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E-grāmata: Story Of Numbers, The

(Indian Inst Of Engineering Science & Technology, Shibpur, India)
  • Formāts: 196 pages
  • Sērija : Iiscpress-wspc Publication 3
  • Izdošanas datums: 27-Jul-2017
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813222946
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Story Of Numbers, ThePalielināt
  • Formāts: 196 pages
  • Sērija : Iiscpress-wspc Publication 3
  • Izdošanas datums: 27-Jul-2017
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813222946
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' this could make an ideal end-of-year prize for a high-school student who is fascinated by all aspects of number. The subsections provide ideas and opportunities for mathematical exploration. This book might also be deemed a suitable resource for first-year undergraduates in that, via independent study, it would allow such students to broaden their knowledge of various number-theoretic ideas. I would recommend it for the purposes given above.'The Mathematical GazetteThis book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvelous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p-adic numbers.Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behavior, which has given rise to the new science of 'Chaos'. Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.The origin of imaginary numbers and the use of complex numbers constitute the next topic. It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics.
Preface vii
About the Author ix
1 Introduction
1(10)
2 Integers
11(58)
2.1 Representation of Integers
11(1)
2.2 Curious Patterns in Numbers
12(6)
2.2.1 Multiplication
13(1)
2.2.2 Combination of multiplication and addition
13(1)
2.2.3 Consecutive integers
14(2)
2.2.4 Pascal's triangle
16(2)
2.3 Iterations
18(4)
2.3.1 Number of even, odd and total digits
18(1)
2.3.2 Sum of squares of the digits
18(1)
2.3.3 Sum of cubes of the digits
19(1)
2.3.4 A fixed point at 1089
20(1)
2.3.5 Kaprekar numbers
20(1)
2.3.6 Collatz conjecture (hailstone numbers)
21(1)
2.4 Prime Numbers
22(16)
2.4.1 Euclidean primes
24(1)
2.4.2 Mersenne primes
24(2)
2.4.3 Double Mersenne primes
26(1)
2.4.4 Fermat primes
26(2)
2.4.5 Pierpoint primes
28(1)
2.4.6 Sophie Germain primes
28(1)
2.4.7 Pillai primes
29(1)
2.4.8 Ramanujan primes
30(1)
2.4.9 Wilson primes
30(1)
2.4.10 Twin primes
30(2)
2.4.11 Carmichael numbers
32(1)
2.4.12 "Emirp"
33(1)
2.4.13 Cyclic primes
33(1)
2.4.14 Prime digit/composite digit primes
34(1)
2.4.15 Almost-all-even-digits primes
34(1)
2.4.16 Palindromic and plateau primes
34(1)
2.4.17 Snowball primes
35(1)
2.4.18 Russian Doll primes
35(1)
2.4.19 Pandigital primes
35(1)
2.4.20 Very large prime numbers with repeated pattern
35(1)
2.4.21 Miscellany
36(2)
2.5 Composite Numbers
38(6)
2.5.1 Highly composite numbers
38(1)
2.5.2 Sierpinski's numbers
39(1)
2.5.3 Perfect and associated numbers
40(2)
2.5.4 Friendly (Amicable) numbers
42(1)
2.5.5 Sociable numbers
43(1)
2.5.6 Untouchable numbers
43(1)
2.5.7 Smith numbers
43(1)
2.6 Sequences
44(6)
2.6.1 Fibonacci (Hemachandra) sequence
44(3)
2.6.2 Padovan sequence
47(1)
2.6.3 Perrin sequence
48(1)
2.6.4 Look-and-say sequence
49(1)
2.7 Pythagorean Triples (Triplets)
50(2)
2.8 Taxicab and Similar Numbers
52(4)
2.8.1 Taxicab numbers
52(3)
2.8.2 Numbers refuting Euler's conjecture
55(1)
2.9 Narcissistic and Similar Numbers
56(3)
2.9.1 Narcissistic numbers
56(1)
2.9.2 Factorians and factorial loops
57(1)
2.9.3 Kaprekar numbers
58(1)
2.9.4 SP and S + P numbers
59(1)
2.10 Some Unassuming Integers
59(3)
2.10.1 Integer 4
59(1)
2.10.2 Integer 7
60(1)
2.10.3 Integers 9, 23, 239
60(1)
2.10.4 Integers 24 and 70
60(1)
2.10.5 Integer 26
61(1)
2.10.6 Integer 77
61(1)
2.11 Very Large Numbers
62(7)
2.11.1 Ogha, Mahaugha, googol and googolplex
63(1)
2.11.2 Measurable infinity
64(1)
2.11.3 Colour combinations of a Rubik cube
64(1)
2.11.4 Archimedes cattle problem
65(1)
2.11.5 Skewes's number
65(1)
2.11.6 Moser number and Graham number
66(3)
3 Real Numbers
69(46)
3.1 Rational Numbers
69(1)
3.2 Irrational Numbers
70(1)
3.3 Transcendental Numbers
71(2)
3.4 Decimal Representation
73(2)
3.5 Continued Fraction Representation
75(3)
3.6 Iterations
78(9)
3.6.1 Square root --- Babylonian
81(1)
3.6.2 Square root --- Indian
82(1)
3.6.3 Gauss's constant
83(1)
3.6.4 Logistic map and Feigenbaum numbers
84(3)
3.7 Special Rational Numbers
87(6)
3.7.1 Unique Egyptian fractions with sum unity
87(1)
3.7.2 A Steinhaus problem
87(1)
3.7.3 Parasite numbers
88(1)
3.7.4 Congruent numbers
89(1)
3.7.5 Bernoulli numbers
90(2)
3.7.6 Curious periodic patterns
92(1)
3.8 Special Irrational and Transcendental Numbers
93(22)
3.8.1 Pythagoras's number: √2
94(1)
3.8.2 Golden sections
94(3)
3.8.3 Vishwanath number
97(1)
3.8.4 Schizophrenic numbers
98(1)
3.8.5 Oldest universal mathematical constant π
99(4)
3.8.6 Base of natural logarithm e
103(5)
3.8.7 Famous formulas having both π and e
108(3)
3.8.8 Apery's constant
111(1)
3.8.9 Euler's constant γ
112(2)
3.8.10 Liouville's number
114(1)
3.8.11 Champernowne's number
114(1)
3.8.12 Hilbert's number
114(1)
4 Imaginary and Complex Numbers
115(28)
4.1 A Brief Early History of √--1
117(2)
4.2 Geometric Representation of Complex Numbers
119(2)
4.3 Euler's Fabulous Formula
121(2)
4.4 Complex Exponentiation and Special Numbers
123(2)
4.5 Fundamental Theorem of Algebra
125(1)
4.6 Gaussian Integers and Gaussian Primes
126(3)
4.7 Riemann Hypothesis
129(5)
4.8 Iterations
134(9)
5 Special Numbers
143(18)
5.1 Hyperreal Numbers
143(3)
5.2 Quaternions
146(3)
5.3 Dual Numbers
149(3)
5.3.1 Arithmetic
149(1)
5.3.2 Function
150(1)
5.3.3 Dual angle and trigonometry
150(1)
5.3.4 Trigonometry
151(1)
5.4 p-Adic Numbers
152(9)
5.4.1 Decimal representation of real numbers and 10-adic numbers
153(1)
5.4.2 p-Adic integers
154(7)
Appendix A
161(4)
A.1 Solution of Equation (1.2)
161(1)
A.2 Brahmagupta's Equation and Its Solution
161(3)
A.3 Solution to Equation (2.26)
164(1)
Appendix B
165(4)
B.1 Sum of Integral Powers of Natural Numbers
165(4)
Appendix C
169(2)
C.1 Origin of Curious Patterns (Section 3.7.6)
169(2)
References 171(4)
Index 175
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