Question

proove root 3 is irrational​

Answer 1

Answer:

Root 3 is irrational is proved by the method of contradiction. If root 3 is a rational number, then it should be represented as a ratio of two integers. We can prove that we cannot represent root is as p/q and therefore it is an irrational number.

Step-by-step explanation:

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Answer 2

Step-by-step explanation:





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Prove that Root 3 is Irrational Number

Is root 3 an irrational number? Numbers that can be represented as the ratio of two integers are known as rational numbers, whereas numbers that cannot be represented in the form of a ratio or otherwise, those numbers that could be written as a decimal with non-terminating and non-repeating digits after the decimal point are known as irrational numbers. The square root of 3 is irrational. It cannot be simplified further in its radical form and hence it is considered as a surd. Now let us take a look at the detailed discussion and prove that root 3 is irrational.

Prove that Root 3 is Irrational Number

The square root of a number is the number that when multiplied by itself gives the original number as the product. A rational number is defined as a number that can be expressed in the form of a division of two integers, i.e. p/q, where q is not equal to 0.

√3  = 1.7320508075688772... and it keeps extending. Since it does not terminate or repeat after the decimal point, √3 is an irrational number. We can learn to prove that root 3 is irrational by following different methods.

Prove That Root 3 is Irrational by Contradiction Method

There are many ways in which we can prove the root of 3 is irrational by contradiction. Let us get one such proof.

Given: Number 3

To Prove: Root 3 is irrational

Proof:

Let us assume the contrary that root 3 is rational. Then √3 = p/q, where p, q are the integers i.e., p, q ∈ Z and co-primes, i.e., GCD (p,q) = 1.

√3 = p/q

⇒ p = √3 q

By squaring both sides, we get,

p2 = 3q2 

p2 / 3 = q2 ------- (1)

(1) shows that 3 is a factor of p. (Since we know that by theorem, if a is a prime number and if a divides p2, then a divides p, where a is a positive integer)

Here 3 is the prime number that divides p2, then 3 divides p and thus 3 is a factor of p.

Since 3 is a factor of p, we can write p = 3c (where c is a constant). Substituting p = 3c in (1), we get,

(3c)2 / 3 = q2

9c2/3 =  q2 

3c2  =  q2 

c2  =  q2 /3 ------- (2)

Hence 3 is a factor of q (from 2)

Equation 1 shows 3 is a factor of p and Equation 2 shows that 3 is a factor of q. This is the contradiction to our assumption that p and q are co-primes. So, √3 is not a rational number. Therefore, the root of 3 is irrational.

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