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maths:number_theory

a brief summary of number theory

Introduction

  • in simple terms, numbers can be:
    • either algebraic or transcedental
    • rational, irrational or complex, although all numbers are part of the set of complex numbers, and real numbers are a subset of complex numbers where the imaginary scalar is zero

Algebraic numbers

  • any complex number (including real numbers) that is a root of a non-zero polynomial
  • all are countable numbers
  • all are computable and therefore definable and arithmetical
  • includes:
    • all rational numbers
    • some irrational numbers
      • numbers sqrt(2) and (cubed root(3))/2 are algebraic since they are roots of polynomials x2 − 2 and 8x3 − 3, respectively.
      • golden ratio φ is algebraic since it is a root of the polynomial x2 − x − 1.
    • some complex numbers
      • for real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic
      • Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers.
  • excludes:
    • transcedental numbers such as π and e
  • there is a unique monic polynomial (with rational coefficients) of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.

Transcedental numbers

  • numbers that are not solutions of any polynomial equation with rational coefficients
  • such as:
    • π and e
    • Liouville numbers
    • the number logab for rational numbers a and b provided b is not of the form b = ac for some rational c
    • almost all complex numbers
  • this concept arose when Euler asserted the above log statement in 1748 although the term transcedental arose in the 17th C when Gottfried Leibniz proved that the sine function was not an algebraic function.

Real numbers R

Rational numbers Q

  • a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q
  • every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers and b > 0.
  • the decimal expansion of a rational number either terminates or eventually begins to repeat the same finite sequence of digits over and over

Integers

  • any fraction where numerator is a rational number and the denominator = 1
  • special integers:
    • Prime numbers
      • an integer is a prime number if the only positive integer that is a divisor is itself or 1
        • two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1
    • Fibonacci numbers
      • a sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1.

Fractions

Irrational numbers

  • the decimal expansion of an irrational number continues without repeating
  • include:
    • surds such as √2
    • certain constants such as π, e, and φ

Complex (imaginary) numbers

  • these were invented in the 16th century to help solve cubic equations
  • z = x + yi where z = complex, x and y are real and i = sqrt(-1)
maths/number_theory.txt · Last modified: 2021/07/24 13:01 by gary1