Natural numbers are the numbers we use for counting: 1, 2, 3, 4 and so on. But what about zero? When we have nothing to count, we write zero. Adding zero to the collection of natural numbers gives us the collection of whole numbers: 0, 1, 2, 3, 4 and so on. The number zero is extremely important in mathematics because it represents nothingness and also acts as the starting point of the number line. Without zero, our number system would be incomplete.
In this chapter, we study the properties of whole numbers: closure property, commutativity, associativity and distributivity. These properties help us perform calculations faster and understand why the rules of arithmetic work. We also learn about the number line, which is a visual tool for representing numbers, and the special numbers zero and one, which behave in unique ways during addition and multiplication.
The concept of the number line helps us understand addition as moving to the right and subtraction as moving to the left. We also learn about patterns in whole numbers, such as the multiplication table patterns, which sharpen our observation skills. This chapter builds the arithmetic foundation needed for algebra, integers and fractions in later chapters.
A number line is a horizontal line marked with numbers at equal distances. We choose a starting point and label it zero. To the right of zero we mark 1, 2, 3 and so on. On the number line:
For example, 2 + 3 means starting at 2 and moving 3 steps to the right to reach 5. The number 6 - 4 means starting at 6 and moving 4 steps to the left to reach 2. Multiplication 3 x 4 can be shown as making 3 jumps of 4 units each to reach 12.
Every whole number has a definite position on the number line. A number on the right of another number is always greater than the number on the left. For example, 9 is to the right of 5, so 9 > 5.
A collection of numbers is said to be closed under an operation if performing that operation on any two numbers of the collection gives a number that also belongs to the collection.
Addition and multiplication are commutative for whole numbers. - Commutative property of addition: a + b = b + a. Example: 7 + 9 = 9 + 7 = 16. - Commutative property of multiplication: a x b = b x a. Example: 3 x 8 = 8 x 3 = 24. - Subtraction and division are NOT commutative: 9 - 5 is not equal to 5 - 9, and 8 divided by 2 is not equal to 2 divided by 8.
Addition and multiplication are associative for whole numbers. - Associative property of addition: (a + b) + c = a + (b + c). Example: (2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9. - Associative property of multiplication: (a x b) x c = a x (b x c). Example: (2 x 3) x 4 = 6 x 4 = 24 and 2 x (3 x 4) = 2 x 12 = 24.
Multiplication distributes over addition: a x (b + c) = (a x b) + (a x c). Example: 3 x (5 + 2) = 3 x 7 = 21 and (3 x 5) + (3 x 2) = 15 + 6 = 21. This property helps us multiply large numbers mentally by breaking them up.
Zero and one are two very special whole numbers.
Whole numbers show interesting patterns that help in quick mental calculation.
2 x 5 = 10, 3 x 5 = 15, 4 x 5 = 20. Observe that the products end alternately in 0 and 5.
The difference between the squares of two consecutive numbers is equal to their sum. For example, 5 squared - 4 squared = 25 - 16 = 9 = 5 + 4.
1 x 9 = 9, 2 x 9 = 18, 3 x 9 = 27. The digits of the products add up to 9.
We can use the properties to make calculations easier.
Whole numbers are used everywhere: counting money, counting people in a queue, numbering pages in a book, and measuring distances in whole units. The successor and predecessor concept from the previous chapter applies to whole numbers too. The successor of 0 is 1, but 0 has no predecessor among whole numbers because there is no whole number before zero.
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is a whole number | a x b is a whole number |
| Commutative | a + b = b + a | a x b = b x a |
| Associative | (a + b) + c = a + (b + c) | (a x b) x c = a x (b x c) |
| Identity | a + 0 = a (additive identity) | a x 1 = a (multiplicative identity) |
| Operation | Example | Result type |
|---|---|---|
| 5 + 7 | Whole number | 12 |
| 4 x 6 | Whole number | 24 |
| 3 - 7 | Not a whole number | -4 |
| 5 divided by 2 | Not a whole number | 2.5 |
| 0 divided by 7 | Whole number | 0 |
| 5 divided by 0 | Not defined | Cannot be done |
Whole numbers extend natural numbers by including zero and give us a complete picture of counting and calculation. We learnt the important properties of closure, commutativity, associativity and distributivity, and discovered the special behaviour of zero and one. The number line provides a visual way to understand operations, and patterns make calculation interesting. These properties will be used again when we study integers, algebra and higher arithmetic.