Prove that -2/3 is rational or irrational..
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Let us assume that √-2/3 is a rational number. So,
√-2/3= a/b (where a and b are prime numbers with HCF = 1)
Squaring both sides
-2/3= a2/b2
⇒ -2b2/3 = a2
⇒ 2 | a2
⇒ 2 | a ------------> (1) [by Theorem -> Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer]
a = 2c for some integer c.
Squaring both sides
a2 = 4c2
2b2 = 4c2 [ Since a2=2b2]
⇒ b2 = 2c2
⇒ 2 | b -------------> (2)
∴ From (1) and (2) we obtain that 2 is a common factor of ‘a’ and ‘b’. But this contradicts the fact.
So our assumption of √2 is a rational number is wrong.
Hence, √2 is an irrational number.
√-2/3= a/b (where a and b are prime numbers with HCF = 1)
Squaring both sides
-2/3= a2/b2
⇒ -2b2/3 = a2
⇒ 2 | a2
⇒ 2 | a ------------> (1) [by Theorem -> Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer]
a = 2c for some integer c.
Squaring both sides
a2 = 4c2
2b2 = 4c2 [ Since a2=2b2]
⇒ b2 = 2c2
⇒ 2 | b -------------> (2)
∴ From (1) and (2) we obtain that 2 is a common factor of ‘a’ and ‘b’. But this contradicts the fact.
So our assumption of √2 is a rational number is wrong.
Hence, √2 is an irrational number.
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