Mathematics - Analysis and Approaches | Number and algebra

👋 Welcome to Number and Algebra: Your Mathematical Toolkit!

Hello future IB Mathematician! This chapter, Number and algebra, is the absolute foundation upon which the rest of your Analysis and Approaches (AA) course is built. Don't worry if algebra feels a bit dusty—we are going to polish those skills until they shine!

Think of algebra as the language of mathematics. By mastering how to manipulate numbers, symbols, and expressions, you gain the power to solve complex problems, model real-world situations, and understand sophisticated mathematical concepts later in the course (like Calculus!).

We'll break down the concepts into manageable steps, focusing on clarity and building confidence, especially in algebraic manipulation and understanding the crucial concepts of indices and logarithms.

1. The Family of Numbers (SL & HL)

1.1 Understanding Different Number Sets

In mathematics, we use specific symbols to categorize numbers. These classifications are vital because they tell us what rules and properties a number follows.

• Natural Numbers (\(\mathbb{N}\)): These are the positive counting numbers.
  Example: \{1, 2, 3, 4, ...\} (Sometimes 0 is included, but usually we start at 1).
• Integers (\(\mathbb{Z}\)): This set includes all Natural Numbers, zero, and the negative whole numbers.
  Example: \{..., -3, -2, -1, 0, 1, 2, 3, ...\}
• Rational Numbers (\(\mathbb{Q}\)): Any number that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
  Example: 0.5 (\(=1/2\)), -3 (\(=-3/1\)), 0.333... (\(=1/3\)). Rational numbers have terminating or repeating decimal expansions.
• Irrational Numbers: These are numbers that CANNOT be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating.
  Example: \(\sqrt{2}\), \(\pi\), \(e\).
• Real Numbers (\(\mathbb{R}\)): This is the set of all Rational and Irrational numbers. This is usually the largest set we deal with in SL mathematics.

🧠 Analogy: Think of these sets like a series of Russian nesting dolls, each contained within the next: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\).

1.2 Working with Scientific Notation and Significant Figures

Mathematics: Analysis and Approaches requires precision. You must be comfortable using Scientific Notation and applying appropriate levels of rounding.

• Scientific Notation (Standard Form): Expressing very large or very small numbers as \(a \times 10^k\), where \(1 \le |a| < 10\) and \(k\) is an integer.
  Example: The distance light travels in one year is approximately \(9.46 \times 10^{15} \text{ metres}\).
• Significant Figures (SF): Unless otherwise specified, all final numerical answers in IB examinations must be given exactly or rounded to three significant figures (3 sf). Using more than 3 sf for intermediate steps, however, is highly recommended to maintain accuracy.

🔑 Key Takeaway: Be precise with rounding and ensure you use your GDC correctly for large or small numbers using the "E" or "\(\times 10^x\)" button.

2. Algebraic Manipulation and Formulas (SL & HL)

2.1 Expanding and Factorizing Expressions

The ability to fluently expand (multiply out brackets) and factorize (put back into brackets) is essential for solving equations and simplifying rational expressions.

Expansion Techniques

The key principle is the Distributive Property: multiply everything inside the bracket by everything outside (or in the other bracket).

Example (FOIL method for quadratics):
\((x + 3)(x - 2) = x(x) + x(-2) + 3(x) + 3(-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6\)

Factorization Techniques

Factorization is the reverse process. Look for the following patterns:

1. Common Factor: Always check for the largest common factor first.
   Example: \(3x^2 + 6x = 3x(x + 2)\)
2. Difference of Two Squares (DOTS): Recognise the pattern \(a^2 - b^2 = (a - b)(a + b)\).
   Example: \(4x^2 - 9 = (2x - 3)(2x + 3)\)
3. Quadratic Trinomials (\(ax^2 + bx + c\)): Find two numbers that multiply to \(ac\) and add up to \(b\).

🚫 Common Mistake: Forgetting to distribute a negative sign!
Incorrect: \(5 - (x + 2) = 5 - x + 2\) (Wrong!)
Correct: \(5 - (x + 2) = 5 - x - 2 = 3 - x\)

2.2 Algebraic Fractions (Rational Expressions)

You must be able to simplify, add, subtract, multiply, and divide algebraic fractions.

• Simplification: Factorize the numerator and denominator and cancel common factors.
• Adding/Subtracting: Find a common denominator (usually the Least Common Multiple, LCM) before combining the numerators.

Step-by-Step Example (Addition):
To calculate \(\frac{2}{x} + \frac{3}{y}\):
1. Common denominator is \(xy\).
2. Adjust fractions: \(\frac{2y}{xy} + \frac{3x}{xy}\)
3. Combine: \(\frac{2y + 3x}{xy}\)

🔑 Key Takeaway: Factorization is your most powerful tool in algebra. Use it to simplify everything before multiplying or adding.

3. Indices (Exponents) and Logarithms (SL & HL)

Indices and logarithms are inverse operations, meaning they undo each other. They allow us to work with growth, decay, and powers very efficiently.

3.1 The Laws of Indices

These rules govern how we handle powers (exponents).

1. Product Rule: When multiplying terms with the same base, add the powers.
   \[a^m \times a^n = a^{m+n}\]
2. Quotient Rule: When dividing terms with the same base, subtract the powers.
   \[\frac{a^m}{a^n} = a^{m-n}\]
3. Power of a Power Rule: When raising a power to another power, multiply the indices.
   \[(a^m)^n = a^{mn}\]
4. Zero Power Rule: Any non-zero base raised to the power of zero is 1.
   \[a^0 = 1\]
5. Negative Power Rule: A negative exponent means the reciprocal of the base raised to the positive power.
   \[a^{-m} = \frac{1}{a^m}\]
6. Fractional Power Rule (Roots): A fractional exponent indicates a root.
   \[a^{\frac{1}{n}} = \sqrt[n]{a} \quad \text{and} \quad a^{\frac{m}{n}} = (\sqrt[n]{a})^m\]

💡 Did you know? The Zero Power Rule makes sense: if you use the quotient rule for \(\frac{a^m}{a^m}\), you get \(a^{m-m} = a^0\). Since any number divided by itself is 1, \(a^0\) must equal 1!

3.2 The Laws of Logarithms

Logarithms answer the question: "What power do I raise the base to, to get this number?"

The relationship between indices and logs is key: \[a^x = y \iff \log_a y = x\]

The most common bases in AA are base 10 (\(\log_{10}\), often written just as \(\log\)) and base \(e\) (\(\log_e\), written as \(\ln\), the natural logarithm).

The Laws of Logs mirror the Laws of Indices:

1. Product Law: Multiplication inside the log becomes addition outside.
   \[\log_a (xy) = \log_a x + \log_a y\]
2. Quotient Law: Division inside the log becomes subtraction outside.
   \[\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\]
3. Power Law: The exponent inside the log can be brought out as a coefficient (This is the most important law for solving exponential equations!).
   \[\log_a x^p = p \log_a x\]

Change of Base Formula

Since your GDC only calculates logs in base 10 or base \(e\) (ln), you often need to change the base for other calculations: \[\log_a b = \frac{\log_c b}{\log_c a}\] Where \(c\) is a convenient base (usually 10 or \(e\)).

🔑 Key Takeaway: Logarithms are defined only for positive inputs. Always check your domain: \(\log_a x\) requires \(x > 0\).

4. Solving Equations and Inequalities (SL & HL)

4.1 Quadratic Equations

A quadratic equation is of the form \(ax^2 + bx + c = 0\). You must be able to solve these using three methods:

1. Factorization: If possible, use the techniques from Section 2.
2. Completing the Square: A method used to derive the vertex form, but also useful for solving when the roots are messy.
3. The Quadratic Formula: This always works! You must know this formula, which is provided in your formula booklet (or you can memorize it for speed): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The term under the square root, \(\Delta = b^2 - 4ac\), is the discriminant. It tells you about the roots:

• If \(\Delta > 0\): Two distinct real roots.
• If \(\Delta = 0\): One repeated real root.
• If \(\Delta < 0\): No real roots (but two complex roots, see Section 5).

4.2 Inequalities

Solving inequalities (using \(<, >, \le, \ge\)) is similar to solving equations, but with one critical rule:

🚨 Critical Rule: If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.

When solving Quadratic Inequalities (e.g., \(x^2 + 2x - 3 > 0\)):

1. Treat it as an equation and find the roots (\(x=-3\) and \(x=1\)).
2. Sketch the corresponding parabola \(y = x^2 + 2x - 3\).
3. Use the sketch to determine which intervals of \(x\) satisfy the inequality (e.g., where the graph is above the x-axis).

Absolute Value Inequalities (e.g., \(|2x - 1| < 5\)):

Absolute value means distance from zero. This type of inequality translates into two separate inequalities: \[-5 < 2x - 1 < 5\] Solving for \(x\): \(-4 < 2x < 6 \implies -2 < x < 3\)

5. HL Extension: Complex Numbers and Proof by Induction

If you are an HL student, you will delve into the exciting world of Complex Numbers (\(\mathbb{C}\)). These numbers arise when the discriminant of a quadratic is negative, allowing us to find roots for *all* quadratic equations.

5.1 Defining Complex Numbers

The foundation of complex numbers is the imaginary unit, \(i\), defined as: \[i = \sqrt{-1} \quad \text{and therefore} \quad i^2 = -1\]

A complex number, \(z\), is generally written in Cartesian form (or standard form): \[z = a + bi\] where \(a\) is the real part (\(\text{Re}(z)\)) and \(b\) is the imaginary part (\(\text{Im}(z)\)).

Operations

• Addition/Subtraction: Treat \(i\) like a variable; combine real parts and imaginary parts separately.
• Multiplication: Use the FOIL method and substitute \(i^2 = -1\).
• Division (Conjugation): To divide complex numbers, you must multiply the numerator and denominator by the complex conjugate of the denominator.
  If \(z = a + bi\), the conjugate is \(\bar{z} = a - bi\). This eliminates the imaginary part from the denominator, leaving only a real number.

5.2 Graphical Representation and Forms

Complex numbers are plotted on the Argand diagram (a plane where the x-axis is the Real axis and the y-axis is the Imaginary axis).

Modulus and Argument

The position of a complex number \(z = a + bi\) is defined by its distance from the origin and the angle it makes with the positive real axis.

• Modulus (\(|z| = r\)): The length of the vector from the origin to \(z\).
  \[r = |z| = \sqrt{a^2 + b^2}\]
• Argument (\(\arg(z) = \theta\)): The angle, usually restricted to \(-\pi < \theta \le \pi\) (or \(0 \le \theta < 2\pi\)), measured counter-clockwise from the positive Real axis.
  \[\theta = \arctan \left(\frac{b}{a}\right) \quad \text{(Careful! Check the quadrant of } z\text{)}\]

Polar (Modulus-Argument) Form

Using the modulus and argument, we can write \(z\) as: \[z = r (\cos \theta + i \sin \theta)\]

Exponential Form (Euler's Formula)

This is the most compact and powerful form, utilizing Euler's famous formula, \(e^{i\theta} = \cos \theta + i \sin \theta\). \[z = r e^{i\theta}\]

Did you know? Euler’s Identity, \(e^{i\pi} + 1 = 0\), famously links the five most fundamental mathematical constants (\(e, i, \pi, 1, 0\)) in one concise equation.

5.3 De Moivre's Theorem (HL)

This theorem simplifies raising a complex number (in polar form) to an integer power \(n\): \[(r (\cos \theta + i \sin \theta))^n = r^n (\cos (n\theta) + i \sin (n\theta))\] In exponential form, this is even simpler: \((r e^{i\theta})^n = r^n e^{in\theta}\). De Moivre's Theorem is also used extensively to find the \(n\)-th roots of complex numbers.

5.4 Proof by Mathematical Induction (HL)

Proof by Induction is a sophisticated HL technique used to prove that a statement or formula is true for all natural numbers \(n \ge 1\). It is a three-step process:

1. Base Case (The First Step): Show the statement is true for the starting value (usually \(n=1\)).
2. Inductive Hypothesis (The Assumption): Assume the statement is true for an arbitrary integer \(k\).
3. Inductive Step (The Proof): Show that if the statement is true for \(k\), it must also be true for \(k+1\). (This is the most challenging step where algebraic manipulation is critical.)

🧠 Analogy: Proof by induction is like setting up dominoes. The Base Case is pushing the first domino over. The Inductive Step proves that if any domino \(k\) falls, the next one \((k+1)\) must also fall. If both conditions are met, all dominoes must fall!

🔑 Key Takeaway (HL): Master the three forms of complex numbers (Cartesian, Polar, Exponential) and be able to convert between them easily. This skill is critical for success in Complex Numbers.