Expressions using Letter-Numbers

Suppose you are building the letter L using matchsticks. For $1$ 'L', you need $2$ matchsticks. For $2$ 'L's, you need $4$ matchsticks ($2 \times 2$). For $3$ 'L's, you need $6$ matchsticks ($3 \times 2$).

Notice the universal pattern: the number of matchsticks is always $2 \times (\text{number of L's})$. If we let the letter $n$ represent the number of 'L's, then:

$$\text{Matchsticks required} = 2 \times n = 2n$$

Here, $n$ is called a letter-number or a variable. It can take any natural number value ($1, 2, 3, 4\dots$).

• Constant: A quantity with a fixed numerical value that never changes ($5, -12, 100$).
• Variable (Letter-Number): A letter (such as $x, y, n, a, b$) that stands for a quantity that can change or vary depending on the situation.

An algebraic expression combines constants and variables using addition, subtraction, multiplication, or division: $$3x + 5, \quad 2a - b, \quad \frac{y}{4} + 7$$

Why do mathematicians write $3x$ instead of $3 \times x$? Because the multiplication sign '$\times$' looks almost identical to the letter '$x$'! To avoid confusion, mathematicians established universal shorthand rules.

1. Between a number and a letter, drop the $\times$ sign: $$5 \times y = 5y$$ (Always write the number first: write $5y$, not $y5$).
2. Between two or more letters, drop the $\times$ sign: $$a \times b = ab, \quad x \times y \times z = xyz$$
3. The number $1$ as a multiplier is invisible: $$1 \times x = x, \quad -1 \times x = -x$$
4. Division is written as a fraction: $$x \div 4 = \frac{x}{4}$$

If you have $3$ apples and someone gives you $2$ more apples, you have $5$ apples ($3a + 2a = 5a$). But if you have $3$ apples and $2$ bananas, you cannot combine them into "5 applenanas"! They remain $3$ apples and $2$ bananas ($3a + 2b$).

• Like Terms: Terms that have the exact same variable factors with identical powers ($3x$ and $7x$; $-2ab$ and $5ab$). Only their numerical coefficients differ.
• Unlike Terms: Terms that have different variables or powers ($4x$ and $4y$; $5a$ and $5a^2$).

Why Like Terms Can Be Added (Distributive Law):
$$3x + 2x = (3 + 2)x = \mathbf{5x}$$ Only the numerical coefficients are added or subtracted; the variable part remains unchanged!

Look at the sequence of odd numbers: $1, 3, 5, 7, 9\dots$

• 1st odd number: $2(1) - 1 = 1$
• 2nd odd number: $2(2) - 1 = 3$
• 3rd odd number: $2(3) - 1 = 5$
• 100th odd number: $2(100) - 1 = \mathbf{199}$

Instead of writing out all one hundred numbers, the single algebraic expression $\mathbf{2n - 1}$ tells you the value of any odd number instantly!

To make $1$ triangle, you need $3$ matchsticks.

To make $2$ connected triangles sharing a common side, you add $2$ more matchsticks ($3 + 2 = 5$).

To make $3$ connected triangles, you add another $2$ matchsticks ($5 + 2 = 7$).

General rule for $n$ connected triangles: $$2n + 1$$

Test it: For $n = 1 \implies 2(1) + 1 = 3$. For $n = 2 \implies 2(2) + 1 = 5$. For $n = 50 \implies 2(50) + 1 = 101$ matchsticks!