Welcome to Algebra: Notation and Manipulation!
Hello future mathematician! This chapter is your foundation stone for all of algebra. Think of algebra as the shorthand language of maths. Instead of writing long sentences, we use symbols (letters) to represent numbers we don't know yet. Mastering notation and manipulation is like learning to speak this language fluently. Don't worry if this seems tricky at first; we will break everything down step-by-step!
Why is this important? These skills allow us to simplify massive calculations and solve complex real-world problems, from finance to physics.
1. The Language of Algebra: Notation and Key Terms
1.1 Standard Algebraic Notation
Algebra uses specific rules to write expressions efficiently:
- Multiplication: We usually hide the multiplication sign between a number and a letter, or between two letters.
Example: \(4 \times y\) is written as \(4y\). We never write \(y4\).
Example: \(a \times b\) is written as \(ab\). - Division: Division is almost always written as a fraction.
Example: \(x \div 5\) is written as \(\frac{x}{5}\). - The number 1: If a variable (letter) is multiplied by 1, we don't write the 1.
Example: \(1x\) is written simply as \(x\).
1.2 Essential Vocabulary
Every subject has its own terms. Here are the must-know algebra words:
- Variable: A symbol (usually a letter like \(x\) or \(y\)) that represents an unknown number.
- Coefficient: The number multiplied by the variable.
Example: In \(7x\), the coefficient is 7. - Term: A single number, a single variable, or variables and numbers multiplied together.
Example: In the expression \(5x + 3y - 2\), the terms are \(5x\), \(3y\), and \(-2\). - Expression: A combination of terms joined by plus or minus signs (it does NOT contain an equals sign).
Example: \(a^2 + 2b - 1\)
Quick Review: Notation
\(p\) means \(1p\).
\(xy\) means \(x \times y\).
\(\frac{a}{2}\) means \(a \div 2\).
2. Substitution: Giving Meaning to Variables
2.1 What is Substitution?
Substitution is the process of replacing the variables in an expression with their given numerical values and then calculating the result.
Analogy: Think of a recipe. The variables are the ingredients (\(x\) is flour, \(y\) is sugar). Substitution is putting the actual measured amounts (numbers) into the mixing bowl.
2.2 Step-by-Step Substitution
Example: Evaluate the expression \(3x + 2y^2\) when \(x = 5\) and \(y = -4\).
- Write down the expression: \(3x + 2y^2\)
- Replace variables with their values (use brackets!):
\(3(5) + 2(-4)^2\) - Calculate using the correct order of operations (BIDMAS/BODMAS):
Remember: Powers come before multiplication.- First, calculate the power: \((-4)^2 = (-4) \times (-4) = 16\).
Expression is now: \(3(5) + 2(16)\) - Second, perform multiplication: \(3 \times 5 = 15\) and \(2 \times 16 = 32\).
Expression is now: \(15 + 32\) - Finally, perform addition: \(15 + 32 = 47\)
- First, calculate the power: \((-4)^2 = (-4) \times (-4) = 16\).
Common Mistake Alert! (Negative Numbers)
When substituting a negative number, always use brackets! If \(y = -4\):
* Correct: \(y^2 = (-4)^2 = 16\)
* Incorrect: \(-4^2 = -16\) (The calculator squares 4 first, then applies the negative sign).
Always bracket the negative value!
3. Simplifying Expressions: Collecting Like Terms
3.1 Identifying Like Terms
We can only add or subtract terms that are Like Terms. Like terms must have the exact same variables raised to the exact same powers.
Analogy: In a fruit shop, you can easily add 3 apples and 5 apples to get 8 apples, but you can't simplify 3 apples plus 5 bananas into a single term.
- Like Terms: \(5x\) and \(-2x\)
- Like Terms: \(4y^2\) and \(9y^2\)
- NOT Like Terms: \(6x\) and \(6x^2\) (The powers are different)
- NOT Like Terms: \(3ab\) and \(3a\) (The variables are different)
3.2 The Simplification Process
When simplifying, the variable part of the term stays exactly the same; we only operate on the coefficients.
Example: Simplify \(5a + 7b - 2a + 3 + b\)
- Identify like terms: Group them together (mentally or by underlining). Remember, the sign (+ or -) belongs to the term that follows it.
- Terms in \(a\): \(+5a\) and \(-2a\)
- Terms in \(b\): \(+7b\) and \(+1b\) (remember \(b\) means \(1b\))
- Constant terms (numbers alone): \(+3\)
- Calculate the coefficients for each group:
- For \(a\): \(5 - 2 = 3\). Result: \(3a\)
- For \(b\): \(7 + 1 = 8\). Result: \(8b\)
- Write the final simplified expression:
\(3a + 8b + 3\)
Key Takeaway: Collecting Terms
You can only combine terms that are identical twins (same variable, same power).
4. The Power of Indices (Exponents)
Indices (or exponents or powers) tell us how many times a base number is multiplied by itself. Understanding the laws of indices is essential for advanced manipulation.
Example: In \(x^5\), \(x\) is the base and 5 is the index (or exponent).
4.1 The Laws of Indices (The Rules of Powers)
These rules apply only when the base numbers are the same.
Law 1: Multiplication (Adding the Indices)
When multiplying terms with the same base, you add the powers.
\(a^m \times a^n = a^{m+n}\)
Example: \(x^2 \times x^4 = x^{2+4} = x^6\)
Why? \(x^2\) is \((x \times x)\) and \(x^4\) is \((x \times x \times x \times x)\). Together, that's 6 \(x\)'s multiplied.
Law 2: Division (Subtracting the Indices)
When dividing terms with the same base, you subtract the powers.
\(a^m \div a^n = a^{m-n}\)
Example: \(y^7 \div y^3 = y^{7-3} = y^4\)
Law 3: Power of a Power (Multiplying the Indices)
When raising a power to another power, you multiply the indices.
\((a^m)^n = a^{m \times n}\)
Example: \((z^3)^5 = z^{3 \times 5} = z^{15}\)
Law 4: The Zero Index
Any non-zero number or variable raised to the power of zero is always 1.
\(a^0 = 1\)
Example: \(100^0 = 1\), and \((4x)^0 = 1\)
Did you know? This comes from the division rule: \(x^3 \div x^3 = 1\). But using Law 2, \(x^{3-3} = x^0\). Since both answers must be the same, \(x^0\) must equal 1.
5. Standard Form (Scientific Notation)
Standard Form is a convenient way to write very large or very small numbers using powers of 10. It is especially useful in science (like calculating the distance to a star or the size of a virus).
5.1 The Rule of Standard Form
A number written in standard form looks like this:
\(A \times 10^n\)
Where:
- A (The front number) must be between 1 and 10 (it can be 1, but must be strictly less than 10). So, \(1 \le A < 10\).
- n (The power of 10) is an integer (a whole number).
5.2 Converting to Standard Form
You need to decide where to place the decimal point to make the number \(A\) fall between 1 and 10, then count how many places you moved the decimal.
Large Numbers (Positive Indices)
Example: Write 45,000,000 in standard form.
- Move the decimal until the number is between 1 and 10: 4.5
- Count the moves: The decimal moved 7 places to the left.
- Standard Form: \(4.5 \times 10^7\)
Small Numbers (Negative Indices)
Example: Write 0.0000062 in standard form.
- Move the decimal until the number is between 1 and 10: 6.2
- Count the moves: The decimal moved 6 places to the right.
- Standard Form: \(6.2 \times 10^{-6}\)
Memory Aid: If you start with a Large number, you get a Positive index. If you start with a Small number, you get a Negative index.
6. Basic Expansion and Factorisation
Expansion and factorisation are opposite processes, like putting on a glove (expansion) and taking it off (factorisation). They rely on the Distributive Law.
6.1 Expansion (The Distributive Law)
Expanding a bracket means multiplying every term inside the bracket by the term outside it.
Formula: \(a(b + c) = ab + ac\)
Example 1: Expand \(4(2x + 5)\)
\(4 \times 2x\) PLUS \(4 \times 5\)
\(8x + 20\)
Example 2 (Watch the signs!): Expand \(-3(y - 7)\)
\(-3 \times y\) PLUS \(-3 \times (-7)\)
\(-3y + 21\)
Quick Tip:
A negative multiplied by a negative always gives a positive! \((-3)(-7) = +21\)
6.2 Factorisation (Reversing the Process)
Factorisation is putting the expression back into brackets. We do this by finding the Highest Common Factor (HCF) of all the terms and placing it outside the bracket.
Example 1: Factorise \(6x + 9\)
- Find the HCF of 6 and 9: The largest number that divides into both is 3.
- Place the HCF outside: \(3(\dots)\)
- Divide each original term by the HCF to find the terms inside:
\(6x \div 3 = 2x\)
\(9 \div 3 = 3\) - Final Factorised Form: \(3(2x + 3)\)
Example 2 (Involving variables): Factorise \(10ab - 15b\)
- HCF of numbers (10 and 15): 5.
- HCF of variables (ab and b): Both terms contain \(b\).
- Total HCF: \(5b\).
- Factorisation: \(5b(\dots)\)
- Divide terms:
\(10ab \div 5b = 2a\)
\(-15b \div 5b = -3\) - Final Factorised Form: \(5b(2a - 3)\)
Key Takeaway: Expansion vs. Factorisation
Expansion uses multiplication to remove brackets. Factorisation uses division (finding the HCF) to insert brackets.