1.1 Types of Numbers

1. Classification of Numbers

Numbers are grouped into sets based on their properties. The Real Number system includes all numbers on the continuous number line.

ℕNatural Numbers
Counting numbers.
Set: ${0, 1, 2, 3, \dots}$ .
ℤIntegers
Whole numbers, including negatives.
Set: ${\dots, - 2, - 1, 0, 1, 2, \dots}$ .
ℚRational Numbers
Can be written as a fraction $\frac{p}{q}$ where p, q are integers. Includes terminating and recurring decimals.
IrrIrrational Numbers
Cannot be written as a simple fraction. Decimals are non-terminating and non-recurring (e.g., $π, \sqrt{2}$ ).

Venn Diagram of Number Sets

2. Special Types of Numbers

Prime Numbers

Numbers with exactly two factors: 1 and themselves.

2, 3, 5, 7, 11, 13, 17, 19, 23...

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

Square Numbers

Result of multiplying an integer by itself ( $n^{2}$ ).

1, 4, 9, 16, 25, 36, 49, 64...

Cube Numbers

Result of multiplying an integer by itself three times ( $n^{3}$ ).

1, 8, 27, 64, 125, 216...

Triangle Numbers

Numbers that can form a triangle. Examples: 1, 3, 6, 10, 15...

Formula: $T_{n} = \frac{n (n + 1)}{2}$

The differences between consecutive triangle numbers increase by 1 (2, 3, 4...).

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Worked Examples

The numbers below are given:

- 3, 0, 1, 2, 4, 7, 9

(a) Write down all the integers in the list.

(b) Write down all the natural numbers in the list.

(d) Write down the square numbers.

(e) Explain why 1 is not classified as a prime number.

(a) Integers: Whole numbers, positive or negative. The list:

- 3, 0, 1, 2, 4, 7, 9

. All are integers.

Answer:

- 3, 0, 1, 2, 4, 7, 9

(b) Natural Numbers: Non-negative integers:

0, 1, 2, 4, 7, 9

(c) Prime Numbers: Numbers greater than 1 with only two factors (1 and itself).

2

is prime (factors 1, 2).

7

is prime (factors 1, 7).

4 and 9 are composite. 0 and 1 are not prime.

Answer:

2, 7

(d) Square Numbers:

1 = 1^{2}

4 = 2^{2}

9 = 3^{2}

Answer:

1, 4, 9

(e) Why not 1?

A prime number must have exactly two distinct factors. 1 has only one factor (itself).

Consider the numbers:

\sqrt{1}, \sqrt{8}, \sqrt{16}, \sqrt{20}

(a) Write down the integers after evaluating the square roots.

(b) Write down the irrational numbers.

(d) Explain why

\sqrt{8}

is not a rational number.

Evaluate roots first:

\sqrt{1} = 1

\sqrt{8} = 2.828 . . .

\sqrt{16} = 4

\sqrt{20} = 4.472 . . .

(a) Integers:

1

and

4

(b) Irrational Numbers:

The surds that don't simplify to integers:

\sqrt{8}

and

\sqrt{20}

(c) Original Squares:

The numbers inside the roots that are perfect squares: 1 and 16.

(d) Why irrational?

8 is not a perfect square. Its square root cannot be written as a fraction

\frac{p}{q}

and has a non-terminating, non-recurring decimal expansion.

(a) Write down the first six natural numbers (starting from 1).

(b) From your list in part (a), identify all the prime numbers.

(d) State whether all square numbers are composite numbers. Give a reason.

(a) First 6 Natural Numbers:

1, 2, 3, 4, 5, 6

(b) Primes:

2, 3, 5.

(c) Squares:

1 = 1^{2}

and

4 = 2^{2}

. Answer: 1, 4.

(d) Composite Squares?

No. 1 is a square number but it is not composite (nor prime).

The first five triangle numbers are:

1, 3, 6, 10, 15

(a) Write down the 6th triangle number.

(b) Identify the odd triangle numbers from the list above.

(d) Write down one triangle number that is also a square number.

(a) 6th Triangle Number:

Add 6 to the 5th number (15):

15 + 6 = 21

(b) Odd ones:

1, 3, 15.

(c) Is 10 square?

No.

3^{2} = 9

and

4^{2} = 16

. There is no integer

n

such that

n^{2} = 10

(d) Triangle and Square:

Looking further in the sequence: 21, 28, 36.

36 is a square number (

6^{2}

). Also 1 is a square. Answer: 1 or 36.

The numbers below are given:

\frac{2}{5}, 0.75, \sqrt{10}, 4, π

(a) Write down all the rational numbers.

(b) Write down all the irrational numbers.

(d) Convert 0.75 into a fraction in its simplest form.

(e) Explain why

π

is irrational.

(a) Rational: Can be fractions.

\frac{2}{5}

0.75 (= \frac{3}{4})

4 (= \frac{4}{1})

(b) Irrational:

Surds and constants:

\sqrt{10}

π

(c) Integer:

(d) Fraction:

0.75 = \frac{75}{100} = \frac{3}{4}

(e) Why Pi?

It cannot be expressed as a ratio of two integers. Its decimal expansion is infinite and non-recurring.

Consider the number 49.

(a) Write 49 as a square number.

(b) Write down all the factors of 49.

(d) State whether 49 is a rational number. Give a reason.

(e) Write down one number greater than 49 that is not a square number.

(a) As Square:

7^{2}

(b) Factors:

1, 7, 49

(c) Prime/Composite:

It has more than 2 factors, so it is Composite.

(d) Rational?

Yes. It can be written as

\frac{49}{1}

. All integers are rational.

(e) Not Square > 49:

Any number like 50, 51, 52... (not 64, 81 etc). Example: 50.

From the list of numbers:

{\sqrt{2}, 8, - 5, \frac{4}{7}, 0.83, π, 25, 0.333 \dots, \sqrt{9}, 11}

, Identify:

(a) The natural numbers

(b) The integers

(d) The irrational numbers

(e) The prime numbers

(a) Natural Numbers (ℕ): Non-negative integers

{0, 1, 2, \dots}

8, 25, \sqrt{9} (= 3), 11

(b) Integers (ℤ): Whole numbers, positive or negative.

Includes all natural numbers plus negatives.

8, - 5, 25, \sqrt{9}, 11

(c) Rational Numbers (ℚ): Can be written as fractions

\frac{p}{q}

Includes integers, terminating decimals, recurring decimals.

8, - 5, \frac{4}{7}, 0.83, 25, 0.333 \dots, \sqrt{9}, 11

(d) Irrational Numbers: Cannot be written as fractions.

\sqrt{2}, π

(e) Prime Numbers: Correctly identify numbers with only 2 factors (1 and itself).

11

is prime.

\sqrt{9} = 3

is prime.

Note: 8 is even, 25 is

5 \times 5

, -5 is negative (primes are positive integers > 1).

Primes:

3, 11

Consider the number

N = \sqrt{72}

(a) Write

N

in the form

k \sqrt{2}

where

k

is an integer.

(b) State whether

N

is rational or irrational.

m

such that

\sqrt{72 m}

is an integer.

(a) Simplify Surd:

Look for square factors in 72.

72 = 36 \times 2

\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}

= 6 \sqrt{2}

. So

k = 6

(b) Rational or Irrational?

\sqrt{2}

is irrational. Any non-zero integer times an irrational is irrational.

N

is irrational.

(c) Rationalize:

\sqrt{72 m} = \sqrt{36 \times 2 \times m} = 6 \sqrt{2 m}

For this to be an integer,

2 m

must be a perfect square.

The smallest integer

m

to make

2 m

a square is

m = 2

(since

2 \times 2 = 4

Answer:

m = 2

(a) List the first 5 triangle numbers.

(b) The

n

-th triangle number is given by

T_{n} = \frac{n (n + 1)}{2}

. Calculate the 50th triangle number.

(a) Listing Triangle Numbers:

Start with 1. Add 2: 3. Add 3: 6. Add 4: 10. Add 5: 15.

1, 3, 6, 10, 15.

(b) Calculation:

Using formula for

n = 50

T_{50} = \frac{50 (51)}{2} = 25 \times 51

25 \times 50 = 1250

25 \times 1 = 25

. Total: 1275.

(c) Proof:

Let the two numbers be

T_{n}

and

T_{n - 1}

T_{n} = \frac{n (n + 1)}{2}

T_{n - 1} = \frac{(n - 1) n}{2}

Sum:

\frac{n^{2} + n}{2} + \frac{n^{2} - n}{2}

= \frac{n^{2} + n + n^{2} - n}{2}

= \frac{2 n^{2}}{2} = n^{2}

n^{2}

is a square number. QED.

10.

Three bells ring at intervals of 12 minutes, 15 minutes, and 20 minutes respectively.

(a) They all ring together at 10:00 AM. Find the time they next ring together.

(b) How many times will they ring together between 10:00 AM and 1:00 PM (inclusive)?

(a) Find LCM:

Multiples of 12: 12, 24, 36, 48, 60...

Multiples of 15: 15, 30, 45, 60...

Multiples of 20: 20, 40, 60...

LCM is 60 minutes.

They ring together every 60 minutes (1 hour).

Next ringing time: 10:00 AM + 1 hour = 11:00 AM.

(b) Count occurrences:

Duration: 10:00 AM to 1:00 PM is 3 hours (180 minutes).

Times:

1. 10:00 AM (Start)

2. 11:00 AM

3. 12:00 PM

4. 01:00 PM

Total: 4 times.

11.

Consider the numbers 168 and 264.

(a) Write 168 as a product of its prime factors.

(b) Find the highest common factor (HCF) of 168 and 264.

k

such that

\sqrt{168 k}

is an integer.

(a) Prime Factorization:

168 = 2 \times 84 = 2^{2} \times 42 = 2^{3} \times 21 = 2^{3} \times 3 \times 7

(b) HCF:

264 = 2 \times 132 = 2^{2} \times 66 = 2^{3} \times 33 = 2^{3} \times 3 \times 11

Common factors:

2^{3}

and

3

H C F = 2^{3} \times 3 = 8 \times 3 = 24

(c) Square Number:

We need

168 k

to be a square.

168 = 2^{3} \times 3 \times 7

Powers must be even.

Multiply by

2

(to get

2^{4}

Multiply by

3

(to get

3^{2}

Multiply by

7

(to get

7^{2}

k = 2 \times 3 \times 7 = 42

12.

Arrange the following numbers in ascending order:

0.6, \frac{5}{8}, 65 %, \frac{2}{3}, {0.6}^{2}

Convert all to decimals:

0.6 = 0.60

\frac{5}{8} = 0.625

65 % = 0.65

\frac{2}{3} \approx 0.666 \dots

{0.6}^{2} = 0.36

Ordering:

0.36 < 0.60 < 0.625 < 0.65 < 0.666

Original format:

{0.6}^{2}, 0.6, \frac{5}{8}, 65 %, \frac{2}{3}