WRITE 5 irrational number BETWEEN 5 and 6
Answers
Answer: Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers.
Statement: The sum of two irrational numbers is sometimes rational or irrational.
Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.
For example, if we add two irrational numbers, say 3√2+ 4√3, a sum is an irrational number.
But, let us consider another example, (3+4√2) + (-4√2 ), the sum is 3, which is a rational number.
So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational number.
Irrational Number Proof
The following theorem is used to prove the above statement
Theorem: Given p is a prime number and a2 is divisible by p, (where a is any positive integer), then it can be concluded that p also divides a.
Proof: Using the Fundamental Theorem of Arithmetic, the positive integer can be expressed in the form of the product of its primes as:
a = p1 × p2 × p3……….. × pn …..(1)
Where, p1, p2, p3, ……, pn represent all the prime factors of a.
Squaring both the sides of equation (1),
a2 = ( p1 × p2 × p3……….. × pn) ( p1 × p2 × p3……….. × pn)
⇒a2 = (p1)2 × (p2)2 × (p3 )2………..× (pn)2
According to the Fundamental Theorem of Arithmetic, the prime factorization of a natural number is unique, except for the order of its factors.
The only prime factors of a2 are p1, p2, p3……….., pn. If p is a prime number and a factor of a2, then p is one of p1, p2 , p3……….., pn. So, p will also be a factor of a.
Hence, if a2 is divisible by p, then p also divides a.
Now, using this theorem, we can prove that √ 2 is irrational.