Properties & Relations of Operations

< 20.7 Classification of Numbers 2 | Topic Index | 21.2 Fundamental Laws of Arithmetic >

Basic arithmetic operations also have properties and are also related to one another. Some of these are &ldquo;natural&rdquo; in the sense that they arise from the very nature of these operations or the way they have been defined. Some others have been &ldquo;conventions&rdquo; imposed by &ldquo;common understanding&rdquo; to ensure that they yield consistent results. We will explore these in this and the next two chapters.

This chapter will explore the various ways in which the operators are related to one another. The next chapter would be on what are called &ldquo;fundamental laws of arithmetic&rdquo; will explore some properties of operations. The last one would be on certain conventions governing the order in which some of these operators need to be performed.

Relations between Operations

The relations between operations are summarised in the table given below.

These relations can be modelled easily with objects &amp; daily transactions. An understanding of these relations would help students to break them up and recombine them in different ways. All these would lead to flexibility in manipulating numbers &amp; operations in performing a computation and achieve a sophisticated level of Number Sense.

The Indian Mathematician, Srinivasa Ramanujan had achieved a very high level of number sense. When Prof Hardy mentioned that 1729 was a boring number, Ramanujan replied immediately that Hardy was mistaken. He told him that it was an interesting number – the smallest number which could be written as the sum of 2 cubes in 2 different ways!

In Computers Everything is Addition

In computer software, using the relations between different operations, all operations (subtraction, multiplication, Division &amp; Exponentiation) are all reduced to additions performed with the binary number system.

Addition &amp; Subtraction

It is very obvious that &ldquo;put together&rdquo; and &ldquo;take away&rdquo; operations are just &ldquo;mirror&rdquo; perspectives of a transaction. Imagine that Ram and Lakshman have 3 and 4 pencils respectively and that Lakshman gives 2 pencils to Ram. From Ram&rsquo;s point of view, it is an addition situation; he had 3 pencils and with the 2 he received from Lakshman he now has 5 pencils. This can be written as 3 + 2 = 5.

From Lakshman&rsquo;s point of view it is a subtraction situation; he had 4 pencils and having given away 2 pencils he now has 2 pencil. This can be written as 4 – 2 =2.

Multiplication &amp; Division

Similarly, multiplication and division can be seen as &ldquo;mirror&rdquo; perspectives of a transaction. If a person enters a room and sees 4 cups with 3 tokens in each, then he can think of them in 2 different ways.

In one he can think of the total number of tokens as 4 X 3, which is a multiplicative situation. This can be written as 4 X 3 = 12.

In another he can think that 12 tokens have been equally shared among 4 cups which is a divisive situation. This can be written as 12&divide;4=3.

< 20.7 Classification of Numbers 2 | Topic Index | 21.2 Fundamental Laws of Arithmetic >