Mathematics Demarcation Year 9 Study Notes

Mathematics Demarcation Year 9

Textbook Reference

• Title: Cambridge IGSCE Core and Extended Mathematics
• Edition: Fifth edition
• Access: Available on Teams under Mathematics, shared, textbooks.

Type of Numbers

1. Natural Numbers

   • Definition: The positive integers starting from 1, used for counting. For example, $1, 2, 3, ext{ etc.}$

2. Integers

   • Definition: The set of whole numbers that can be positive, negative, or zero. Mathematically represented as $ext{… , -3, -2, -1, 0, 1, 2, 3, …}$.

3. Prime Numbers

   • Definition: A natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include $2, 3, 5, 7, 11$.

4. Square Numbers

   • Definition: The product of a number multiplied by itself. For example, $1^2 = 1, 2^2 = 4, 3^2 = 9$.

5. Cube Numbers

   • Definition: The product of a number multiplied by itself two times. For example, $1^3 = 1, 2^3 = 8, 3^3 = 27$.

6. Common Factors

   • Definition: A number that divides two or more numbers without leaving a remainder. The common factors of 12 and 18 are $1, 2, 3, 6$.

7. Common Multiples

   • Definition: A number that is a multiple of two or more numbers. The common multiples of 4 and 6 include $12, 24, 36$.

8. Rational Numbers

   • Definition: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For example, $\frac{1}{2}, 3, -\frac{4}{5}$.

9. Irrational Numbers

   • Definition: Numbers that cannot be expressed as a simple fraction, having non-repeating, non-terminating decimal expansions. Examples include $ext{π}$, $ext{e}$, and $ext{√2}$.

10. Reciprocals

    • Definition: The reciprocal of a number $x$ is $\frac{1}{x}$. For example, the reciprocal of 4 is $\frac{1}{4}$.

Sets

1. Venn Diagram

   • Definition: A diagram that shows all possible logical relations between a finite collection of different sets. It uses circles to represent sets and their interrelations.

2. Set Language

   • Explanation: Involves terminology and symbols used to define and describe sets, including operations such as union ($\bigcup$), intersection ($\bigcap$), and difference ($A - B$).

Power and Roots

1. Squares

   • Concept: The square of a number is that number multiplied by itself, denoted as $x^2$.

2. Square Roots

   • Definition: The square root of a number $x$ is a value that, when multiplied by itself, gives the number $x$, denoted as $ext{√}x$.

3. Cubes

   • Concept: The cube of a number is that number multiplied by itself twice, denoted as $x^3$.

4. Cube Roots

   • Definition: The cube root of a number $x$ is a value that, when multiplied by itself twice, gives the number $x$, denoted as $ext{∛}x$.

5. Other Powers and Numbers

   • Overview: Any integer raised to a power can be represented as $x^n$, where $n$ is an integer. Common powers include squares and cubes, but this can extend to fractional and negative powers as well.

The Four Fraction Operations

1. Addition

   • Explanation: To add fractions, one must have a common denominator. The sum $\frac{a}{b} + \frac{c}{d}$ can be calculated as $\frac{ad + bc}{bd}$.

2. Subtraction

   • Explanation: Subtracting fractions also requires a common denominator. The difference $\frac{a}{b} - \frac{c}{d}$ is computed as $\frac{ad - bc}{bd}$.

3. Division

   • Explanation: To divide fractions, multiply by the reciprocal of the divisor. For example, $\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} imes \frac{d}{c}$.

4. Multiplication

   • Explanation: To multiply fractions, simply multiply their numerators and denominators. For example, $\frac{a}{b} imes \frac{c}{d} = \frac{ac}{bd}$.