The Story of Numbers

The following sections are included:

• Dedication
• Preface
• About the Author
• Contents

Mathematics is a grand edifice of the human intellect constructed over a period of a few millennia, cutting across different civilisations. In natural and biological sciences, old theories make way for new ones, but in mathematics, new developments are added, sometimes connecting previous separate areas, without replacing them. Freeman Dyson in a lecture classified mathematicians into two groups, namely, birds and frogs. Birds take an overview of different areas and establish connections among them, whereas frogs continue to find new gems in one restricted area. The activities of both groups have enriched the subject. The ever increasing field of modern mathematics starts with counting. For counting we use positive integers 1, 2, 3, … , which are aptly called natural numbers. Everybody notices one Sun, two eyes, five fingers and so on in nature. Leopold Kronecker said, “God made the natural numbers, everything else is man’s handiwork” [9].

The following sections are included:

• Representation of Integers
• Curious Patterns in Numbers
  • Multiplication
  • Combination of multiplication and addition
  • Consecutive integers
  • Pascal’s triangle
• Iterations
  • Number of even, odd and total digits
  • Sum of squares of the digits
  • Sum of cubes of the digits
  • A fixed point at 1089
  • Kaprekar numbers
  • Collatz conjecture (hailstone numbers)
• Prime Numbers
  • Euclidean primes
  • Mersenne primes
  • Double Mersenne primes
  • Fermat primes
  • Pierpoint primes
  • Sophie Germain primes
  • Pillai primes
  • Ramanujan primes
  • Wilson primes
  • Twin primes
  • Carmichael numbers
  • “emirp201D;
  • Cyclic primes
  • Prime digit/composite digit primes
  • Almost-all-even-digits primes
  • Palindromic and plateau primes
  • Snowball primes
  • Russian Doll primes
  • Pandigital primes
  • Very large prime numbers with repeated pattern
  • Miscellany
• Composite Numbers
  • Highly composite numbers
  • Sierpinski’s numbers
  • Perfect and associated numbers
  • Friendly (Amicable) numbers
  • Sociable numbers
  • Untouchable numbers
  • Smith numbers
• Sequences
  • Fibonacci (Hemachandra) sequence
  • Padovan sequence
  • Perrin sequence
  • Look-and-say sequence
• Pythagorean Triples (Triplets)
• Taxicab and Similar Numbers
  • Taxicab numbers
  • Numbers refuting Euler’s conjecture
• Narcissistic and Similar Numbers
  • Narcissistic numbers
  • Factorians and factorial loops
  • Kaprekar numbers
  • SP and S + P numbers
• Some Unassuming Integers
  • Integer 4
  • Integer 7
  • Integers 9, 23, 239
  • Integers 24 and 70
  • Integer 26
  • Integer 77
• Very Large Numbers
  • Ogha, Mahaugha, googol and googolplex
  • Measurable infinity
  • Colour combinations of a Rubik cube
  • Archimedes cattle problem
  • Skewes’s number
  • Moser number and Graham number

In Figure 2.1 we have seen that integers indicated by discrete points on a line, called the number line, have gaps between them. The concept of numbers is extended to include all the points on the number line. All such numbers to be discussed in this chapter are called real numbers and the number line will be referred to as the real number line. The real numbers have two basic classifications, viz., rational and irrational. Just like reference [7] for prime numbers, a real number dictionary and online internet-based reference for special real numbers is also available.

The square of both positive and negative real numbers is positive. Thus, the square root of a negative real number cannot be a real number. On the part of mathematicians, even when they encountered such square roots of negative real numbers during their work, there was a strong reluctance to accept such quantities as numbers. They rather preferred to declare that the problems, generating such quantities, were wrong or infeasible and did not deserve any consideration. In school mathematics, we encounter such square roots of negative quantities while solving quadratic algebraic equations of the form ax² + bx + c = 0, with a, b and c real numbers satisfying the discriminant Δ = b²-4ac < 0. As both the roots of such an equation turned out to be non-real, mathematicians of an early era did not hesitate to declare that the equation itself was meaningless (not real). The trouble really started when initial progress in the closed form solution of cubic algebraic equations was achieved in early sixteenth century. In a cubic equation, at least one root must always be real. The other two may be real or not. So, even with the appearance of square root of negative real numbers, these equations containing a real root could not be declared “non-equations”. The rich history of this early confusion (and development) and final acceptance of the square root of negative real numbers as “imaginary” numbers and extending the number system to include complex numbers (a combination of real and imaginary parts) is available in the brilliant exposition of Nahin [24]. Famous contemporary mathematician Mazur [22] wrote about the role of the imagination and imagery in both poetry and mathematics to help non-mathematicians in imagining imaginary numbers.

In Chapters 2–4, we have discussed integers, real numbers, imaginary and complex numbers, all being applied to various fields of mathematics. Some special numbers have also been created by mathematicians with some special applications in mind. Such special numbers include hyperreal numbers, quaternions, octonions, dual numbers and p-adic numbers. The letter p in the p-adic numbers stands for a prime number p. Here, we include brief discussions on these special numbers [47].

The following sections are included:

• Appendix A
  • Solution of Equation (1.2)
  • Brahmagupta’s Equation and Its Solution
  • Solution to Equation (2.26)
• Appendix B
  • Sum of Integral Powers of Natural Numbers
• Appendix C
  • Origin of Curious Patterns (Section 3.7.6)

The following sections are included:

• References
• Index