Mathematics in the Modern World — Condensed Exam Notes

Chapter 1 Nature of Mathematics

• Mathematics reveals and models patterns (e.g., Fibonacci sequence $1,1,2,3,5, ext{…}$).

• Fibonacci rule: $F<em>n = F</em>{n-1}+F<em>{n-2}\ ;\ F</em>0=0, F_1=1$.

• Closed form (Binet): $F_n = \dfrac{\bigl(\tfrac{1+\sqrt5}{2}\bigr)^n-\bigl(\tfrac{1-\sqrt5}{2}\bigr)^n}{\sqrt5}$.

• Golden ratio $\phi\approx1.618$ appears in nature, human body, design.

Chapter 2 Mathematics as a Language

• Mathematical language is precise, concise, powerful; uses symbols, operations, logical structure

• Expressions (no truth value) vs. sentences (truth–valued).

• Conventions: $a\cdot b = ab$, number precedes variables, variables ordered alphabetically.

• Set basics: finite/infinite, empty $\varnothing$, subset criteria; notation $x\in A$.

• Relations: sets of ordered pairs; a function maps each domain element to exactly one range element.

• Binary operations (+, −, ×, ÷) obey closure, commutative, associative, identity, inverse, distributive laws.

• Logic building blocks: connectives ((\land,\lor,\rightarrow,\leftrightarrow)), quantifiers $\forall,\exists$; negation rules.

Chapter 3 Reasoning

• Inductive reasoning: form conjectures from specific cases; susceptible to counter-examples.

• Deductive reasoning: derive necessary conclusions from general premises; yields proofs.

Chapter 4 Statistics & Data

• Data types: qualitative vs. quantitative (discrete/continuous).

• Measurement scales: nominal, ordinal, interval, ratio.

• Central tendency: mean $\bar x$, median $\tilde x$, mode.

• Dispersion: range, interquartile range $IQR=Q<em>3-Q</em>1$, variance $s^2$, standard deviation $s$.

• Distribution shapes: symmetric (mean = median), right-skew (mean > median), left-skew (mean < median); Pearson skew $Sk=\dfrac{3(\bar x-\tilde x)}s$.

• Presentation: frequency tables, bar/pie charts (qualitative); histogram, stem-leaf, box-whisker (quantitative).

Chapter 5 Data Management Tools

• Normal distribution: bell curve, parameters $\mu,\sigma$; standard normal $Z=\dfrac{X-\mu}{\sigma}$, total area 1.

• Use $Z$-table for probabilities; symmetry: P(Z>a)=P(Z< -a).

• Correlation (Pearson): $r = \dfrac{n\sum XY-\sum X\sum Y}{\sqrt{\bigl(n\sum X^2-(\sum X)^2\bigr)\bigl(n\sum Y^2-(\sum Y)^2\bigr)}}$; $-1\le r\le1$.

• Determination: $r^2$ (percent variance explained).

• Simple linear regression: line $\hat y = a + bx$ where $b=\dfrac{\sum XY-\tfrac{\sum X\sum Y}{n}}{\sum X^2-\tfrac{(\sum X)^2}{n}}$, $a=\bar y-b\bar x$.

Chapter 6 Commercial Mathematics

• Simple interest: $I=Prt$, maturity $M=P(1+rt).$

• Compound interest: $M=P\bigl(1+\tfrac j m\bigr)^{mt}$; present value $P=\dfrac{M}{(1+ i)^n}$.

• Stocks: dividend yield $=\dfrac{\text{annual dividend}}{\text{price}}\times100\%$; Gordon model $P<em>0=\dfrac{D</em>1}{r-g}$ (constant growth).

• Bonds: coupon $CP=Fr$; price $B_0=\dfrac{CP}{2}\dfrac{1-(1+\tfrac r2)^{-2t}}{\tfrac r2}+\dfrac{F}{(1+\tfrac r2)^{2t}}$.

Chapter 7 Mathematics of Graphs

• Graph: vertices + edges; degree = edges per vertex.

• Simple, null, directed graphs; paths vs. circuits.

• Euler’s Formula (planar connected): $f=e-v+2$.

• Euler circuit: traverses every edge once.

• Coloring: 2-colorable ⇔ no odd cycle; Four-Color Theorem for planar regions.

Chapter 8 Linear Programming (2-var)

• Formulate objective (max $P=ax+by$ / min $C=ax+by$) subject to linear constraints + $x,y\ge0$.

• Feasible region = polygon from inequalities; optimal value at a vertex (graphical method).

Chapter 9 Logic

• Proposition: declarative sentence true or false.

• Compound propositions via (\land,\lor,\lnot,\rightarrow,\leftrightarrow); truth tables determine tautology (always T), contradiction (always F), contingency (mix).

• Conditional forms: converse, inverse, contrapositive; logical equivalence $p\equiv q$ if $p\leftrightarrow q$ is tautology.

• Inference rules (e.g., Modus Ponens, Modus Tollens) validate arguments symbolically.

• Euler diagrams test categorical syllogisms (All P are Q, Some P are Q, etc.).