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.