Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics

Front Matter....Pages i-ix
Alice in Logiland — in which we meet Alice, Tweedledee and Tweedledum, and Logic …....Pages 1-13
Unique Factorisation — in which trivial arithmetic reveals a glimpse of hidden depths....Pages 14-20
Numbers—in which we abandon logic to achieve understanding, then use logic to deepen understanding....Pages 21-35
The Real Numbers—in which we find holes in the number line and pay the price for repairs....Pages 36-51
A Variety of Versions and Uses of Induction—in which another triviality plays the lead....Pages 52-71
Permutations—in which ALICE is transformed....Pages 72-88
Nests—in which the rationals give birth to the reals and the scene is set for arithmetic in ℝ....Pages 89-97
Axioms for ℝ—in which we invent Arithmetic, Order our numbers and Complete our description of the reals....Pages 98-110
Some Infinite Surprises—in which some wild sets are tamed, and some nearly escape....Pages 111-140
Sequences and Series— in which we discover very odd behaviour in even the smallest infinite set....Pages 141-160
Graphs and Continuity—in which we arrange a marriage between Intuition and Rigour....Pages 161-176
Back Matter....Pages 177-205