[01] Preliminary and Fundamental Concepts in Number Theory

Karl Friedrich Gauss

he once said, “Mathematics is the queen of the sciences and arithmetic the queen of mathematics”

Number Theory

Along with geometry, it is regarded as one of the oldest branches of mathematics that developed as early as the Greek and Egyptian civilizations.

integers

The theory of numbers is concerned, at least in its elementary aspects, with the properties of the _______, particularly the positive integers 1, 2, 3, ... (also known as the natural numbers).

set of natural/counting numbers (positive integers)

State the formal name of this set: ℕ = { 1, 2, 3,... }

set of integers

State the formal name of this set: ℤ = { ..., -3, -2, -1, 0, 1, 2, 3,... }

set of rational numbers

ℚ { m/n | m, n ∈ Z, n /= 0 }

set of real numbers

ℝ

summation index

In the summation of a_i from i = 1 to n, what is the variable i called?

Well-Ordering Principle (WOP)

Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a ∈ S such that a ≤ b, for all b ∈ S

Archimedean Property

If a, b ∈ N, then there exists n ∈ N such that na ≥ b.

Pigeonhole Principle

If k + 1 or more objects are placed into k boxes, then at least one

box contains two or more of the objects.

Generalized Pigeonhole Principle

If m objects are placed into n boxes, then some box contains

at least the ceiling function of m/n objects

Joseph Louis Lagrange

The summation notation was first introduced in 1772 by the French mathematician

Dirichlet Box Principle

The Pigeonhole Principle is also known as the ___________

Gustav Peter Lejeune Dirichlet

The Pigeonhole Principle is named after the German mathematician _______ who used it extensively in his work on number theory.