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# 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 $ac$

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 $a$

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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Edzai Conilias Zvobwo is passionate about empowering Africans through mathematics, problem-solving techniques and media. As such, he founded MathsGee. Through this organisation, he has helped create an ecosystem for disseminating information, training, and supporting STEM education to all African people. A maths evangelist who teaches mathematical thinking as a life skill, Edzai’s quest has seen him being named the SABC Ambassador for STEM; he has been invited to address Fortune 500 C-suite executives at the Mobile 360 North America; was nominated to represent Southern Africa at the inaugural United Nations Youth Skills Day in New York; was invited to be a contributor to the World Bank Group Youth Summit in 2016; has won the 2014 SADC Protocol on Gender and Development award for his contribution to women’s empowerment in education; and has partnered with local and global firms in STEM interventions.