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Real Numbers

    Introduction

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10th class > Real Numbers > Introduction

Introduction

In the previous grades, we have encountered and irrational numbers. Here, we continue our discussion on real numbers. We begin with two very important properties of positive integers: namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

Euclid’s division algorithm, deals with divisibility of integers. Stated simply, it says:

Any positive integer 'a' can be divided by another positive integer 'b' in such a way that it leaves a remainder 'r' that is smaller than b.

Many of us recognise this as the usual long division process. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them and use it mainly to compute the of two positive integers.

The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. We already know that:

Every number can be expressed as a product of primes in a unique way.

This important fact is the Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics.

We use the Fundamental Theorem of Arithmetic for two main applications:

  • First, we use it to prove the irrationality of many of the numbers we have studied, such as: 2, 3 and 5.

  • Second, we apply this theorem to explore when exactly the decimal expansion of a rational number, say pq (where q ≠ 0) is terminating and when it is nonterminating- it is repeating.

We do so by looking at the prime factorisation of the denominator q of pq. We will see that the of q will completely reveal the nature of the decimal expansion of pq.

So let's get started.