Whole Numbers and Integers

1. Natural Numbers and Whole Numbers

Natural numbers are counting numbers: 1, 2, 3, 4, ... (denoted by N).
Whole numbers include zero: 0, 1, 2, 3, 4, ... (denoted by W).

Key point: Every natural number is a whole number, but 0 is a whole number that is NOT a natural number.

2. Properties of Whole Numbers

Commutative Property

Associative Property

• Addition: (a + b) + c = a + (b + c)
• Multiplication: (a x b) x c = a x (b x c)

Worked Example: Simplify 47 + 68 + 53 using properties.

47 + 68 + 53 = (47 + 53) + 68 = 100 + 68 = 168.

Distributive Property

a x (b + c) = (a x b) + (a x c)

Worked Example: Find 25 x 104 using the distributive property.

25 x 104 = 25 x (100 + 4) = (25 x 100) + (25 x 4) = 2500 + 100 = 2600.

Identity Elements

• Additive identity: a + 0 = a (zero is the identity).
• Multiplicative identity: a x 1 = a (one is the identity).

Common Mistake: Thinking 0 is the multiplicative identity. No! Any number multiplied by 0 is 0, not the number itself.

3. Introduction to Integers

Integers include all whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...

Denoted by Z (from German 'Zahlen' meaning numbers).

Number Line Representation

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

• Numbers to the right are greater.
• Numbers to the left are smaller.
• Every integer has a position on the line.

4. Ordering Integers

• For any two integers, the number on the right is larger.
• Positive integers > 0 > negative integers.
• Example: -5 < -2 < 0 < 3 < 7.

Exam Focus (2 marks): 'Arrange in ascending order: -8, 3, -1, 0, 7, -4.'

Ascending order: -8 < -4 < -1 < 0 < 3 < 7.

5. Addition of Integers

Rule 1 (same sign): Add absolute values, keep the sign.
Rule 2 (different signs): Subtract smaller absolute value from larger, take the sign of the larger.

Worked Example: Find (-6) + (-9).

Both negative: |-6| + |-9| = 6 + 9 = 15. Sign is negative. Answer = -15.

Worked Example: Find 12 + (-7).

Different signs: |12| = 12, |-7| = 7. 12 - 7 = 5. Sign of larger (12) is positive. Answer = 5.

Common Mistake: Writing (-6) + (-9) = +15. Remember: same sign means the answer keeps that sign.

6. Subtraction of Integers

Change subtraction to addition of the opposite: a - b = a + (-b).

Worked Example: Find (-8) - (-3).

(-8) - (-3) = (-8) + 3 = -5.

7. Additive Inverse

The additive inverse of an integer is the number that, when added to it, gives 0.

• Additive inverse of 5 is -5, because 5 + (-5) = 0.
• Additive inverse of -7 is 7, because (-7) + 7 = 0.

Exam Focus (3 marks): 'Find: 34 + (-17) + (-12) + 25.'

Step 1: Group positives: 34 + 25 = 59.
Step 2: Group negatives: (-17) + (-12) = -29.
Step 3: 59 + (-29) = 30.

8. Multiplication and Division of Integers

The same rules apply to division.

Common Mistake: Thinking (-3) x (-4) = -12. The product of two negatives is POSITIVE.

9. Comparison Table: Whole Numbers vs Integers

10. Self-Test

1. State the property used: 45 x 12 = 12 x 45.
2. Find using the distributive property: 15 x 99.
3. Add: (-18) + 7 + (-5) + 23.
4. Subtract: (-12) - (-18) - 6.
5. Write the additive inverse of: (a) 23 (b) -15 (c) 0.
6. Multiply: (-8) x 6 x (-2).
7. Arrange in descending order: -3, 5, -7, 0, 2, -1.
8. Simplify using properties: 125 x 8 x 4 x 25.

11. Answers to Self-Test

1. Commutative property of multiplication.
2. 15 x (100 - 1) = 1500 - 15 = 1485.
3. (-18) + (-5) = -23; 7 + 23 = 30; 30 + (-23) = 7.
4. (-12) + 18 - 6 = 6 - 6 = 0.
5. (a) -23 (b) 15 (c) 0.
6. (-8) x 6 = -48; -48 x (-2) = 96.
7. 5 > 2 > 0 > -1 > -3 > -7.
8. (125 x 8) x (4 x 25) = 1000 x 100 = 1,00,000.