Proof 2 • Contradiction with Rational and Irrational Numbers • P2 Ex1A • 💡

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[@t.o8386]

hi sir, for the question at 1:00 do you have to contradict it in that order or could you also do "assume if a is a rational number and b is a rational number then a-b is irrational"? thanks!

[@BicenMaths]

That isn’t what we call the negation of the original statement - your statement would be trying to prove something else, by changing what kinds of numbers a and b can be. The way I suggest is the only way I think!

[@t.o8386]

@@BicenMaths okay thank you!!

[@benlynch7812]

Hey Sir, I do not quite understand 5:54 - we are told A is rational and B is irrational, and assume A - B is rational, yet since A is rational and we find B to be rational in the negation argument, surely A - B is rational ?

[@benlynch7812]

Nevermind! Re-reading what was happening carefully made me understand!

[@AbiaSohail]

Hi Sir, When we do these proof questions do we have to explain the proof or is the maths enough to get the marks?

[@BicenMaths]

A conclusion statement at the end is usually necessary - but it doesn't need to be particularly detailed!

[@koobs13]

On the last example about irrational numbers, how does a and b having common factors prove that root 2 is irrational? thanks

[@BicenMaths]

If a and b had common factors, then we would be able to do simplify this to another fraction, let's say c/d. We could then do the same process we did to a and b, but this time to c and d - and would draw the conclusion that c and d also have common factors, meaning they could also cancel! This logic with then continue infinitely, with a fraction that can continuously be simplified, which we know is not possible - hence it can't be written in that form so is irrational. (Sorry for the delay in replying!)

[@crepecake5290]

at 6:43 how do we know that the fraction representing b cannot be simplified further? because then that would mean its rational right?

[@BicenMaths]

We're not actually concerned with them being able to simplify or not with this question, just if it is an integer over an integer - and if it is, then it is rational!

[@crepecake5290]

@@BicenMaths ohhh ok thanks

[@dayan5669]

at 8:25, sure we have disproven that B is irrational for our assumption, but in the original question it also states that B is irrational aswell, so does that not mean that we have also disproven the original statement?

[@BicenMaths]

We have only proven that b is rational *given that* we assumed the opposite thing in the first place, so that doesn’t undermine the original statement.

[@dayan5669]

@@BicenMaths but our original statement states that b is irrational, so how have we proven the statement is true if we’ve proved that b is rational

[@BicenMaths]

We proved b was rational given that we assumed something opposite - this produced the contradiction with b being rational, and therefore meant our assumption was wrong, so the original statement is true!

[@dayan5669]

@@BicenMathsah right i see, tysm

[@fatm786]

Proof 2 • Contradiction done!

[@salmankhan-pl1bu]

are all year 2 playlists complete

[@BicenMaths]

Yes, they are 👍🏼

[@salmankhan-pl1bu]

@@BicenMaths i had completed my maths eam using all your year 1 vidoes. i got 94 per cent. Hope to do the same with year 2

[@BicenMaths]

94%!!! That’s incredible! I love hearing positive stories like this - well done, and thank you for sharing! 👏🏼