why -1/√2 is an irrational number?
Answers
Answer:
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
Step-by-step explanation:
Let us assume that √2 is a rational number.
Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0
⇒√2=p/q
On squaring both the sides we get,
⇒2=p²/q²
⇒2q²=p² —————–(i)
p²/2= q²
So 2 divides p
p is a multiple of 52
⇒p=2m
⇒p²=4m² ————-(ii)
From equations (i) and (ii), we get,
2q²=2m²
⇒q²=2m²
⇒q² is a multiple of 2
⇒q is a multiple of 2
Hence, p,q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number
√2 is an irrational number .
As we know that any number divided by an irrational number is irrational,
Therefore -1/√2 is an irrational number