Whole Numbers

Natural numbers are 1, 2, 3, 4, ... . If we include zero (0) in the collection of natural numbers, we get the collection of whole numbers. Thus, whole numbers are 0, 1, 2, 3, 4, ... . Every natural number is a whole number, but 0 is a whole number which is not a natural number. The smallest whole number is 0. There is no largest whole number.

A number line is a straight line on which numbers are marked at equal intervals or distances. To represent whole numbers on a number line, we draw a line, mark a point on it as 0. Then, we mark points to the right of 0 at equal distances and label them 1, 2, 3, and so on. The distance between any two consecutive numbers is called a unit distance.

To add two whole numbers on a number line, we start from the first number and move to the right by the number of units equal to the second number. For example, to add 3 and 4, start at 3 and move 4 units to the right. You will reach 7. So, 3 + 4 = 7.

To subtract two whole numbers on a number line, we start from the first number and move to the left by the number of units equal to the second number. For example, to subtract 5 from 8 (i.e., 8 - 5), start at 8 and move 5 units to the left. You will reach 3. So, 8 - 5 = 3.

To multiply two whole numbers on a number line, we start from 0 and make jumps of equal size. For example, to multiply 3 by 2 (i.e., 3 × 2), start from 0 and make 2 jumps, each of 3 units to the right. You will reach 6. So, 3 × 2 = 6.

If we add or multiply any two whole numbers, the result is always a whole number. We say that whole numbers are closed under addition and multiplication. For example, 2 + 3 = 5 (a whole number), 4 × 5 = 20 (a whole number). Whole numbers are not closed under subtraction (e.g., 2 - 5 is not a whole number) or division (e.g., 2 ÷ 5 is not a whole number).

For any two whole numbers, the order in which we add or multiply them does not change the result. We say that addition and multiplication are commutative for whole numbers. For example, 2 + 3 = 3 + 2 = 5. Also, 4 × 5 = 5 × 4 = 20. Subtraction and division are not commutative.

For any three whole numbers, the way we group them for addition or multiplication does not change the result. We say that addition and multiplication are associative for whole numbers. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9. Also, (2 × 3) × 4 = 2 × (3 × 4) = 24. Subtraction and division are not associative.

This property relates multiplication and addition. For any three whole numbers a, b, and c, multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers separately and then adding the products. For example, 2 × (3 + 4) = (2 × 3) + (2 × 4). L.H.S. = 2 × 7 = 14. R.H.S. = 6 + 8 = 14. L.H.S. = R.H.S.

Additive Identity: When 0 is added to any whole number, the sum is the number itself. Hence, 0 is called the additive identity for whole numbers. For example, 5 + 0 = 5. Multiplicative Identity: When any whole number is multiplied by 1, the product is the number itself. Hence, 1 is called the multiplicative identity for whole numbers. For example, 7 × 1 = 7.