It's a contradiction because we're assuming that *E* is the one and only smallest rational positive number. However, E/2 (which is obviously smaller than E) is also rational, therefore it's a contradiction. Hope I helped
Because if n > 0 then n/2 is also > 0 for all rational numbers of n. Try it on a calculator. Keep dividing a positive number by 2 and you’ll see it’s always > 0.
@@aashsyed1277 Hey, I used to a 16 year old when I wrote that comment and wasn't that good at English and I myself can't understand what 16 year old me meant to say! LOL
No because you apply the same logic to that number as well, nothing changed. There was nothing special about ε that ε/2 doesn't have. Call ε/2 some other number and repeat the process.
+Emily Zhang A rational number is any number that can be expressed as the quotient or fraction p/q of two integers. Therefore, if you assume ϵ is a rational number, it's numerator and denominator (p and q) must be integers (otherwise, ϵ wouldn't be a rational number).
Don't look for me P would be =1 and q=100,000,000, therefore E is a rational number, but this is only true because E can be expressed as a division of two integrers, try the same technique with sqrt(2) and see what happens
Numbers which can be written in the form of p/q where p and q are integers, q≠0 are called as rational numbers. It's not like those numbers having non integer numerator or denominator can not be rational. Since we can write 0.001/0.1 as 1/100 where now p and q both are integers and q≠0, so 0.001/0.1 is a rational number. And we cannot do this with √2 or any other irrational number, even though we can write √2 as √2/1, or 2/√2 in both the cases we have p/q form, but either numerator is not an integer or denominator. We can write numbers having terminating or non terminating but recurring decimal expansion in the form of p/q with p and q integers and q not equal to 0, so these are rational, on the other hand numbers with non terminating non recurring decimal expansions cannot be written in that form so irrational.
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Well explained, thank you!
Why do we have to divide by 2. I'm new to doing proof, thus my reason for asking.
Excellent!
Thanks a lot!
too good. Just too good!
Awesome.
The thing I don't get.. E/2 < E, that is true, it doesn't meant that E/2 is less than 0, so how is it contradiction..? That is what is confusing me
It's a contradiction because we're assuming that *E* is the one and only smallest rational positive number. However, E/2 (which is obviously smaller than E) is also rational, therefore it's a contradiction. Hope I helped
@@filiplass7442 Yeah i think it did lol. Thank you very much!
@@IoniB You're welcome :) Have a nice day!
Because if n > 0 then n/2 is also > 0 for all rational numbers of n.
Try it on a calculator. Keep dividing a positive number by 2 and you’ll see it’s always > 0.
@@SuperYtc1 Ahh alright, thank you!
Can we say that if E is a rational no E^2 is rational as well E^2
I don't understand.
@@aashsyed1277 Hey, I used to a 16 year old when I wrote that comment and wasn't that good at English and I myself can't understand what 16 year old me meant to say! LOL
could you do root 4 + root 5
Staedtler should pay you!
But then what if that number you found is the smallest? it could well be the smallest, no?
No because you apply the same logic to that number as well, nothing changed. There was nothing special about ε that ε/2 doesn't have. Call ε/2 some other number and repeat the process.
Why do p and q have to be integers? If E was 1/100 000 000, then couldn't p=0.000000001 and q=0.1? That way, p and q aren't integers D:
+Emily Zhang A rational number is any number that can be expressed as the quotient or fraction p/q of two integers. Therefore, if you assume ϵ is a rational number, it's numerator and denominator (p and q) must be integers (otherwise, ϵ wouldn't be a rational number).
Don't look for me P would be =1 and q=100,000,000, therefore E is a rational number, but this is only true because E can be expressed as a division of two integrers, try the same technique with sqrt(2) and see what happens
Numbers which can be written in the form of p/q where p and q are integers, q≠0 are called as rational numbers. It's not like those numbers having non integer numerator or denominator can not be rational. Since we can write 0.001/0.1 as 1/100 where now p and q both are integers and q≠0, so 0.001/0.1 is a rational number. And we cannot do this with √2 or any other irrational number, even though we can write √2 as √2/1, or 2/√2 in both the cases we have p/q form, but either numerator is not an integer or denominator. We can write numbers having terminating or non terminating but recurring decimal expansion in the form of p/q with p and q integers and q not equal to 0, so these are rational, on the other hand numbers with non terminating non recurring decimal expansions cannot be written in that form so irrational.
Proof: "you can divide by 2"....
So much ado about nothing...
ha, q can not equal 0