Fundamental Laws of Arithmetic

< 21.1 Properties & Relations of Operations | Topic Index | 21.3 Order of Operations >

We now explore properties of some of the operations and the rules to be followed when more than one operation occurs in the same expression.

In the 19th century, ideas in math developed at a rapid rate using the internal logic of the discipline. Mathematicians & philosophers of math also started coming across “paradoxes” which they found difficult to explain with the current understanding that they had.

Mathematicians felt the need for organising the entire discipline of math in a logical structure. They had an example before them of Euclid’s development of the discipline of plane geometry, starting with some simple axioms & postulates.

Hence the formulated rules of operation when multiple numbers were operated upon by multiple operators.

These rules are considered very fundamental to arithmetic operations and are called “Fundamental Laws of Arithmetic”.

They are listed in the table below.

These laws are intuitive and can be verified visually or with objects. These laws are applicable for numbers of any type – rational, irrational, imaginary etc.

Axioms of Arithmetic

These can even be called the “axioms” of arithmetic in the sense they cannot be proved but have to be accepted. They are similar to the axioms with which the theorems of plane geometry start.

There is another way of looking at these laws. Any entity which obeys these laws can be considered as a number. In fact, this could be taken as a general definition of a number. It also ensures that the rules of operations of different kinds of numbers yield consistent results. You will recall that we derived integer operation rules both by extending patterns and by using distributive law.

Fundamental Laws of Algebra

Even in school level algebra we operate only on numbers, except that the numbers are sometimes represented with letters of the alphabet. Hence the same laws apply in algebra also. In fact in the above table, we have represented the laws in algebraic notation.

This is different from the Fundamental Theorem of Algebra which is beyond the K-8 curriculum.

< 21.1 Properties & Relations of Operations | Topic Index | 21.3 Order of Operations >