Academic Notes: Topic 1 - Number and Algebra

NUMBERS AND ROUNDING

• Number Sets:
    - Natural Numbers: $\mathbf{N} = {0, 1, 2, 3, \ldots}$ consist of non-negative integers and are the basic counting numbers, crucial for elementary arithmetic.
    - Integers: $\mathbf{Z} = {0, \pm 1, \pm 2, \ldots}$ represent whole numbers, encompassing both positive and negative values.
    - Rational Numbers: $\mathbf{Q} = { \frac{a}{b}: a, b \in \mathbf{Z}, b \neq 0}$ indicate numbers expressible as the quotient of two integers, including fractions and finite decimals.
    - Real Numbers: $\mathbf{R} = \text{rational} + \text{irrational}$ includes all points on the number line, representing both terminating and non-terminating decimals.

• Interval Notation: The notation convention allows for compact representation of sets:
    - $[a,b]$ denotes inclusivity with both endpoints included,
    - $(a,b)$ signifies exclusivity,
    - $[a, + \infty)$ captures all real numbers greater than or equal to $a$, which is particularly useful in calculus and analysis.

• Rounding: In final IB exams, numerical answers must be expressed in either exact form or rounded to $3$ significant figures (s.f.) to enhance clarity and precision in the communication of results, essential for mathematical correctness in application.

• Scientific Form: This format expresses numbers as $a \times 10^k$, where 1 \leq a < 10 and $k \in \mathbf{Z}$, which aids in simplifying representations of very large or very small values, facilitating ease of calculations in scientific contexts.

SEQUENCES AND SERIES

• Notation: The nth term of a sequence is denoted as $u_n$ while the sum of the first $n$ terms (partial sum) is represented by $S_n$. This standard notation is widely accepted and facilitates discussions in various branches of mathematics.

• Summation: The expression $\sum_{n=1}^k u_n$ signifies the aggregate of sequence terms from index $1$ up to $k$, a critical operation for both analytical and computational endeavors related to sequences and series.

• Recursive Relation: A sequence may be specified recursively, with subsequent terms defined in relation to previous terms. This is exemplified by the Fibonacci sequence: $u_{n+1} = u_n + u_{n-1}$, starting from $u_1=1$ and $u_2=1$, illustrating the inductive generation of terms.

ARITHMETIC SEQUENCES (A.S.)

• Definition: An arithmetic sequence is characterized by a constant common difference given by $d = u_n - u_{n-1}$, essential for identifying arithmetic progressions in various applications.

• General Formula: The nth term of an arithmetic sequence can be computed using the formula $u_n = u_1 + (n - 1)d$, enabling easy calculation of any term using the first term and the common difference.

• Sum Formulas: The total sum for the first $n$ terms can be calculated using two primary formulas:
    - $S_n = \frac{n}{2}(u_1 + u_n)$, which averages the first and last terms, and
    - $S_n = \frac{n}{2}[2u_1 + (n - 1)d]$, which expresses the sum in terms of the first term and common difference, offering flexibility in their application.

• Arithmetic Mean: Given consecutive terms $a, b, c$, the arithmetic mean is defined as $b = \frac{a+c}{2}$, which is fundamental for understanding measures of center in data analysis.

GEOMETRIC SEQUENCES (G.S.)

• Definition: A geometric sequence is identified by a constant common ratio denoted as $r = \frac{u_n}{u_{n-1}}$, which is pivotal for analyzing exponential growth or decay phenomena in various fields, including finance and science.

• General Formula: The nth term can be explicitly formulated as $u_n = u_1 r^{n-1}$, showing how each term relates multiplicatively to the first term and the ratio, useful for direct calculation.

• Sum Formulas: Two primary formulas exist for the sums of the first $n$ terms:
    - $S_n = \frac{u_1 (r^n - 1)}{r - 1}$ for cases where $r \neq 1$, and
    - $S_n = \frac{u_1(1 - r^n)}{1 - r}$ for r < 1. These formulas are essential in applications involving compound interest and growth models.

• Infinite Sum: When |r| < 1, the series converges, and the value is calculated as $S_{\infty} = \frac{u_1}{1 - r}$, which is significant for evaluating limits in converging sequences.

• Geometric Mean: For terms $a, b, c$, the geometric mean is expressed as $b = \pm \sqrt{ac}$, which is crucial for understanding proportional relationships in various mathematical contexts.

FINANCIAL APPLICATIONS

• Compound Interest: The future value (FV) is computed using the formula $FV = PV(1 + \frac{r}{100k})^{kn}$, where $PV$ is the present value, $r$ is the interest rate, and $k$ represents compounding periods per year, foundational for understanding investment growth.

• Depreciation: Future value can also reflect value loss over time, defined as $FV = PV(1 - \frac{r}{100})^n$, integrating into financial modeling for depreciation assessments.

• Real Value (RV): To account for the effects of inflation, real value is computed using $RV = \frac{FV}{(1 + \frac{a}{100})^n}$, with $a$ representing the annual inflation rate, enhancing financial planning and analysis.

• GDC Usage: In financial calculations, the graphing display calculator mode supports variables including $n, I \%, PV, PMT, FV, P/Y, C/Y$, optimizing assessments for both educational purposes and practical applications.

THE BINOMIAL THEOREM

• Factorials: A factorial, denoted $n! = 1 \times 2 \times 3 \times \ldots \times n$, where $0! = 1$, is fundamental in combinatorial mathematics, serving as a basis for permutation and combination analysis.

• Combinations: The number of ways to select $r$ items from a set of $n$ is calculated as: $\binom{n}{r} = \frac{n!}{r!(n - r)!}$, a cornerstone concept in probability statistics.

• Expansion: The expansion of $(a + b)^n$ results in the polynomial:
    - $a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \ldots + b^n$, which assists in algebraic manipulations and theoretical explorations.

• General Term: The general form for any term in the expansion can be described as $T_{r+1} = \binom{n}{r} a^{n-r} b^r$, illustrating the structure of polynomial expansions clarity for practical use.

METHODS OF PROOF (HL ONLY)

• Direct Deduction: This method involves logical reasoning that leads from premises to a conclusion, forming the basis for valid arguments in mathematical discourse.

• Contrapositive: The logical equivalence $(A \Rightarrow B) \text{ is equivalent to } (\text{not } B \Rightarrow \text{not } A)$ is crucial for establishing truths within logical frameworks.

• Contradiction: By assuming the opposite of a given assertion, arriving at an absurdity validates the original claim’s correctness, a vital method in formal proofs.

• Counterexample: Identifying a singular instance that disproves a statement effectively renders it false, showcasing the utility of counterexamples in mathematical reasoning.

• Mathematical Induction: Induction involves:
    - 1. Base Case: Proving the initial case $n=1$,
    - 2. Inductive Assumption: Assuming the statement holds for $n=k$,
    - 3. Inductive Step: Demonstrating truth for $n=k+1$, thereby establishing the assertion across integers conclusively.

• Pigeonhole Principle: This principle posits that if $n+1$ objects are distributed among $n$ containers, at least one must contain multiple objects, useful for various combinatorial arguments.

SYSTEMS OF LINEAR EQUATIONS

• Types of Solutions: Solutions to a system can exist as unique (intersecting lines), non-existent (parallel lines), or infinitely many (coincident lines), which are central concepts in linear algebra applications.

• Gaussian Elimination: A systematic algorithm using elementary row operations on an augmented matrix to achieve row-echelon form, thereby facilitating solutions for systems of equations, a fundamental technique in linear algebraic computations.

COMPLEX NUMBERS (HL ONLY)

• Definitions: The imaginary unit is defined by $i^2 = -1$; complex numbers are expressed in Cartesian form as $z = x + iy$, allowing complex calculations across many domains.

• Operations: Key operations include the modulus defined as $|z| = \sqrt{x^2 + y^2}$ and the conjugate as $\bar{z} = x - iy$, which are essential in the manipulation of complex expressions.

• Powers of i: The powers of $i$ cycle through: $i^1=i, i^2=-1, i^3=-i, i^4=1$, establishing a periodic pattern that aids in computation.

• Fundamental Theorem of Algebra: Asserts that every polynomial of degree $n$ possesses exactly $n$ complex roots, reinforcing the importance of complex numbers in polynomial factorization and roots analysis.

• Polar and Euler Forms: The polar form $z = r(\cos(\theta) + i \sin(\theta)) = r \text{cis}(\theta)$ represents complex numbers via magnitude and angle, while the Euler form $z = re^{i\theta}$ streamlines multiplication/exponentiation operations.

• De Moivre’s Theorem: This theorem provides a formula for powers: $z^n = (r \text{cis}(\theta))^n = r^n \text{cis}(n \theta)$, important for deriving powers and roots in complex analysis.

• Roots of Unity: The $n$ complex roots of the equation $z^n = 1$ can be determined via the formula $z_k = \text{cis}\left(\frac{2k\pi}{n}\right)$ for $k = 0, 1, \ldots, n-1$, exhibiting symmetry in the complex coordinate plane and revealing that their sum equals $0$.