Algebra & Number Systems

TL;DR

You'll learn about different types of numbers and their properties, especially real numbers. You'll master polynomials, including finding their zeroes and applying key theorems like the Division Algorithm. Finally, you'll tackle pairs of linear equations, learning how to solve them graphically and algebraically.

1. The Mental Model

Think of numbers as building blocks and algebra as the rules for combining and manipulating these blocks. You're learning the fundamental language of math, which helps you describe relationships and solve problems. You'll move from basic number types to solving equations that model real situations.

2. The Core Material

Number Systems: The Real Numbers

You'll primarily work with Real Numbers. This huge group includes:
* Natural Numbers (N): Counting numbers (1, 2, 3, ...).
* Whole Numbers (W): Natural numbers plus zero (0, 1, 2, 3, ...).
* Integers (Z): Whole numbers and their negatives (... -2, -1, 0, 1, 2, ...).
* Rational Numbers (Q): Numbers that can be written as a fraction p/q, where p and q are integers and q is not zero (e.g., 1/2, -3, 0.75). Their decimal representations are either terminating or non-terminating repeating.
* Irrational Numbers: Numbers that cannot be written as p/q. Their decimal representations are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$).

Key Concept: The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. You use this for finding HCF and LCM.
* HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Found by taking the smallest power of each common prime factor.
* LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers. Found by taking the highest power of all prime factors involved.
* Relationship: For any two positive integers a and b, HCF(a, b) $\times$ LCM(a, b) = a $\times$ b.

Polynomials

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
* Degree: The highest power of the variable in a polynomial.
* Linear Polynomial: Degree 1 (e.g., ax + b)
* Quadratic Polynomial: Degree 2 (e.g., ax² + bx