Topic 1.1Foundation Topic

Types of Number

🌳Number Classification Tree

The hierarchy of number sets from Real numbers down to special subsets:

Basic sets

Natural Numbers, Integers, and Whole Numbers

Understand the hierarchy of whole numbers: natural numbers (counting numbers), integers (including negatives and zero), and how they relate to rational numbers.

Key Concepts & Rules

•Natural numbers (ℕ): 1, 2, 3, 4, 5... (positive whole numbers, no zero)
•Integers (ℤ): ..., -3, -2, -1, 0, 1, 2, 3... (whole numbers including negatives)
•Whole numbers: 0, 1, 2, 3... (natural numbers plus zero)
•Every natural number is an integer, but not every integer is natural
•Zero is an integer but NOT a natural number
•Negative numbers are integers but not natural or whole

Worked Examples: Natural Numbers, Integers, and Whole Numbers

From the list -3, √7, 0, 12, 2/5, identify: (a) a natural number, (b) an integer, (c) a rational number, (d) an irrational number.

-3

ℤ ℚ

√7

Irrational

ℤ ℚ

ℕ ℤ ℚ

2/5

ℚ

(a) Natural: 12 (positive whole number, no fractions/decimals)

(b) Integer: -3 or 0 or 12 (whole numbers including negatives and zero)

(c) Rational: 2/5 (can be expressed as a fraction a/b)

(d) Irrational: √7 (cannot be expressed as a simple fraction)

Note: 12 is natural, integer, AND rational!

Is √49 rational or irrational? Justify your answer.

√49 = 7

7 is an integer

All integers are rational

✓ Rational

Step 1: Calculate √49 = 7

Step 2: 7 is an integer (whole number)

Step 3: All integers can be written as fractions (7 = 7/1)

Conclusion: √49 is rational

Trick: Perfect squares have rational roots!

Classify each number: -2, 0, 7, 2.5, π

-2Integer, Rational (ℤ, ℚ)

0Whole, Integer, Rational

7Natural, Integer, Rational (ℕ, ℤ, ℚ)

2.5Rational only (ℚ)

πIrrational only

-2: Integer (negative whole), Rational (can be -2/1)

0: Whole number, Integer, Rational (0/1 = 0)

7: Natural (counting), Integer, Rational

2.5: Rational only (5/2), not integer

π: Irrational (3.14159... non-terminating, non-recurring)

Numbers can belong to multiple sets!

Find a rational and an irrational number between 3 and 4.

Rational

3.5

Rational

3.5 = 7/2

Irrational

π ≈ 3.142

Rational example: 3.5 (or 7/2)

Other rationals: 3.1, 3.25, 11/3, 3.999...

Irrational example: π ≈ 3.14159...

Other irrationals: √10 ≈ 3.162..., e ≈ 2.718... (but e < 3)

Infinitely many rationals AND irrationals between any two numbers!

Fractions and decimals

Rational and Irrational Numbers

Distinguish between rational numbers (can be written as fractions, including terminating and recurring decimals) and irrational numbers (non-terminating, non-recurring like π and √2).

Key Concepts & Rules

•Rational (ℚ): Can be written as a/b where a, b are integers, b ≠ 0
•Terminating decimals are rational (e.g., 0.25 = 1/4)
•Recurring decimals are rational (e.g., 0.333... = 1/3)
•Irrational: Non-terminating AND non-recurring decimals
•Examples of irrational: π, √2, √3, √5, e
•Perfect square roots are rational (√49 = 7), non-perfect are irrational

Worked Examples: Rational and Irrational Numbers

Prove that the recurring decimal 0.454545... is rational by converting it to a fraction.

Let x = 0.454545...

100x = 45.454545...

100x - x = 45.454545... - 0.454545...

99x = 45

x = 45/99 = 5/11

Step 1: Let x = 0.454545...

Step 2: Multiply by 100 (2 recurring digits): 100x = 45.454545...

Step 3: Subtract: 100x - x = 45.454545... - 0.454545...

Step 4: 99x = 45, so x = 45/99 = 5/11

Conclusion: Since it can be written as a fraction, it is rational

All recurring decimals are rational numbers!

State whether each is rational or irrational: (a) 0.333..., (b) √5, (c) 4.25

0.333...

= 1/3

Rational

√5

≈ 2.236...

Irrational

4.25

= 17/4

Rational

(a) 0.333...: Recurring decimal = 1/3 → Rational

(b) √5: Not a perfect square, non-terminating, non-recurring → Irrational

(c) 4.25: Terminating decimal = 17/4 → Rational

Key: Recurring or terminating = Rational; Non-terminating AND non-recurring = Irrational

Show that √2 + √3 is irrational. (Hint: Assume it's rational and find a contradiction)

Proof by Contradiction:

Assume √2 + √3 = p/q (rational)

Square both sides...

Contradiction!

Assume: √2 + √3 = a/b where a, b are integers (rational)

Square both sides: (√2 + √3)² = a²/b²

Expand: 2 + 2√6 + 3 = a²/b²

Simplify: 5 + 2√6 = a²/b²

Rearrange: 2√6 = a²/b² - 5 = (a² - 5b²)/b²

So: √6 = (a² - 5b²)/(2b²) which would make √6 rational

Contradiction: √6 is known to be irrational!

Sum of two irrationals can be rational OR irrational!

A number is chosen from {1, 2, 3, ..., 30}. What is the probability it is prime?

Prime numbers from 1-30:

2357111317192329

10 primes out of 30 numbers

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Count: 10 prime numbers

Total: 30 numbers in the set

Probability:

\frac{10}{30} = \frac{1}{3}

Note: 1 is NOT prime! 2 is the only even prime.

Building blocks

Prime Numbers, Factors, and Multiples

Identify prime numbers (exactly two factors), find all factors of a number, list multiples, and express numbers as products of prime factors using factor trees.

Key Concepts & Rules

•Prime: Natural number with exactly two distinct factors: 1 and itself
•1 is NOT prime (only one factor)
•2 is the only even prime number
•Factor: A number that divides exactly into another number
•Multiple: The result of multiplying a number by an integer
•Prime factorization: Breaking down into product of prime numbers

Worked Examples: Prime Numbers, Factors, and Multiples

Express 84 as a product of its prime factors.

84 = 2 × 2 × 3 × 7

Method: Divide by smallest prime until you reach 1

84 ÷ 2 = 42

42 ÷ 2 = 21

21 ÷ 3 = 7

7 ÷ 7 = 1

Answer:

84 = 2 \times 2 \times 3 \times 7 = 2^{2} \times 3 \times 7

Always write in index form for final answer!

Find all factors of 24.

Factor pairs: (1,24), (2,12), (3,8), (4,6)

Method 1 - Systematic: Check 1, 2, 3, 4... up to √24 ≈ 4.9

1 × 24 = 24 ✓

2 × 12 = 24 ✓

3 × 8 = 24 ✓

4 × 6 = 24 ✓

Answer: 1, 2, 3, 4, 6, 8, 12, 24 (8 factors total)

Tip: Find pairs by dividing 24 by each integer!

Find the first five multiples of 7.

→

7 × 1, 7 × 2, 7 × 3, 7 × 4, 7 × 5

Definition: Multiples are the "times table" of a number

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

Answer: 7, 14, 21, 28, 35

Multiples go on forever: 7, 14, 21, 28, 35, 42, 49...

Express 28 as the sum of two prime numbers.

Primes less than 28:

23571113171923

5 + 23 = 28

Both prime ✓

11 + 17 = 28

Both prime ✓

Primes less than 28: 2, 3, 5, 7, 11, 13, 17, 19, 23

Find pairs that sum to 28:

5 + 23 = 28 ✓

11 + 17 = 28 ✓

Answer: 5 + 23 = 28 or 11 + 17 = 28

Goldbach's Conjecture: Every even number > 2 is sum of two primes!

Applications

HCF, LCM, and Special Numbers

Calculate Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using prime factorization or Venn diagrams. Recognize square and cube numbers.

Key Concepts & Rules

•HCF: Largest number that divides two or more numbers exactly
•LCM: Smallest number that is a multiple of two or more numbers
•HCF from prime factors: multiply common primes with lowest powers
•LCM from prime factors: multiply all primes with highest powers
•Square number: n² (result of multiplying integer by itself)
•Cube number: n³ (result of multiplying integer by itself three times)

Worked Examples: HCF, LCM, and Special Numbers

Find the HCF and LCM of 24 and 60.

Step 1 - Prime factorize:

24 = 2^{3} \times 3

60 = 2^{2} \times 3 \times 5

Step 2 - HCF (lowest powers of common factors):

H C F = 2^{2} \times 3 = 4 \times 3 = 12

Step 3 - LCM (highest powers of all factors):

L C M = 2^{3} \times 3 \times 5 = 8 \times 3 \times 5 = 120

HCF = intersection, LCM = union of prime factors

Find the LCM and HCF of 12 and 18.

12 = 2² × 3

Factors: 2, 2, 3

18 = 2 × 3²

Factors: 2, 3, 3

HCF

Common: 2 × 3 = 6

LCM

All: 2² × 3² = 36

Prime factors:

12 = 2^{2} \times 3

18 = 2 \times 3^{2}

HCF (common factors, lowest powers):

H C F = 2^{1} \times 3^{1} = 2 \times 3 = 6

LCM (all factors, highest powers):

L C M = 2^{2} \times 3^{2} = 4 \times 9 = 36

Check: HCF × LCM = 12 × 18 ✓

What is the smallest square number greater than 100?

10²

= 100

→

11²

= 121 ✓

→

12²

= 144

10² = 100 (not greater than 100)

Next integer: 11

11² = 121

Answer: 121

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...

Write the next three square numbers after 36.

6²

7²

8²

9²

Given: 36 = 6²

Next: 7² = 49

8² = 64

9² = 81

Answer: 49, 64, 81

Perfect squares: n² where n is a whole number