Real Numbers

Instructor  Sourav Singh
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INTRODUCTION

What are real numbers?

• Real numbers are the set of numbers that include all rational and irrational numbers.
Real numbers are denoted by the symbol R.

Rational numbers are

•

Those numbers that can be written in the form  p q , where p and q are integers and q ≠ 0.

• They consist of both fractions (such as  1 2 ,  −3 4 ,  5 6 ), as well as integers (such as −3, −2, −1, 0, 1, 2, and 3), as every integer can be represented as fractions with denominator 1.

For example:
– The integer 5 can be expressed as the fraction  5 1 .
– The integer −3 can be expressed as the fraction  −3 1 .

• Rational numbers can be written in decimal forms also which can be both terminating ( 1 5  = 0.2) or non-terminating ( 1 3  = 0.3̅).

INTRODUCTION

•Irrational numbers, on the other hand, cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. They include numbers like π (pi), √2 (the square root of 2). Irrational numbers are infinite in nature and have an infinite number of non-repeating decimal places.

CHART FOR REAL NUMBERS

FUNDAMENTAL THEOREM OF ARITHMETIC

We have studied prime numbers and composite numbers. Let’s review that prime numbers are those which have exactly two factors – 1 and itself and composite numbers are those which have more than two factors. Also, we know 1 is neither prime nor composite. Any composite number can be written as the product of two numbers which can be either prime or composite. Now, if it is composite it can be split up further till a stage will come when we will get all the factors as prime numbers.  Let us use the factor tree method of prime factorization to help you understand better.

FUNDAMENTAL THEOREM OF ARITHMETIC

• So, the fundamental theory of arithmetic which is also known as the  unique factorization theorem or the unique-prime-factorization theorem , states that every composite number can be expressed uniquely as a product of prime numbers, and this factorization is unique regardless of the order in which the prime factors are listed. This means that every composite number can be written in the form:

n =  p1a1  ×  p2a2  ×  p3a3  × … ×  pkak .

where  p1,  p2,  p3, … ,  pk  are prime numbers and  a1,  a2,  a3, … ,  ak  are positive integers.

FUNDAMENTAL THEORY OF ARITHMETIC

  • THEOREM –  Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
  • The fundamental theorem of arithmetic can be proven by contradiction. Suppose that there exists a positive integer greater than 1 that cannot be expressed as a product of primes, or that it can be expressed in two different ways as a product of primes.
  • Let n be the smallest positive integer that has this property. Since n cannot be prime (since it can be expressed as a product of primes in one of the assumed ways), it must be composite. Therefore, there exist positive integers a and b, both greater than 1, such that n = ab.
  • Since a and b are both smaller than n, they can be expressed as a product of primes, by the assumption that all smaller integers can be expressed in this way. Therefore, we can write:

> Therefore, we can write:
a =  p1,  p2,  p3, … ,  pk ;  b =  q1,  q2,  q3, … ,  ql ,
where  p1,  p2,  p3, … ,  pk  and  q1,  q2,  q3, … ,  ql  are all prime numbers.

> But then we have:
n = ab = ( p1  ×  p2  × … ×  pk ) × ( q1  ×  q2  × … ×  ql ) ,
which shows that n can also be expressed as a product of primes, contradicting our assumption that it cannot.

Therefore, our initial assumption must be false, and every positive integer greater than 1 can indeed be expressed as a unique product of primes. This completes the proof of the fundamental theorem of arithmetic.

H.C.F. AND L.C.M. OF NUMBERS

Let’s see how to find H.C.F. and L.C.M. of 2 numbers with the following example:-

Find the H.C.F. and L.C.M. of 84 and 180.

NOTE (Read this for explanation)

After writing the numbers as product of prime factors, make the digits in both the numbers same. If we see in this example, 84 has 3 prime factors, 2, 3 and 7 and 180 has 2, 3 and 5 as prime factors. To make the digits same we will multiply 5 and 7 in the prime factorization of 84 and 120 respectively to make the digits same.

REVISITING IRRATIONAL NUMBERS

We know what are irrational numbers. In this section, we will prove the irrationality of  √2 ,  √3 ,  √5  and  √p  in general where  p  is a prime number.

The Fundamental Theorem of Arithmetic is one of the theorems that we make use of in our proof.  The following theorem, whose proof is based on the Fundamental Theorem of Arithmetic, is necessary before we can demonstrate that  √2  is irrational.

•  THEOREM 2 – Let  p  be a prime number and  a  be a positive integer. If  p  divides  a2 , then  p  divides  a .

> Let the prime factorisation of  a  be as follows:  a =  p1 ,  p2 ,  … ,  pn , where  p1 ,  p2 ,  … ,  pn  are primes, not necessarily distinct.
Therefore,  a2  = ( p1 ,  p2 ,  … ,  pn ) × ( p1 ,  p2 ,  … ,  pn ) = ( p12 ), ( p22 ),  … ,  ( pn2 ) .

Now, we are given that  p  divides  a2 . Therefore, from the Fundamental Theorem of Arithmetic, it follows that  p  is one of the prime factors of  a2 .
However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of  a2  are  p1 ,  p2 ,  … ,  pn .
So  p  is one of  p1 ,  p2 ,  … ,  pn .  Now, since  a =  p1 ,  p2 ,  … ,  pn ,  p  divides  a .

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