prove that √2 is irrational number

Описание: in this video we will learn how to prove that √2 is irrational number
To prove that √2 is irrational, we use proof by contradiction .
we Assume √2 is rational, meaning it can be expressed as a fraction p/q where p and q are integers with no common factors.
squaring both sides,
2 = p²/q²,
which implies p² = 2q².
This means p² is even, and therefore p must also be even.
Let p = 2k, where k is another integer. Substituting this back into the equation, we get (2k)² = 2q²,
or 4k² = 2q²,
which simplifies to 2k² = q².
This means q² is also even, and thus q must be even.
However, if both p and q are even, they share a common factor of 2, contradicting the initial assumption that p and q have no common factors.
Therefore, √2 cannot be expressed as a fraction and is irrational.