Number Systems and Algebra Fundamentals

Subject: Mathematics
Topic: 1
Cambridge Code: 0580

---

Number Systems

Real Numbers

Types of real numbers:

Natural numbers: 1, 2, 3, ... (N) Whole numbers: 0, 1, 2, 3, ... (W) Integers: ..., -2, -1, 0, 1, 2, ... (Z) Rational numbers: Can be expressed as p/q where p, q ∈ Z, q ≠ 0 (Q) Irrational numbers: Cannot be expressed as fractions (√2, π, e) Real numbers: All rational and irrational numbers (R)

Surds

Surd - Root of number that cannot be simplified to rational number

Properties: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Rationalizing denominators:

• Multiply by conjugate
• Example: $\frac{1}{1+\sqrt{2}} = \frac{1-\sqrt{2}}{(1+\sqrt{2})(1-\sqrt{2})} = \frac{1-\sqrt{2}}{1-2} = \sqrt{2}-1$

Indices (Powers)

Laws of indices: $a^m \times a^n = a^{m+n}$ $a^m ÷ a^n = a^{m-n}$ $(a^m)^n = a^{mn}$ $a^0 = 1, a ≠ 0$ $a^{-n} = \frac{1}{a^n}$ $a^{\frac{1}{n}} = \sqrt[n]{a}$ $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

---

Algebraic Expressions

Expanding and Factoring

Expanding brackets:

• Single: $3(2x + 5) = 6x + 15$
• Double: $(a + b)(c + d) = ac + ad + bc + bd$
• Difference of squares: $(a + b)(a - b) = a^2 - b^2$

Factoring:

• Common factor: $6x + 9 = 3(2x + 3)$
• Difference of squares: $x^2 - 4 = (x - 2)(x + 2)$
• Trinomial: $x^2 + 5x + 6 = (x + 2)(x + 3)$
• Grouping: $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$

Algebraic Fractions

Simplifying: $\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2, x ≠ 2$

Operations: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ $\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

---

Linear Equations

Solving Linear Equations

Single variable: $2x + 5 = 13$ $2x = 8$ $x = 4$

Simultaneous equations (two unknowns):

Method 1: Substitution $y = 2x + 1$ $3x + y = 11$ $3x + (2x + 1) = 11$ $5x = 10$ $x = 2, y = 5$

Method 2: Elimination $2x + 3y = 8$ $x - 3y = 1$ Adding: $3x = 9$, so $x = 3$

---

Quadratic Equations

Solving Quadratics

Quadratic formula: $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$ For $ax^2 + bx + c = 0$

Discriminant: $Δ = b^2 - 4ac$

• $Δ > 0$: Two distinct real roots
• $Δ = 0$: One repeated real root
• $Δ < 0$: No real roots

Factoring: $x^2 + 5x + 6 = 0$ $(x + 2)(x + 3) = 0$ $x = -2 \text{ or } x = -3$

Completing the Square

$x^2 + 6x - 7 = 0$ $x^2 + 6x + 9 - 9 - 7 = 0$ $(x + 3)^2 = 16$ $x + 3 = ±4$ $x = 1 \text{ or } x = -7$

---

Polynomials

Polynomial - Sum of terms with variable powers

Degree: Highest power of variable

Operations:

• Addition: Combine like terms
• Subtraction: Distribute negative
• Multiplication: Distributive property
• Division: Long division or synthetic

Polynomial Division

Example: $(x^3 + 2x^2 - 5x + 3) ÷ (x - 1)$

Using synthetic division or polynomial long division yields: $x^2 + 3x - 2 + \frac{1}{x-1}$

Factor Theorem

If f(a) = 0, then (x - a) is a factor of f(x)

Example: If $f(x) = x^3 - 6x^2 + 11x - 6$ and $f(1) = 0$ Then $(x - 1)$ is a factor

---

Logarithms

Logarithm - Inverse of exponential function

Definition

$\log_a x = y \text{ if and only if } a^y = x$

Common bases:

• $\log_{10}$ (common logarithm)
• $\ln$ or $\log_e$ (natural logarithm)

Laws of Logarithms

$\log_a(xy) = \log_a x + \log_a y$ $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$ $\log_a(x^n) = n\log_a x$ $\log_a(a) = 1$ $\log_a(1) = 0$

Change of Base

$\log_a x = \frac{\log_b x}{\log_b a}$

Example: $\log_2 16 = \frac{\log_{10} 16}{\log_{10} 2} = \frac{1.204}{0.301} ≈ 4$

---

Exponential Functions

Exponential function: $f(x) = a^x$ where $a > 0, a ≠ 1$

Properties

• Always positive: $a^x > 0$ for all x
• Passes through $(0, 1)$: $a^0 = 1$
• Base determines growth/decay
• Asymptote at y = 0 (or shifted)

Solving Exponential Equations

Example: $2^x = 8$ $2^x = 2^3$ $x = 3$

Using logarithms: $3^x = 10$ $x\log 3 = \log 10$ $x = \frac{\log 10}{\log 3} = \frac{1}{0.477} ≈ 2.10$

---

Key Points

1. Real numbers include rational and irrational
2. Indices follow specific laws for multiplication/division
3. Algebraic expressions can be expanded and factored
4. Linear equations have one solution (usually)
5. Quadratic equations have up to 2 solutions
6. Discriminant determines number of roots
7. Polynomials follow operations rules
8. Logarithms are inverse of exponentials
9. Laws of logarithms simplify calculations
10. Exponential functions model growth/decay

---

Practice Questions

1. Simplify surds and rationalize denominators
2. Apply laws of indices
3. Expand and factor expressions
4. Solve linear equations
5. Solve quadratic equations (all methods)
6. Complete the square
7. Polynomial operations
8. Solve logarithmic equations
9. Solve exponential equations
10. Apply change of base formula

---

Revision Tips

• Practice expanding and factoring frequently
• Memorize laws of indices and logarithms
• Understand discriminant concept
• Practice all quadratic-solving methods
• Learn synthetic division
• Work with both exponential and logarithmic forms
• Check solutions in original equations