Mathematics for GIS Professionals

Axioms of Mathematics

Axioms of arithmetic

An axiom is a statement that is self-evidently true and in mathematics it is a statement or proposition on which an abstractly defined structure is based. All the rules of arithmetic are based on seven mathematical axioms.

Arithmetic is a subset of mathematics and not all mathematical relationship adhere to the arithmetic axioms.  Kurt Gödel an Austrian, and later American, logician, mathematician, and philosopher proved this through his incompleteness theorems of mathematical logic that established inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.

It is important to take note that all the mathematical rules applied in geographical analysis are based on the axioms to be discussed in this chapter. Mathematics is an abstract language that uses numbers and symbols to communicate relationships and patterns. The four basic arithmetic operations are addition, subtraction, multiplication and division.

Below is a list of axioms of arithmetic:

Commutative Law

 
Axiom Explanation
 $a + b = b + a$  For any numbers $a$ and $b$ it is known that $a$ plus $b$ is equal to $b$ plus $a$
$a * b = b * a$  For any numbers $a$ and $b$ it is known that $a$ times $b$ is equal to $b$ times $a$

 

Associative Law

 
Axiom Explanation
$(a + b) + c = a + (b + c) $ For any numbers  $a$, $b$ and $c$ it is known that the sum of $a$ and $b$ plus $c$  is equal to the sum of $b$ and $c$ plus $a$
$(ab)c = a(bc)$ For any numbers  $a$, $b$ and $c$ it is known that the product of $a$ and $b$ times $c$  is equal to the product of $b$ and $c$ times $a$

Distributive Law

 
Axiom Explanations
$a * (b + c) = a * b + a * c $ For any numbers  $a$, $b$ and $c$ it is known that the product  of $a$ and $b + c$ is equal to the sum of $ab$ and $ac$

Additive Identity

 
Axiom Explanation
$a + 0 = a $ There exists a number $0$ called the additive identity such that for any number  $a$, the sum of $a$ and $0$ is equal to $a$

 

Multiplicative Identity

 
Axiom Explanation
$a * 1 = a $ There exists a number $1$ called the multiplicative identity such that for any number  $a$, the product of $a$ and $1$ is equal to $a$

Additive Inverse

 
Axiom Explanation
$a + c = 0 $ For any number  $a$ there exists a number $c$ called the additive inverse such that, the sum of $a$ and $c$ is equal to $0$

 

Transitive law of multiplication

 
Axiom Explanations
$a * b = a * c $ If then $b=c$ If the product of $a$ and $b$ is equal to the product of $a$ and $c$ for any numbers  $a$ , $b$ and $c$ then $b$ is equal to $c$

 

 

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