Chapter 4 section 1-2 cont’d
Axioms
Our book does not mention axioms but it should. Axioms are statements that are asserted to be true for purposes of constructing a theory. For example:
Axiom: Given a line \(L\), and a point \(P\) not on \(L\), there is exactly one line through \(P\) parallel to \(L\).
Axiom: An empty set exists.
Axioms in this course
- Existence of integers, natural numbers, rational numbers, and real numbers.
- Properties of addition, multiplication such as commutative and associative laws, including closure.
- Properties of \(>\) and \(<\)
The Division Algorithm
The Division Algorithm: Given \(a,b\in\mathbb{Z}\) with \(b>0\), there are unique integers \(q\) and \(r\) with \(0\le r<b\) so that \(a=bq+r\).
The Fundamental Theorem of Arithmetic
Theorem: Every natural number greater than one is a product of prime numbers, and this factorization into primes is unique up to rearranging the terms.
Some fundamental definitions: divisibility
Definition: Suppose \(a\) and \(b\) are integers. We say that \(a\) divides \(b\), written \(a|b\), if \(b=ac\) for some \(c\in\mathbb{Z}\). In this case we also say that \(a\) is a divisor of \(b\) and that \(b\) is a multiple of \(a\).
GCD and LCM
Definition: The greatest common divisor of integers \(a\) and \(b\), written \(\mathrm{gcd}(a,b)\), is the largest integer that divides both \(a\) and \(b\).
Definition: The least common multiple of integers \(a\) and \(b\), written \(\mathrm{lcm}(a,b)\), is the smallest integer that is a multiple of both \(a\) and \(b\).