Numbers and the Number System - Grade 8

Natural / Counting Numbers (\mathbb{N})

Definition: These are the positive whole numbers starting from 1. (\mathbb{N} = {1, 2, 3, …})

Whole Numbers (\mathbb{N_0})

Definition: These include all natural numbers plus zero. (\mathbb{N_0} = {0, 1, 2, 3, …})

The Number One

Uniqueness: One is unique.
Multiplication/Division: Any number multiplied or divided by one remains the same.
- Example: 5 \times 1 = 5, 12 \div 1 = 12
Denominator: Any number can be written with one as the denominator.
- Example: 3 = \frac{3}{1}, \frac{2}{5} = \frac{2}{5}

The Number Zero

Addition/Subtraction: Adding or subtracting zero from any number leaves the number unchanged.
- Example: 8 + 0 = 8, 6 - 0 = 6
Multiplication: Any number multiplied by zero equals zero.
- Example: 3 \times 0 = 0
Division: Zero divided by any non-zero number is zero.
- Example: 0 \div 11 = 0
Division by Zero: Dividing any number by zero is undefined.
- Example: 4 \div 0 = \text{undefined}

Factor

Definition: A factor is a whole number that divides exactly into another number.
Rules:
- 1 is a factor of every number.
- Every number is a factor of itself.
- Example: Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Multiple

Definition: A multiple is a number into which another number can divide exactly.
Obtaining Multiples: Multiplying a number by a natural number.
Rule:
- Each number has itself as its first multiple.
- Example: The first 4 multiples of 5 are: 5, 10, 15, 20.

Prime Number

Definition: A prime number has only two factors.
Factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, etc.
The Number 2 : 2 is the only even prime number.

Prime Factor

Definition: Prime factors are factors of a number that are prime numbers themselves.
Example: Factors of 12 are 1, 2, 3, 4, 6, and 12, but its prime factors are 2 and 3.

Composite Number

Definition: A composite number has three or more factors.
Rules:
- It cannot be 1.
- It cannot be a prime number.
- Examples: 4, 6, 8, 9, 10, etc.
Note: The number 1 is neither prime nor composite.

Rules for Divisibility

Divisible by 2: The last digit must be divisible by 2.
Divisible by 3: The sum of the digits must be divisible by 3.
Divisible by 4: The last two digits must be divisible by 4.
Divisible by 5: The last digit must be 0 or 5.
Divisible by 6: The number must be even and divisible by 3.
Divisible by 8: The last three digits must be divisible by 8.
Divisible by 9: The sum of the digits must be divisible by 9.
Divisible by 10: The last digit must be 0.

Writing a Number as a Product of Prime Factors

Method: Divide the number by prime factors only, starting with the smallest possible prime number.
Example: Write 1260 as a product of prime factors:
- 1260 = 2 \times 2 \times 3 \times 3 \times 5 \times 7
- In exponential notation: 1260 = 2^2 \times 3^2 \times 5 \times 7

Perfect Square

Definition: A perfect square (or square number) is a number whose square root is a whole number.
Examples: The first four perfect squares are 1, 4, 9, 16.

Square Root (\sqrt{})

Definition: The square root of a number is another number which, when squared, equals the first number.
- Example: \sqrt{49} = 7
Generally: The square root will usually be smaller than the perfect square.

How to Find the Square of a Number

Method: Multiply the number by itself.
Fractions: Convert mixed fractions to improper fractions before squaring.
Examples:
- (2)^2 = 2 \times 2 = 4
- (\frac{1}{3})^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}
- (0.5)^2 = 0.5 \times 0.5 = 0.25

Combining Square Roots and Other Operations

Rule: The square root acts as a bracket.
Order of Operations: If there is a "+" or "-" sign under the square root, perform the operation under the square root first, then calculate the square root and complete the sum if necessary.
Examples:
- \sqrt{16} - 1 = 4 - 1 = 3

Perfect Cube

Definition: A perfect cube is obtained by multiplying an integer by itself three times.
Example: The perfect cube of 4 is 4 \times 4 \times 4 = 64

Cube Root

Definition: The cube root of a number is another number which, when cubed, will equal the first number.
Example: \sqrt[3]{64} = 4
Generally: The cube root will usually be smaller than the perfect cube.

How to Find the Cube of a Number

Rule: Multiply the number by itself, then multiply the answer by the first number again.
Examples:
- (1)^3 = 1 \times 1 \times 1 = 1
- (\frac{2}{3})^3 = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}

How to Find the Square Root or Cube Root of a Fraction

Step 1: If the number is a mixed number, change it to an improper fraction.
Step 2: Find the square root or cube root of the numerator and the denominator separately.
Examples:
- \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}
- \sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2}

How to Find the Square Root or Cube Root of a Decimal Number

Step 1: Ignore the decimal places and find the square root or cube root of the number that remains.
- Example: \sqrt{0.000121} becomes \sqrt{121} = 11
- Example: \sqrt[3]{0.000000064} becomes \sqrt[3]{64} = 4
Step 2: Group the numbers after the decimal in pairs (for square root) or triplets (for cube root).
Rule: The number of groups determines the number of decimal places in the answer.
Examples:
- 0.000121 has 2 groups (00, 01, 21), therefore the answer must have 3 decimal places.
  - \sqrt{0.000121} = 0.011
- 0.000000064 has 2 groups (000, 000, 064), therefore the answer must have 3 decimal places.
  - \sqrt[3]{0.000000064} = 0.004

How to Find the Square Root of a Perfect Square by Factors

Step 1: Divide the perfect square by prime factors.
- Example: Find \sqrt{784}
Step 2: Pair off the factors.
Step 3: Take one number from each pair and multiply them together.
- 784 = 2 \times 2 \times 2 \times 2 \times 7 \times 7 \implies \sqrt{784} = 2 \times 2 \times 7 = 28
Step 4: Check the answer on a calculator.

How to Find the Cube Root of a Perfect Cube by Factors

Step 1: Divide the perfect cube by prime factors.
- Example: Find \sqrt[3]{91125}
Step 2: Group the factors in triplets.
Step 3: Take one number from each group and multiply them together.
- 91125 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \implies \sqrt[3]{91125} = 3 \times 5 \times 5 = 75
Step 4: Check the answer on a calculator.

Number System

Natural/Counting Numbers (\mathbb{N}): {1, 2, 3, …}
Whole Numbers (\mathbb{N_0}): {0, 1, 2, 3, …}
Integers (\mathbb{Z}): {…, -3, -2, -1, 0, 1, 2, 3, …}
Rational Numbers (\mathbb{Q}):
- Definition: Can be written as \frac{a}{b}, where a and b are both integers and b \neq 0.
- Includes:
  - All natural numbers
  - All whole numbers
  - Integers
  - Fractions
  - Roots of perfect squares and cubes
  - Recurring decimals: (an infinite repeating pattern of numbers) e.g. 0.\overline{9674532}
  - Terminating decimals: (decimals that end) e.g. 2.475
Irrational Numbers (\mathbb{Q'}):
- Includes:
  - Never-ending, non-repeating decimal numbers (e.g. 2.12368…)
  - \pi
  - Any square root of a non-perfect square (e.g. \sqrt{8}, \sqrt{3}, \sqrt{10})
  - Any cube root of a non-perfect cube (e.g. \sqrt[3]{9}, \sqrt[3]{10})
  - The square of an irrational number will fall between the square of two rational numbers (perfect squares / cubes).
    - Example: 4 < (\sqrt{10})^2 < 9 \implies 2 < \sqrt{10} < 3
Real Numbers (\mathbb{R}):
- Definition: The set of all the rational and irrational numbers together.
Non-real Numbers (\mathbb{R'}):
- Definition: The even root of any negative number.
  - Examples: \sqrt{-2}, \sqrt[4]{-16}
Complex Numbers (\mathbb{C}):
- Definition: Combination of real and non-real (imaginary) numbers.

The Number System Diagram

Illustration of the relationship between Complex, Real, Non-Real, Rational, Irrational, Integer, Whole, and Natural numbers.

Table of Numbers, Squares, and Cubes

Numbers 1-10 along with respective Square Root and Square.
Numbers 1-10 along with respective Cube Root and Cube.
List of Prime Numbers 2-97.