Algebraic Notation & Vocabulary | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Algebraic Notation

What is algebraic notation?

• When working with algebraic expressions, specific conventions and symbols are used to represent operations.
  • Algebraic notation is the system of symbols and rules used in algebra to represent mathematical ideas.
• For addition and subtraction in numbers, we use the symbols + and - respectively. In algebra, these symbols are also used.
  • For example: a + b, c - d
• However, in multiplication, no specific symbol is used. Multiplication is usually denoted by placing terms next to each other.
  • For instance: ab (represents a × b)
  • And for division, fractions are commonly used.
  • Example: a/b (means a ÷ b)
  • Example: 3ab (means 3 × a × b)
• Combinations of operations are also possible in algebraic expressions.
  • Example: (a × b) + c/3
  • The order of operations still applies in algebra: multiplication and division take precedence over addition and subtraction.
• Exponents (indices) and roots are handled similarly to numerical expressions.
  • Example: a² represents a × a
  • Example: 4a² represents 4 × a × a
• Brackets are used to indicate the order of operations.
  • For example: (a + b) × c indicates that the addition in the brackets should be calculated first before multiplying by c.

How will I need to use algebraic notation?

• Comprehend the meanings of algebraic expressions, especially in contexts where formula rearrangement is necessary.
• Practical instances may involve scenarios where you must convert textual information into algebraic form, eventually leading to the formulation of solvable equations.

Algebraic Vocabulary

• Understanding the meanings of "term" and "factor" is essential as they are fundamental elements in algebra.
• It's important to differentiate between an expression, equation, formula, and inequality to grasp algebra and proofs thoroughly. You may need to identify each correctly.

What is a term?

• A term is a basic unit in algebra, which can be:
  • a letter (variable) on its own, for example, x.
  • a number on its own, for example, 20.
  • a number multiplied by a letter, for example, 5x.
• The number in front of a letter in a term is known as a coefficient.
  • For instance, in the term 6x, the coefficient of x is 6.
  • In the term -5y, the coefficient of y is -5.
• Terms that consist only of numbers (without any letters) are termed as constants.
• Terms can involve powers and multiple letters, like 6xy, 4x², ab³c, and more.

What is a factor?

• Exploring Term Factors:
  • A factor divides a term precisely, showing how the term can be broken down into smaller components that multiply to give the original term.
  • For instance, all factors of 4xy include 1, 2, 4, x, 2x, 4x, y, 2y, 4y, xy, 2xy, and 4xy.
• Understanding Factorization:
  • Factorization is the process of expressing a number or algebraic expression as a product of its factors.
  • It helps in simplifying complex expressions and understanding their building blocks.
• Common Factors and Highest Common Factor:
  • A common factor divides two or more terms and is shared between them.
  • The highest common factor is the greatest factor that is common to all the terms.
  • For instance, the common factors of 6xy and 4x are 2, x, and 2x.

What is an Expression?

• An expression is an algebraic statement without an equals sign, indicating there is nothing to solve.
• An expression consists of terms that are combined using addition, subtraction, multiplication, or division.
  • Terms are elements like 2x, 5y, b2 - 2cd, and more. 
  • A single term can also be considered an expression.
• Expressions can be simplified for easier understanding. 
  • For instance, x + x + x can be simplified to 3x.

What is an equation?

• An equation is a mathematical statement containing an equals sign, equating the left-hand side to the right-hand side.
• In an equation, both sides are of equal value. For example, 2x = 10 implies that 2x and 10 are of the same value.
• To solve an equation, one finds the values of the variables that make both sides equal. The solution, like x = 5, resolves the equation.

What is a formula?

• A formula is a worded rule, definition, or relationship between different quantities, written in shorthand using letters. 
  • For example, the formula w = mg represents weight 𝑤w as mass m multiplied by gravitational acceleration 𝑔g. 
• It is common to substitute numbers into a formula, but a formula on its own cannot be solved. 
• To turn a formula into an equation, additional information is needed. 
  • For instance, if w = 50 and m = 5 in the formula w = mg, the equation 50 = 5g can be formed. 

What is an inequality?

• An inequality is a mathematical statement that compares two expressions and shows their relationship regarding size.
  Here's how different inequality signs work:
  • x > y: x is greater than y
  • x ≥ y: x is greater than or equal to y
  • x < y: x is less than y
  • x ≤ y: x is less than or equal to y
• If we have x ≥ 8, this implies that x can be 8 or any number greater than 8. 
  • It can be interpreted as "8 or more" or "at least 8".
• When solving inequalities, the solutions typically result in new inequalities. For example:
  • If we have x + 10 < 15, solving this gives x < 5. Therefore, x can be any number less than 5.