Basic Properties of Integers

2024-05-2809:12 Status:Complete

Laws of Addition and Multiplication

• Closure: $a+b$ and $a\cdot b$ are integers, whenever $a$ and $b$ are integers.
• Commutative laws: $a+b=b+a$ and $a\cdot b=b\cdot a$ for all integers $a$ and $b$.
• Associative laws: $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot =a\cdot (b\cdot c)$ for all integers $a$ and $b$.
• Distributive law: $(a+b)\cdot c=ac+bc$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for all integers $a,b$ and $c$.
• Identity elements: $a+0=a$ and $a\cdot 1=a$ for all integers of $a$.
• Additive inverse: For every integer $a$ there is an integer solution $x$ to the equation $a+x=0$; this integer $x$ is called the additive inverse of $a$ and is denoted by $-a$. By $b-a$ we mean $b-(-a).$
• Cancellation law: If $a,b$ and $c$ are integers with $a\cdot c=b\cdot c$ with $c\ne 0$, then $a=b$.

Further Laws and Properties

By ordering the positive integers’ in the order: $1,2,\mathrm{3...}$ the following definition stands: If $a$ and $b$ are integers, then $a<b$ if $b-a$ is a positive integer. ($a<b=b>a$).

From this, the following laws can be found:

• Closure for the Positive Integers: $a+b$ and $a\cdot b$ are positive integers whenever $a$ and $b$ are positive integers.
• Trichotomy law: For every integer $a$, either $a>0$, $a=0$, or $a<0$.

Basic properties of ordering numbers can be proved using these axioms.

• The Well-Ordering Property: Every nonempty set of positive integers least. (The set of positive integers is Well-Ordered because there is the lowest element, while the entire integer doesn’t as there is no lower bound).

With the following definition: $[x]$ = largest integer less than or equal to $x$ where $x\in \mathbb{R}$, then, $[x]\le x<[x]+1$. This is also known as the Floor and ceiling functions.

An example of proofs that use these principles is the Proof that √2 is irrational.