Understanding Algebraic Notation
This topic covers the fundamental shorthand used in algebra. Mastering this notation is essential as it forms the language for all subsequent algebraic manipulation, simplification, and problem-solving in the ESAT.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Juxtaposition implies multiplication: writing variables or a number and a variable next to each other, like `5x` or `ab`, means they are multiplied.
- A number in front of a variable is a coefficient, representing repeated addition. For example, `4y` is shorthand for `y + y + y + y`.
- A superscript number is an exponent (or index), indicating repeated multiplication. For instance, `a³` means `a × a × a`.
- Division is typically written as a fraction. The expression `x/y` is the standard way to write `x ÷ y`.
- By convention, algebraic terms are written with the numerical coefficient first, followed by variables in alphabetical order, e.g., `7ab²` is preferred over `b²a7`.
Formulae
na = a + a + ⋯ + a (n times) To understand the meaning of a coefficient as repeated addition.
an = a × a × ⋯ × a (n times) To understand the meaning of an exponent as repeated multiplication.
(ab)n = an × bn When raising a product of terms to a power, the power applies to each factor inside the bracket.
Definitions
- Variable
- A letter or symbol used to represent an unknown or changeable quantity.
- Coefficient
- The numerical factor that multiplies the variable(s) in an algebraic term. In `6x²`, the coefficient is 6.
- Exponent
- Also known as an index or power, it indicates how many times a base number or variable is multiplied by itself.
- Term
- A single component of an algebraic expression, which can be a number, a variable, or a product of numbers and variables, like `5`, `x`, or `8ab²`.
Worked example
The cost of a pencil is `p` pence. The cost of a pen is twice the square of the cost of a pencil. Write a simplified algebraic expression for the total cost of 5 pencils and 2 pens.
- 1
First, establish the cost of each item in algebraic notation.
- 2
Cost of one pencil = `p`.
- 3
The cost of a pen is 'twice the square of p'.
The square of p is `p2`.
Twice this is `2 × p2`, which is written as `2p2`.
- 4
Next, calculate the cost for the quantities required.
- 5
Cost of 5 pencils = `5 × p = 5p`.
- 6
Cost of 2 pens = `2 × (2p2) = 4p2`.
- 7
The total cost is the sum of these two amounts:
`5p + 4p2`.
Answer: 4p2 + 5p
Common mistakes
- ×Confusing repeated addition with repeated multiplication, for example, mixing up `2x` (which is `x+x`) and `x2` (which is `x*x`).
- ×Incorrectly applying powers to terms with coefficients. For example, writing `(4x)2` as `4x2` instead of the correct `16x2`.
- ×Misinterpreting written phrases. 'The square of 3y' is `(3y)2 = 9y2`, whereas '3 times the square of y' is `3y2`.
No-calculator tips
- ✓When reading a problem, actively translate phrases into algebraic symbols as you go. 'A number cubed' becomes `x3`, '5 less than y' becomes `y-5`.
- ✓Always apply the standard ordering convention (coefficient first, then alphabetical variables) to make expressions cleaner and easier to simplify later on.
- ✓When faced with a complex fraction, remember the fraction bar acts like a set of brackets for everything above and below it. For example, `(x+2)/(y-1)`.