Numbers: A Very Short Introduction

Peter M. Higgins, Oxford University Press 2011
  1. How not to think about numbers
  2. The unending sequence of primes
  3. Perfect and not so perfect numbers
  4. Cryptography: the secret life of primes
  5. Numbers that count
  6. Below the waterline of the number iceberg
  7. To infinity and beyond!
  8. Numbers but not as we know them
Notable People:
Natural numbers - counting numbers of 1,2,3...
Binary system - using two numerals of 0 and 1, as the basic symbols for counting.
Prime number - has exactly two (divisors)factors, 1 and the number itself.
Composite number - is composed of smaller numbers.
Squared - a number raised to the power 2, for two dimensions thus creating a square.
Cubed - a number raised to the power 3, for three dimensions thus creating a cube.
Triangular number - 10 which is comprised of 1+2+3+4.
Twin Primes - a pair of twin of 11 and 13, two consecutive prime numbers.
Proper factors - of 15 would be the factors not including itself, 1,2,3,5.
Abundant Number - 12 which is less than the sum of its proper factors, 1,2,3,4,6.
Sieve of Eratosthenes - method to find all primes up to the number N by sequentially crossing out multiples of the prime number you are on, which are themselves composite numbers, until you get to a number whose square root is larger than N(because the square roots result indicates that the number is not prime and all numbers smaller than the result have been crossed out due to the preceding primes being multiplied to find their composites. The remaining prime numbers will be odd(except for 2), and have yet to be crossed out) Once you have found a prime, you may optimize by going to the prime^2 and crossing out multiples of the prime, this works because any number below prime^2 has already been crossed out via the other primes.

Our modern number system allows us to express numbers in an efficient and uniform manner, which is great for comparisons and arithmetical operations; ultimately for counting due to positional value in numeric representation.
The ancients thought about number patterns in terms of special geometric properties of the number.
Factorization reveal something of the number's inner structure.
Primes are the building blocks of multiplication.
The distribution of primes is non erratic.
Searching for prime divisibility can employ novel methods of testing divisibility for smaller primes, ex. numbers divisible by 3 if and only if the sum of the digits are divisible by three. 87 is 8+7=15 which is 1+5=6 which 6=2*3 proving divisibility. These methods are further optimizations, which can be used in parallel to quickly find the result of the question.

Chapter 2 the unending sequence of primes