Algebraic Notation
Algebraic Notation
Algebra uses letters to represent unknown values, variables (quantities that can change), and general numbers (to express rules that hold for any value). The notation follows precise conventions — knowing what each expression means, and how to write expressions correctly, is the foundation for all further algebra.
The transition from arithmetic to algebra replaces specific numbers with letters, but the operations remain the same. The notation is designed to be compact: multiplication signs between letters or between a number and a letter are omitted, and division is written as a fraction.
Notation Conventions
| Algebraic form | Meaning | Note |
|---|---|---|
| \( 3x \) | \( 3 \times x \) | Coefficient written first, no × sign |
| \( ab \) | \( a \times b \) | Letters written alphabetically; × omitted |
| \( x^2 \) | \( x \times x \) | Power notation replaces repeated multiplication |
| \( \frac{x}{3} \) | \( x \div 3 \) | Division written as fraction |
| \( 2(x + 3) \) | \( 2 \times (x + 3) \) | Brackets: multiply everything inside |
| \( -x \) | \( -1 \times x \) | Coefficient of \(-1\) is written as \(-x\), not \(-1x\) |
Critical Distinctions
Several pairs of expressions look similar but mean very different things. Confusing them is one of the most common sources of error in early algebra:
- \( 2x \) means \( 2 \times x \) — double \( x \). \( x^2 \) means \( x \times x \) — \( x \) squared. For \( x = 5 \): \( 2x = 10 \) but \( x^2 = 25 \).
- \( 3x^2 \) means \( 3 \times x^2 \) — three times \( x \) squared. \( (3x)^2 \) means \( (3x) \times (3x) = 9x^2 \) — the bracket is squared entirely.
- \( x + x = 2x \) (adding), but \( x \times x = x^2 \) (multiplying). Adding the same letter twice is not the same as squaring it.
Worked Examples
Interpret \( 5ab^2 \), \( \frac{2x+1}{y} \) and \( -3m^2n \).
\[ 5ab^2 = 5 \times a \times b \times b \] \[ \frac{2x+1}{y} = (2x+1) \div y quad \text{(the whole numerator is divided by } y \text{)} \] \[ -3m^2n = -3 \times m \times m \times n \]Write algebraic expressions for: the cost of \( n \) pens at £2 each; the perimeter of a square with side \( s \); half of \( (x + y) \).
Cost: \( £2n \) (or \( 2n \) pence if working in pence).
Perimeter: \( 4s \) (four sides, each of length \( s \)).
Half of \( (x+y) \): \( \dfrac{x+y}{2} \) (brackets are needed to show the whole sum is halved).
Write algebraic expressions for three consecutive integers starting from \( n \), and simplify their sum.
The three integers: \( n,quad n+1,quad n+2 \).
\[ \text{Sum} = n + (n+1) + (n+2) = 3n + 3 = 3(n+1) \]This shows that the sum of any three consecutive integers is always a multiple of 3 — an algebraic proof of a number pattern.
Key Takeaways
- In algebra, multiplication is implied between a number and a letter (e.g. \( 3x \)) or between two letters (e.g. \( ab \)) — no × sign is written.
- Division is written as a fraction. If the numerator contains more than one term, brackets are implied.
- \( 3x^2 \) and \( (3x)^2 \) are different: the power applies to \( x \) only in the first, and to the entire product \( 3x \) in the second.
- \( x + x = 2x \) (addition collects terms), but \( x \times x = x^2 \) (multiplication creates a power).
- Letters in a product are written in alphabetical order by convention; the coefficient (number) always comes first.