Concept Notes 1: The Emergence of Variables & Algebraic Notation
Algebra is the language of generalization. Instead of calculating specific numerical answers one by one, algebra uses letter-numbers (variables) to express mathematical patterns that hold true universally.
1. Variables vs. Constants
- Constant: A quantity having a fixed numerical value. Examples: $5, -18, \frac{3}{4}, 0, 100$.
- Variable (Letter-Number): A symbol (usually represented by letters such as $x, y, z, n, a, b$) that can represent any number from a given replacement set. Its value is not fixed—it varies according to the problem context.
Example: The Matchstick Square Pattern
To construct $1$ isolated square, we need $4$ matchsticks. To construct $2$ connected squares, we need $4 + 3 = 7$ matchsticks. To construct $3$ connected squares, we need $4 + 3 + 3 = 10$ matchsticks. For $n$ connected squares, the total number of matchsticks required is:
Here, $n$ is the variable, representing the number of squares. For any integer $n \ge 1$, the formula instantly generates the exact stick count!
2. Standard Algebraic Conventions
To make mathematical text concise, algebra eliminates unnecessary multiplication signs:
- Multiplication between a number and a letter:
$4 \times x$ is written simply as $\mathbf{4x}$. (The numerical factor is always written in front of the variable!)
- Multiplication between two letters:
$a \times b$ is written as $\mathbf{ab}$.
- Repeated multiplication of the same letter:
$x \times x$ is written as $\mathbf{x^2}$ (read as "$x$ squared"). $y \times y \times y$ is written as $\mathbf{y^3}$ (read as "$y$ cubed").
- Division:
$x \div 5$ is written as the fraction $\mathbf{\frac{x}{5}}$.