Fundamental Laws of Arithmetic

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.

As long as arithmetic operations were performed with just 2 numbers, the result was easy to obtain. But situations arose when multiple numbers & multiple operations were involved. Hence rules were formulated to make the operations easy to carry out and yield consistent results.

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.