Comprehensive Notes on Numbers, Indices, Surds, and Logarithms

REAL NUMBERS

• Learning Objectives:

  • Represent a real number on the number line.

  • Understand the relationships between natural numbers, integers, rational numbers, and irrational numbers as subsets of real numbers.

  • Classify numbers within the real number system.

• Classification of Real Numbers:

  • Natural or Counting Numbers ($N$): The set starting from 1 ($1, 2, 3, 4, \dots$).

  • Whole Numbers ($W$): The set of natural numbers including zero ($0, 1, 2, 3, 4, \dots$).

  • Integers ($Z$): Positive and negative whole numbers including zero ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$).

  • Rational Numbers ($Q$):

    • Can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

    • Includes integers.

    • Includes terminating decimals (finite), such as $4.5 = \frac{9}{2}$ or $1.254$.

    • Includes repeating and non-terminating decimals (infinite), such as $1.333333\dots = 1.\dot{3}$ or $-3.134134134134 = -3.\dot{1}3\dot{4}$.

  • Irrational Numbers ($Q'$ or $\overline{Q}$):

    • Cannot be written as a fraction.

    • All real numbers that do not belong to the rational group.

    • Includes non-perfect squares.

    • Includes non-repeating and non-terminating decimals.

    • Examples: $\pi = 3.14159\dots$, $e = 2.71828182845\dots$, $\sqrt{13} = 3.605551275\dots$.

• Venn Diagram Relationships:

  • The set of Real Numbers ($R$) is the union of Rational and Irrational numbers: $(Q \cup Q') = R$.

  • The hierarchy of subsets is: $N \subset W \subset Z \subset Q$.

• Properties of Real Numbers:

  • Closure Property: If $a$ and $b$ are real numbers, then $a + b$ is a real number and $ab$ is a real number.

  • Commutative Property: $a + b = b + a$ and $ab = ba$.

  • Associative Property: $a + (b + c) = (a + b) + c$ and $a(bc) = (ab)c$.

  • Identity Property:

    • Addition: $a + 0 = a$

    • Multiplication: $a \times 1 = a$

  • Inverse Property:

    • Addition: $a + (-a) = 0$

    • Multiplication: $a \times \frac{1}{a} = 1$ (where $a \neq 0$)

  • Distributive Property: $a(b + c) = ab + ac$

  • Zero Property of Multiplication: The product of $0$ and any number results in $0$. For any real number $a$, $a \times 0 = 0$. Examples: $0 \times -4 = 0$, $m \times 0 = 0$, $27x^2 y^4 \times 0 = 0$.

• Number Line Representation:

  • Both rational and irrational numbers exist on the number line.

  • The origin is at $0$.

  • Numbers to the right of the origin are larger; numbers to the left are smaller.

• Intervals of Real Numbers:

  • Open Interval: $(a, b) \implies a < x < b$

  • Closed Interval: $[a, b] \implies a \leq x \leq b$

  • Half-open / Half-closed Interval: $(a, b] \implies a < x \leq b$ or $[a, b) \implies a \leq x < b$

• Set Notation and Symbols:

  • $\{ \}$: Set (a collection of elements).

  • $A \cap B$: Intersection (elements belonging to both sets).

  • $A \cup B$: Union (all elements from both sets).

  • $A \subset B$: Subset (Set $A$ is included in Set $B$).

  • $A \subsetneq B$: Proper subset (Subset where $A \neq B$).

  • $A \not\subset B$: Not a subset.

  • $a \in A$: Element (member of the set).

  • $a \notin A$: Not an element.

  • $\emptyset$: Null or empty set.

COMPLEX NUMBERS

• Learning Objectives:

  • Understand a complex number consists of a real part and an imaginary part.

  • Find $Re(z)$ and $Im(z)$.

  • Use the imaginary unit $i$ where $i^2 = -1$.

  • Perform arithmetic (add, subtract, multiply, divide) with complex numbers.

  • Find the conjugate of a complex number.

  • Solve quadratic equations with negative discriminants ($b^2 - 4ac < 0$).

  • Plot complex numbers on an Argand diagram.

• Definitions:

  • A complex number $z$ is of the form $z = a + bi$.

  • $a$ is the Real Part, denoted $Re(z)$.

  • $b$ is the Imaginary Part, denoted $Im(z)$.

  • $bi$ is an imaginary number; $i$ is the imaginary unit ($i = \sqrt{-1}$, $i^2 = -1$).

• Powers of $i$:

  • $i^1 = i$

  • $i^2 = -1$

  • $i^3 = -i$

  • $i^4 = 1$

  • (The cycle repeats every 4 powers).

• Quadratic Equations ($b^2 - 4ac < 0$):

  • If the discriminant is negative, the roots are complex or imaginary: $\alpha = a + bi$ and $\beta = a - bi$.

  • The graph of such a quadratic does not touch the x-axis.

• Operations with Complex Numbers:

  • Addition: Add real parts and imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$.

  • Subtraction: Distribute the negative sign and add like parts: $(a + bi) - (c + di) = (a - c) + (b - d)i$.

  • Multiplication: Use the distributive property (FOIL): $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$. Always replace $i^2$ with $-1$.

  • Division: Rationalize the denominator by multiplying the numerator and denominator by the complex conjugate of the denominator.

• Complex Conjugates:

  • If $z = a + bi$, then the conjugate is $\bar{z} = a - bi$.

  • The product of a number and its conjugate is always a real constant: $z\bar{z} = (a + bi)(a - bi) = a^2 + b^2$.

• Argand Diagram:

  • A coordinate plane where the horizontal axis is the Real axis and the vertical axis is the Imaginary axis.

  • $z = a + bi$ is plotted as the point $(a, b)$.

INDICES, SURDS, AND LOGARITHMS

• Indices (Exponents):

  • Terminology: In $3^4$, 3 is the "Base" and 4 is the "exponent", "index", or "power".

  • Rules of Indices:

    • Multiplication: $a^m \times a^n = a^{m+n}$

    • Division: $a^m \div a^n = a^{m-n}$

    • Power of a Power: $(a^m)^n = a^{mn}$

    • Power of a Product: $(ab)^m = a^m b^m$

    • Power of a Quotient: $(\frac{a}{b})^m = \frac{a^m}{b^m}$

    • Zero Index: $a^0 = 1$ (for $a \neq 0$)

    • Negative Index: $a^{-n} = \frac{1}{a^n}$ and $(\frac{a}{b})^{-m} = (\frac{b}{a})^m$

    • Fractional/Rational Index: $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

• Surds:

  • A surd is an irrational number of the form $\sqrt[n]{a}$, where $a$ is not a perfect $n$th power.

  • Properties of nth Roots:

    1. $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$

    2. $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

    3. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$

    4. $\sqrt[n]{a^n} = a$ (if $n$ is odd)

    5. $\sqrt[n]{a^n} = |a|$ (if $n$ is even)

  • Rationalizing Surds: The process of removing radicals from the denominator by multiplying by a rationalizing factor (often the conjugate).

    • For $\sqrt{x}$, the factor is $\sqrt{x}$.

    • For $a + \sqrt{b}$, the factor is $a - \sqrt{b}$.

    • For $\sqrt[n]{a^m}$, the factor is $\sqrt[n]{a^{n-m}}$ (if $n > m$).

• Logarithms:

  • Transformation: Index form $x = a^y \iff y = \log_a x$.

  • Constraint: Logarithms of negative numbers and zero are not defined in the set of Real numbers.

  • Properties of Logarithms ($a, x, y > 0, a \neq 1$):

    1. Product Rule: $\log_a (xy) = \log_a x + \log_a y$

    2. Quotient Rule: $\log_a (\frac{x}{y}) = \log_a x - \log_a y$

    3. $\log_a 1 = 0$

    4. $\log_a a = 1$

    5. Power Rule: $\log_a x^c = c \log_a x$

    6. $\log_a a^c = c$

    7. Inverse Property: $a^{\log_a x} = x$

    8. Change of Base: $\log_a x = \frac{\log_b x}{\log_b a}$

SOLVING EQUATIONS

• Solving Index Equations:

  • Method 1: Comparing bases (inspection). If $a^x = a^y$, then $x = y$.

  • Method 2: Taking logarithms of both sides and using power properties.

  • Method 3: Substitution. Use a variable (e.g., $u = 3^x$) to convert exponential equations into quadratic equations (e.g., $9^x - 10(3^x) + 9 = 0 \implies u^2 - 10u + 9 = 0$).

• Solving Radical Equations:

  1. Isolate the surd term on one side.

  2. Square both sides and simplify.

  3. Repeat if multiple surds exist.

  4. Extraneous Solutions: Always check solutions against the original equation. Values that do not satisfy the original equation are extraneous and must be discarded.

• Solving Logarithmic Equations:

  • Use log properties to combine terms into a single logarithm.

  • Convert to index form to solve for the variable.

  • Domain Check: Solutions must result in positive arguments for all logarithms in the original equation. Discard any that result in a log of a negative number or zero.