1. Core Concepts: Number Sets

Think of number sets like nested boxes. The smaller boxes fit inside the larger ones.

• Natural Numbers (ℕ): Your counting numbers. {1, 2, 3, ...}

• Integers (ℤ): All whole numbers, including zero and negatives. {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational Numbers (ℚ): Any number that can be written as a fraction p/q, where 'p' and 'q' are integers and 'q' is not zero.

  • This includes all integers (e.g., 5 = 5/1), terminating decimals (e.g., 0.5 = 1/2), and recurring decimals (e.g., 0.333... = 1/3).

• Irrational Numbers (ℚ'): Numbers that cannot be written as a simple fraction. Their decimal form goes on forever without repeating.

  • Famous examples: π (pi), √2, √3, √5.

• Real Numbers (ℝ): The biggest box. It contains all rational and irrational numbers.

Venn Diagram Summary: (This is a crucial visual to remember!)

2. Rational Decimals: Terminating vs. Recurring

This is a common exam question. Here's how to tell them apart without doing long division.

1. Simplify the fraction first! (e.g., 6/30 simplifies to 1/5).

2. Look at the prime factors of the denominator.

   • Terminating: If the prime factors are only 2s and/or 5s, the decimal will terminate (end).

     • Example: 3/8 -> Denominator is 8 = 2 x 2 x 2. (Only 2s). So, it terminates (0.375).

     • Example: 7/20 -> Denominator is 20 = 2 x 2 x 5. (Only 2s and 5s). So, it terminates (0.35).

   • Recurring: If the denominator has any prime factor other than 2 or 5 (like 3, 7, 11, etc.), the decimal will recur (repeat).

     • Example: 5/6 -> Denominator is 6 = 2 x 3. (Has a 3). So, it recurs (0.8333...).

3. Surds (කරණ / சேடு)

A surd is simply a root of a number that is irrational. For your exam, this mainly means the square root of a number that is not a perfect square (e.g., √20).

Key Skills for the Exam:

a) Simplifying Surds

Trick: Find the largest perfect square that is a factor of the number.

• Example: Simplify √48

  1. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

  2. The largest perfect square factor is 16.

  3. Rewrite: √48 = √(16 x 3)

  4. Split: √16 x √3

  5. Simplify: 4√3

b) Adding and Subtracting Surds

Tip: Think of surds like algebraic terms. You can only add or subtract 'like' surds.

• YES: 3√5 + 2√5 = 5√5 (Like 3x + 2x = 5x)

• NO: 3√5 + 2√7 (Cannot be simplified)

Exam Trick: You often need to simplify the surds before you can add or subtract them.

• Example: Simplify √20 - √5

  1. Simplify √20: √20 = √(4 x 5) = 2√5

  2. Now substitute back: 2√5 - √5

  3. Simplify: √5

c) Rationalizing the Denominator

This means getting the surd out of the denominator. Method: Multiply the top and bottom of the fraction by the surd in the denominator.

• Example: Rationalize 5/√3

  1. Multiply top and bottom by √3: (5 x √3) / (√3 x √3)

  2. Since √3 x √3 = 3, the answer is (5√3) / 3.

     
     

Exam Tips & Common Mistakes

• Mistake 1: Forgetting to simplify a fraction before checking if it's terminating or recurring. 3/6 is terminating because it's 1/2.

• Mistake 2: Adding surds incorrectly. √2 + √3 is NOT √5. You cannot combine them.

• Exam Tip: When a question asks for a calculation involving surds, simplify everything first. The final answer is often much simpler than the initial problem. For example, (√20)/2 - √5 simplifies to 2√5 / 2 - √5, which is √5 - √5 = 0.

• Remember: Always show your simplification steps clearly. Marks are often awarded for the correct method, even if you make a small calculation error.