Mathematics (Specification A) | Use of symbols

📚 International GCSE Maths: Equations, Formulae and Identities
Chapter: Use of Symbols (Algebraic Language)

Hello future mathematician! This chapter is your foundation for all of algebra. Don't worry if letters and numbers mixing feels strange—algebra is simply a powerful system of shortcuts. Once you master the rules of using symbols, you unlock the ability to solve complex problems quickly!

The goal here is simple: Learn to read, write, and understand the basic language of Maths.

1. Variables: The Building Blocks of Algebra

In maths, symbols—usually letters like \(x\), \(y\), \(a\), or \(t\)—are used to represent numbers that are unknown or can change. These letters are called variables.

What is a Variable?

• A Variable is a symbol (a letter) that represents an unknown number or a quantity that changes.
• It's the opposite of a Constant, which is a fixed number (like 5 or \(-2\)).

🍎 Analogy: The Empty Box
Think of a variable like an empty box. We call the box \(x\). We don't know what number is inside yet, but whatever it is, that letter \(x\) represents it. If we find out that \(x\) is 7, we can put 7 in the box!

Terms and Coefficients

An algebraic expression is built from terms. A term is a single number, a single variable, or variables and numbers multiplied together.

• In the term \(5x\), the 5 is the coefficient.
• The Coefficient is the number multiplied by the variable.
• Example: In \(3a^2\), 3 is the coefficient.

2. The Rules of Algebraic Notation (The Shortcuts)

To write algebra efficiently, we use specific shortcuts. You must learn these rules to write mathematical statements correctly.

Rule 1: Omitting Multiplication Signs

We almost never use the \(\times\) symbol in algebra because it looks too much like the variable \(x\).

• \(3 \times a\) is written as \(3a\). (The number always comes first!)
• \(x \times y\) is written as \(xy\).
• \(2 \times a \times 5\) is simplified by multiplying the constants: \(2 \times 5 \times a = 10a\).

Rule 2: Multiplying by One

If a variable is only multiplied by 1, we omit the number 1 entirely.

• \(1 \times x\) is written as \(x\). (We assume the coefficient is 1.)
• \(-1 \times y\) is written as \(-y\). (We must keep the negative sign!)

⚠ Common Mistake Alert!
Students sometimes forget the 'invisible 1'. If you see \(x + 5\), remember it means \(1x + 5\).

Rule 3: Division Notation

In algebra, we primarily use fraction lines (vinculum) to represent division.

• \(x \div 4\) is written as \(\frac{x}{4}\).
• \((a + b) \div 2\) is written as \(\frac{a + b}{2}\).

Rule 4: Powers (Indices)

When a variable is multiplied by itself, we use indices (exponents).

• \(a \times a\) is written as \(a^2\) (a squared).
• \(y \times y \times y\) is written as \(y^3\) (y cubed).
• \(3 \times x \times x\) is written as \(3x^2\).

3. Expressions, Equations, Formulae, and Identities

These terms are often confused, but they have very precise meanings based on the symbols they contain.

3.1. Expression

A collection of terms joined by addition or subtraction signs. It does NOT contain an equals sign (=).

• Examples: \(5x + 2y\), or \(a^2 - 3\).
• What you can do: Simplify them or evaluate them (substitution).
• Key Takeaway: Expressions can be written, but not solved.

3.2. Equation

A statement that shows two expressions are equal. It MUST contain an equals sign (=).

• Examples: \(3x + 1 = 10\), or \(y - 4 = 2y\).
• What you can do: Solve them to find the specific value of the variable(s) that makes the statement true.
• Key Takeaway: The equals sign (=) is the central symbol. It means "is the same as."

3.3. Formula (Plural: Formulae)

A special type of equation that shows the relationship between different quantities, usually defined by an abbreviation (e.g., Area, Volume, Speed).

• Example: The formula for the area of a circle: \(A = \pi r^2\).
• Example: Speed is distance divided by time: \(S = \frac{D}{T}\).
• Key Takeaway: A formula links specific variables to calculate a known result.

3.4. Identity

An equation that is true for all possible values of the variables. We use the symbol \(\equiv\) (three lines) to denote an identity.

• Example: \(a(b + c) \equiv ab + ac\) (This is the Distributive Law, and it's always true).

4. Using Symbols: Substitution

Substitution is the process of replacing variables in an expression or formula with their given numerical values.

Step-by-Step Guide to Substitution

When you substitute, you are "evaluating" the expression (finding its final value).

Example: Evaluate \(4a + 3b - 2\) when \(a = 5\) and \(b = -2\).

1. Write down the expression:
   \(4a + 3b - 2\)
2. Replace the variables with their numerical values (use brackets!):
   \(4(5) + 3(-2) - 2\)
3. Calculate using BIDMAS/BODMAS:
   First, the multiplication terms:
   \((4 \times 5) + (3 \times -2) - 2\)
   \(20 + (-6) - 2\)
4. Finish the calculation:
   \(20 - 6 - 2 = 12\)

Using brackets when substituting negative numbers is essential to avoid errors.

Substitution with Powers

Remember that the power only applies to the term it is next to.

Example: Evaluate \(x^2 + 5x\) when \(x = -3\).

Substitution: \((-3)^2 + 5(-3)\)
Calculation:
\((-3) \times (-3) = 9\)
\(5 \times (-3) = -15\)
Result: \(9 + (-15) = 9 - 15 = -6\)

Key Takeaway: Substitution is the process of translating algebraic language back into pure numerical math, always following the order of operations (BIDMAS/BODMAS).

5. Simplifying Expressions: Collecting Like Terms

Simplifying an expression means making it shorter and easier to read by adding or subtracting terms that are similar.

The Rule of Like Terms

You can only combine Like Terms. Like Terms are terms that have the exact same variable part, including the power.

🍌 Analogy: Fruits!
Imagine you have 5 Apples (\(5a\)) and 3 Bananas (\(3b\)). If someone gives you 2 more Apples (\(+2a\)), you can only combine the Apples. You cannot combine the Apples and Bananas.

• \(5a + 3b + 2a\) simplifies to \(7a + 3b\).

Identifying Like and Unlike Terms

• Like Terms: \(4x\) and \(12x\). (Both have \(x^1\))
• Unlike Terms: \(4x\) and \(12y\). (Different variables)
• Unlike Terms: \(4x\) and \(12x^2\). (Different powers)
• Like Terms: \(7ab\) and \(-2ab\). (Both have \(ab\))

Step-by-Step Simplification

Example 1 (Single Variable): Simplify \(6x + 8 - 2x - 3\).

1. Identify like terms:
   • \(x\) terms: \(+6x\) and \(-2x\)
   • Constant terms: \(+8\) and \(-3\)
2. Collect and calculate the groups:
   • \(6x - 2x = 4x\)
   • \(8 - 3 = +5\)
3. Write the final simplified expression:
   \(4x + 5\)

Example 2 (Multiple Variables and Powers): Simplify \(5a + 3b^2 - 2a + b^2 - 1\).

1. Group the terms (include the sign in front of the term):
   • \(a\) terms: \(5a\) and \(-2a\)
   • \(b^2\) terms: \(+3b^2\) and \(+b^2\) (Remember the invisible coefficient of 1!)
   • Constant term: \(-1\)
2. Calculate:
   • \(5a - 2a = 3a\)
   • \(3b^2 + 1b^2 = 4b^2\)
3. Final Expression:
   \(3a + 4b^2 - 1\)

Don't worry if this seems tricky at first—the key is careful organisation and remembering to always take the sign with the term!