What you'll learn
Trigonometry, Calculus, Statistics and More — Applied to Real Problems
Every chapter applies advanced mathematical expressions to real-world problems in science, engineering, finance and data analysis — not just abstract theory.
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Trigonometry sin, cos, tan and their inverses - angles, triangles, waves and the unit circle.
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Logarithms log(), ln(), log2() - decibels, pH, earthquake scales and exponential growth.
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Statistics Mean, variance, standard deviation and the normal distribution - from raw data to insight.
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Sequences & Series Arithmetic and geometric progressions, sigma notation and compound calculations.
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Complex Numbers Imaginary numbers, Euler's formula, polar form and applications in engineering.
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Calculus Foundations Rates of change, numerical differentiation and approximating integrals - all with expressions.
Inside the guide
20 Chapters Covering Advanced Mathematical Expressions
Each chapter builds directly on the Beginner's Guide and ends with practice exercises. Every exercise has a full worked solution.
Ch. 1 Integer Arithmetic: Modulo, GCD and Primes
Modular arithmetic, greatest common divisors, prime testing and cryptography basics.
Ch. 2 Number Bases and Conversions
Binary, octal and hexadecimal — how computers represent numbers and why it matters.
Ch. 3 Precision, Floating Point and Exact Arithmetic
The full picture of floating-point, catastrophic cancellation, and when precision matters most.
Ch. 4 Angles, Radians and the Unit Circle
The fundamental relationship between degrees, radians, and points on the unit circle.
Ch. 5 sin, cos, tan and Their Inverses
Right-triangle trigonometry, identities, and inverse functions — with worked engineering examples.
Ch. 6 The Laws of Sines and Cosines
Solving any triangle — not just right-angled ones — with practical surveying and navigation examples.
Ch. 7 Logarithms in Depth
log(), ln(), log2() — change of base, logarithmic scales and the relationship to exponents.
Ch. 8 Exponential Growth and Decay
Population models, radioactive decay, continuous compounding — and how to linearise them.
Ch. 9 Logarithmic Scales in Practice
Decibels, pH, the Richter scale, and why logarithms make large ranges manageable.
Ch. 10 Descriptive Statistics
Mean, median, mode, variance, standard deviation and range — calculated as single expressions.
Ch. 11 The Normal Distribution
Bell curves, z-scores and probability calculations without a statistics table.
Ch. 12 Combinatorics: Permutations and Combinations
Factorials, nCr, nPr and the probability calculations that underpin them.
Ch. 13 Sequences and Series
Arithmetic and geometric progressions, the sum of series, and financial applications.
Ch. 14 Complex Numbers
Imaginary numbers, the complex plane, Euler's formula e^(iπ) + 1 = 0, and polar form.
Ch. 15 Vectors and Dot Products
2D and 3D vector arithmetic, magnitude, direction and the geometric interpretation.
Ch. 16 Limits and Continuity
What limits are, how to approximate them numerically, and where they break down.
Ch. 17 Numerical Differentiation
Rates of change and the derivative concept — approximated precisely with expression calculators.
Ch. 18 Numerical Integration
Riemann sums, the trapezoid rule and Simpson's rule — all as typed expressions.
Ch. 19 Optimisation
Finding maxima and minima numerically — with applications in engineering and economics.
Ch. 20 Building Complex Expressions
Multi-step formulas, expression design principles, and advanced real-world worked examples.
Everything included
Everything Included in the Advanced Guide
Reference material, practice exercises and full worked solutions throughout — so you build real mathematical ability, not just familiarity.
✓ 20 Chapters Beginner's end to advanced techniques
✓ Worked Examples Every technique applied to real problems
✓ Practice Exercises At the end of every chapter
✓ Full Solutions Every exercise answered with explanation
✓ Function Reference Complete list of all supported functions
✓ Formula Library Key formulas from every chapter
✓ Glossary Every advanced term defined
✓ Instant PDF Download immediately, any device
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