PROOF √𝟐 IS IRRATIONAL

What method/proof (begging with ‘c’) will make us verify that root to is indeed irrational?

What method/proof (begging with ‘c’) will make us verify that root to is indeed irrational?

A proof by contradiction.

Step 1

If √2 is rational then, it can be written as

√2 = 𝑝 /q, 𝑝, 𝑞 ∈ ℤ and 𝒑/ 𝒒 is in its simplest form.

• Also a side note that if p and q are a factor of 2 THEN (in the end answer) it should say p/q CANT be written in its . form!!!!

• Which contradicts our ‘assumption’ that root 2 is rational SO IT MUST BE IRRATIONAL .

Step 2

• To get rid of a surd you have to square it.

• Since our squared the left YOU MUST do it to the right.

• Note how big p AND q receives a square!!!

Step 3

Step 4

From second step, we see how 2q²= p², which means that 2 is a factor of p SO then p can be equal to 2k where k, is an element of z.

• THIS MEANS THAT P MUST BE EVEN!!!

• Now WE KNOW by subbing in p for 2k that q² is equal to 2k² so his must mean 2 is ALSO A factor of q and is even!!!

Last step (5)

What does this proof tell us we can do with any irrational surd?

The same method can be used to prove any irrational surd.

• PROOF BY CONTRADICTION