Equating rational and irrational parts: proof and worked problems
Learn when coefficients of an irrational number can be equated, prove that cube root of two cannot satisfy a rational quadratic, and avoid division by a possibly zero coefficient.
Mathematics beyond memorizing formulas
Explore algebra, number theory, probability, geometry, and calculus through worked examples. Learn which conditions suggest a method, why that method works, and how to turn your reasoning into a written proof.
Written by Koki Saito, a physician and graduate of the University of Tokyo Faculty of Medicine, drawing on his own entrance-exam experience.
About the author →Start with a worked guide
How to simplify nested square roots, with three worked examples
Recognize a sum and product inside a nested radical, keep the correct sign, and understand when rational radicands cannot remove the nesting.
Conditional probability: the formula, information protocols, and worked examples
Identify the conditioning event, distinguish independence from chronology, and solve family, sampling, noisy-report, bag, and tournament problems.
Integration by parts: formulas, proof, and four useful patterns
Derive integration by parts from the product rule and practice logarithmic, polynomial-trigonometric, polynomial-exponential, and cyclic integrals.
Triple-angle identities: remember, derive, and apply them
Check the signs and coefficients of sine and cosine triple angles, derive the tangent formula, and solve polynomial, complex-number, and integration examples.
University of Tokyo science mathematics 2020: problems and solutions
Complete solutions for cyclic inequalities, triangle-area regions, a rotating parametric region, subset-product sums, cone sweeps, and ellipse normals.
University of Tokyo science mathematics 2015: problems and solutions
Complete English explanations of six problems on swept regions, random strings, tangent curves, Fibonacci recurrences, binomial parity, and concentrating integrals.
The fundamental theorem of calculus in integral equations
Differentiate integrals with variable limits, retain initial conditions, and solve three examples including an integral recurrence.