)
Commutative Property
The commutative property of whole numbers says that when you add or multiply two whole numbers, the order of numbers doesn't matter; you'll get the same result either way.
Example:
Consider two whole numbers, `4` and `7`.
Solution:
`1`. Adding them: `4 + 7 = 11` and `7 + 4 = 11`
`2`. Multiplying them: `4 × 7 = 28` and `7 × 4 = 28`
Associative Property
The associative property of whole numbers states that when you add or multiply three or more numbers, the grouping doesn't matter; you'll get the same result.
Example:
Let's take three whole numbers, `2, 3,` and `5`.
Solution:
`1`. Adding them: `(2 + 3) + 5 = 2 + (3 + 5) = 10`
`2`. Multiplying them: `(2 × 3) × 5 = 2 × (3 × 5) = 30`
Distributive Property
The distributive property of whole numbers tells us how to distribute multiplication over addition or subtraction.
Example:
Consider three whole numbers, `4, 6,` and `8`.
Solution:
`1`. Distributive property of multiplication over addition: `4 × (6 + 8) = (4 × 6) + (4 × 8)`
`2`. Distributive property of multiplication over subtraction: `4 × (8 - 6) = (4 × 8) - (4 × 6)`
Additive Identity
The additive identity property of whole numbers states that when you add zero to any whole number, you get the same whole number.
Example:
Let's take a whole number, `10`.
Solution:
Adding it with zero: `10 + 0 = 10`
Multiplicative Identity
The multiplicative identity property of whole numbers states that when you multiply any whole number by `1`, you get the same whole number.
Example:
Consider a whole number, `15`.
Solution:
Multiplying it by `1: 15 × 1 = 15`
Multiplication by Zero
Multiplying any whole number by zero always results in zero.
Example:
Let's take a whole number, `20`.
Solution:
Multiplying it by zero: `20 × 0 = 0`
Division by Zero
Division by zero is undefined in mathematics; it cannot be performed.
Example:
Consider a whole number, `30`.
Solution:
Dividing it by zero: `30 ÷ 0 =` Undefined
Whole Numbers on Number Line
Whole numbers and natural numbers can be represented visually on a number line. On the number line, natural numbers are all the positive integers starting from `1` and extending infinitely to the right. Whole numbers, on the other hand, include all the natural numbers along with zero. Thus, on the number line, whole numbers occupy the space to the right of zero, including zero itself.
Smallest Whole Number
The smallest whole number is `0`. Whole numbers begin from `0` according to whole numbers definition. Zero, despite having no numerical value, serves as a placeholder and falls between positive and negative numbers on a number line. It is neither positive nor negative.
Solved Examples
Example `1`: Identify the whole numbers among the following numbers: ` -3, \frac{1}{2}, 55, 0, 103.2 `.
Solution:
Among the given numbers, the whole numbers are `55` and `0`.
Example `2`: Find the sum of the whole numbers `124, 357,` and `619`.
Solution:
To find the sum, we add the given whole numbers:
`124 + 357 + 619 = 1100`
Therefore, the sum of the whole numbers is `1100`.
Example `3`: Use the distributive property of multiplication to simplify the expression: ` 8 \times (12 - 4) `.
Solution:
Applying the distributive property:
` 8 \times (12 - 4)`
`= 8 \times 12 - 8 \times 4 `
` = 96 - 32 `
` = 64 `
Therefore, ` 8 \times (12 - 4) = 64 `.
Example `4`: Prove the commutative property of addition for the whole numbers `215` and `327`.
Solution:
According to the commutative property of addition:
` a + b = b + a `
Given ` a = 215 ` and ` b = 327 `:
` 215 + 327 = 327 + 215 `
` 542 = 542 `
Hence, the commutative property of addition is verified.
Example `5`: Determine the product of the whole numbers `18` and `0`.
Solution:
The product of any whole number and `0` is always `0`.
So, ` 18 \times 0 = 0 `.
Therefore, the product of `18` and `0` is `0`.
Practice Problems
Q`1`. Which of the following numbers is a whole number?
- `-8`
- `2.5`
- `0`
- `-3/4`
Answer: c
Q`2`. Evaluate the expression ` 7 \times (10 - 3) ` using the distributive property.
- `56`
- `49`
- `35`
- `42`
Answer: b
Q`3`. Which property is illustrated by the equation ` 15 + (8 + 6) = (15 + 8) + 6 `?
- Associative Property of Addition
- Commutative Property of Addition
- Associative Property of Multiplication
- Commutative Property of Multiplication
Answer: a
Q`4`. If ` x ` and ` y ` are whole numbers, what is the product of ` x ` and `0`?
- ` x `
- ` y `
- `0`
- ` xy `
Answer: c
Q`5`. Which of the following numbers is not a whole number?
- `21`
- `0`
- `5`
- `3/4`
Answer: d
Frequently Asked Questions
Q`1`. What is a whole number?
Answer: Whole numbers are a set of numbers that include all the natural numbers (counting numbers) from `0` onwards, along with `0` itself. In mathematical notation, the set of whole numbers is denoted as ` \{0, 1, 2, 3, \ldots\} `. Any number that belongs to this set is a whole number.
Q`2`. What is the difference between whole numbers and natural numbers?
Answer: The main difference between whole numbers and natural numbers is that natural numbers start from `1` and go onwards `\{1, 2, 3, \ldots\}`, while whole numbers include `0` along with all the natural numbers `\{0, 1, 2, 3, \ldots\}`.
Q`3`. Are negative numbers considered whole numbers?
Answer: No, negative numbers are not considered whole numbers. Whole numbers consist of non-negative integers, starting from `0` and going towards positive infinity.
Q`4`. What properties do whole numbers possess?
Answer: Whole numbers possess several properties, including closure property (addition and multiplication), commutative property (addition and multiplication), associative property (addition and multiplication), distributive property, additive identity, multiplicative identity, and multiplication by zero.
Q`5`. How do you add and multiply whole numbers?
Answer: Adding and multiplying whole numbers follow the same rules as adding and multiplying integers. In addition, you simply add the numbers together. For multiplication, you multiply the numbers together. If there are multiple numbers to add or multiply, you can use properties like the associative and commutative properties to simplify the calculations.