Prove that 3+2root 7 is an irrational number
Answers
ATQ, We have to prove that 3 + 2√7 is an irrational number.
Step 1: Prove √7 is irrational.
Step 2: With the help of Step 1, prove 3 + 2√7 is an irrational number.
Let us assume that √7 is a rational number. This implies that √7 can be expressed in the form p/q where q ≠ 0 and p & q are co-primes.
⇒ √7 = p/q
Squaring on both sides:
⇒ (√7)² = (p/q)²
⇒ 7 = p²/q²
⇒ 7q² = p² → Equation (1)
→ 7 divides p².
∴ 7 divides "p" as well. → Relation (1)
Theorem applied: If p is a prime number and divides q², then p divides 'q' as well where 'q' is a positive integer
Let us take p = 7c where c is any positive integer.
From equation 1;
⇒ 7q² = p²
⇒ 7q² = (7c)²
⇒ 7q² = 49c²
⇒ q² = 7c²
→ 7 divides q².
∴ 7 divides "q" as well. → Relation (2)
From Relation 1 & 2 we can say that both p & q have factors other than 1 & themselves. This contradicts the fact that they are co-primes. This is due to the incorrect assumption that √7 is a rational number.
Now, lets prove that 3 + 2√7 is an irrational number.
Let us assume that 3 + 2√7 is a rational number. This implies that it can be expressed in the form p/q where q ≠ 0 and p & q are co-primes.
⇒ 3 + 2√7 = p/q
⇒ 2√7 = (p/q) - 3
⇒ 2√7 = (p - 3q)/q
⇒ √7 = (p - 3q)/2q
Here, √7 is an irrational number, and (p - 3q)/2q is a rational number. But we know that;
Irrational number ≠ Rational number.
This contradicts our assumption that 3 + 2√7 is a rational number.
∴ 3 + 2√7 is an irrational number.
Answer:
3-2√7 is an irrational number
Step-by-step explanation:
Given : The term 3-2√7
To find : Prove that 3-2√7 is an irrational
Solution :
Now we have given the term 3-2√7 consider 3-2√7 as a rational number then we can write it in the form of a/b where a and b are co prime
3-2√7=a/b
-2√7=a/b-3
-2√7=(a-3b)/b
√7=(-a+3b)/2b
So this prove that (3b-a)/2b is a rational number.
But √7is an irrational number so it contradicts our assumption
So our assumption is wrong
3-2√7 is an irrational number
Hence prove
So 3-2√7 is an irrational number