Maths WorkshopMaths problems with full solutions

Irrationality

43 problems · Proofs of irrationality for roots, logarithms and fundamental constants.

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Ex.11Product of irrational and nonzero rational

#irrationality #product #rational

Ex.12Sum of irrational and rational

#irrationality #sum #rational

Ex.23Irrationality of √2 + √3

#irrational #proof by contradiction

Ex.36Irrationality of ∛2

#cube root #irrationality #unique factorization

Ex.37Between any two rationals lies an irrational

#density #irrationals #betweenness

Ex.38Irrationality of √p for prime p

#sqrt p #prime #irrationality

Ex.39Irrationality of √2

#sqrt2 #irrationality #proof by contradiction

Ex.40Irrationality of √3

#sqrt3 #irrationality #mod 3

Ex.41Linear independence of logs

#logarithm #independence #irrationality

Ex.57Irrationality of ⁿ√2

#nth root #irrationality #prime factorization

Ex.58Irrationality of log₁₀ 2

#logarithm #irrationality #base 10

Ex.59Irrationality of log₂ 3

#logarithm #irrationality #base change

Ex.60A square root in disguise

#irrationality #quadratic #conjugates

Ex.61Ratios involving irrationals

#ratio #irrationality #algebraic

Ex.71Decimal expansions and rationality

#decimal expansion #rational #periodic

Ex.72Irrationality of a cubic root expression

#cubic root #irrationality #algebraic

Ex.73Irrationality of √2 + √3

#algebraic #sum of roots #irrationality

Ex.74A sum of radicals

#radicals #irrationality #algebraicity

Ex.84Rational root theorem and irrationality

#rational root theorem #irrationality #cubic

Ex.89Sum of irrationals in disguise

#irrationality #sum #structure

Ex.90Irrationality of π

#pi #irrationality #Lambert

Ex.91Continued fractions and irrationality

#continued fractions #irrationality #sqrt2

Ex.92Irrationality of √n for non-square n

#sqrt n #non-square #irrationality

Ex.95x³ + 2y³ + 4z³ = 0

#infinite descent #parity #irrationality

Ex.98Liouville numbers

#Liouville #transcendental #irrationality

Ex.99A general irrationality criterion

#criterion #irrationality #polynomial

Ex.115Gelfond–Schneider theorem (statement)

#Gelfond-Schneider #transcendence #Hilbert 7th

Ex.122Fractional parts of multiples of √2

#fractional parts #density #irrational

Ex.247Sum of irrationals can be rational

#irrationality #sum #counterexample

Ex.493Between any two reals lies a rational

#density #rationals #betweenness

Ex.494√2 is algebraic

#algebraic #sqrt2 #minimal polynomial

Ex.1026Rational root theorem and irrationality

#rational root theorem #algebraic #irrationality

Ex.1027Uncountability of irrationals

#uncountability #Cantor #irrationals

Ex.1028Rational coefficients and irrational roots

#rational coefficients #irrational roots #conjugates

Ex.1029Degree of algebraic numbers

#algebraic degree #minimal polynomial #sqrt2

Ex.1030Rational, algebraic, transcendental

#hierarchy #algebraic #transcendental

Ex.1752$e$ is not a quadratic irrational

#e #quadratic irrational #contradiction

Ex.1753Liouville's theorem and examples

#Liouville #approximation #transcendental

Ex.1754Independence of square roots

#independence #square roots #linear independence

Ex.2173Algebraic independence of $e$ and $\pi$

#algebraic independence #e #pi

Ex.2174Irrationality of e+ π and πe

#irrationality #e #pi #open problem

Ex.2175Euler's number and transcendence

#e #transcendence #Euler

Ex.2176Schanuel's conjecture (statement)

#Schanuel #conjecture #transcendence