Mathematics for the Liberal Arts Student Book

1	Counting	1
1.1	One, two, three	1
1.2	Some counting problems and estimates	5
1.3	A fundamental counting principle	8
1.4	Permutations	13
1.5	Two complications	19
1.6	Combinations	25
1.7	Notes	31
2	Probability	33
2.1	What are the odds?	33
2.2	Measuring likelihood	37
2.3	Independent trials	44
2.4	Expectation	48
2.5	Conditional probability	54
2.6	Notes	59
3	Statistics	61
3.1	Analysis of data	61
3.2	Population and sample	67
3.3	What if it were?	70
3.4	Liars	75
3.5	Notes	77
4	Geometry	79
4.1	Area	79
4.2	The Pythagorean theorem	84
4.3	Squaring the circle	90
4.4	Numbers and points	93
4.5	Plotting more points	96
4.6	Plotting still more points	100
4.7	Geometric sensitivity	105
4.8	Paths	113
4.9	Geometric means	121
4.10	Counting again	123
4.11	Notes	130
5	Logic	133
5.1	Think of the possibilities	133
5.2	What's my number?	134
5.3	The liar paradox	137
5.4	Subject and predicate	139
5.5	Syllogisms	142
5.6	Notes	146
6	Exponential growth	149
6.1	The power of powers	149
6.2	Doubling time	152
6.3	Half-life	154
6.4	Explosions	156
6.5	Rates of interest	161
6.6	Logarithms	172
6.7	Notes	173
7	An average chapter	175
7.1	The arithmetic mean	175
7.2	Weighted arithmetic means	179
7.3	The geometric mean	184
7.4	The harmonic mean	189
7.5	Comparing the means	195
7.6	The Farey mean	198
7.7	Notes	204
8	What are natural numbers made of?	205
8.1	The building block of addition	205
8.2	How can I build thee? Let me count the ways	207
8.3	Building blocks for subtraction	211
8.4	The Euclidean algorithm	214
8.5	The building blocks of multiplication	217
8.6	Notes	223
9	Changing bases	225
9.1	Earlier number systems	225
9.2	Base five	230
9.3	Base twelve	236
9.4	Base two	239
9.5	Notes	244
10	Clock arithmetic	247
10.1	The twelve-hour clock	247
10.2	Arithmetic of even and odd; casting out nines	253
10.3	Zero divisors	256
10.4	Pigeonholes and inverses	258
10.5	The perfect shuffle	261
10.6	Fermat's little theorem	265
10.7	Notes	268
11	Secret writing	269
11.1	Simple substitution	269
11.2	The Gold-Bug	272
11.3	Letters are numbers	275
11.4	Block encoding	278
11.5	Trap-door functions	282
11.6	Notes	284
12	Infinite sets	287
12.1	Finite and infinite	287
12.2	Decimal representations of real numbers	289
12.3	Comparing sizes of sets	291
12.4	More comparisons	295
12.5	More infinities	299
12.6	Notes	302
13	Number theory selections	305
13.1	Primes and divisibility	305
13.2	Some rules for divisibility	308
13.3	A general divisibility rule	312
13.4	Sums of divisors	315
13.5	Deficiency and abundancy	318
13.6	Perfection	321
13.7	Amicability	324
13.8	How are primes distributed?	326
13.9	Sums of squares	333
13.10	Pythagorean triples	336
13.11	Notes	339
A	Mathematics encounters	341
A.1	Temperature conversion	342
A.2	The Greek ladder method	343
A.3	The collapsing compass	344
A.4	Catalan's conjecture	345
A.5	Egyptian fractions	346
A.6	The Pythagoreans	347
A.7	Triangle numbers	348
A.8	The four-color problem	349
A.9	Morley's Theorem	350
B	Notation and arithmetic	351
B.1	The alphabet and punctuation of mathematics	351
B.2	Arithmetic of the integers	358
B.3	Arithmetic of fractions	363
B.4	Arithmetic of exponents	370
B.5	Arithmetic of equations	375
B.6	A standard deck of cards	379
	Index	381