Proof that Square Root 2 is Irrational
ฝัง
- เผยแพร่เมื่อ 17 ก.ค. 2024
- This video is housed in our WCoM Basics: College Algebra playlist, but it's important for all mathematicians to learn.
Tori proves using contradiction that the square root of 2 is irrational.
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I love black boards. Lately, I attended classes where the teacher used a computer screen projection and this was so bad for learning. Black boards force the teacher to teach at the same pace that he or she writes. It allows the students to take notes and ask questions.
What about white boards???
@@bobbysteakhouse7022 chalk is much better, it's satisfying
th-cam.com/video/ozFc8TwHv1k/w-d-xo.html
At school the teacher would have 2 options if you were disruptive:
1. Throw the chalk at you.
2. Throw the larger chalk rubber at you.
Yes , Absolutely agree
Great explanation! I was watching a lecture for another class and the instructor mentioned this as an example of proving by contradiction and he definitely didn't spend 6 minutes talking it out, so much appreciated. I just subscribed, haha.
th-cam.com/users/akdemyformathsbyawaneeshsir
زبنت
غايا
ظ رنم
نننهنننننتفا اث
بثىق
This is the best description of this on TH-cam. Thank you
It was a very helpful video..I've been looking for explanations for this theorem, but I didn't understand any of them...thank u so much..
Got confused at some point but I finally got it, good video.
I now know how everything comes about, keep posting more
Greatly explained mam .You highlighted each and every important point .Thank you very much .Your video widely helped me
Good morning madam
Super explanation and very simple way to understand
I am from Tamil Nadu.
Thanks
Really it's so helpful....I can't expected that even I will understood your language or not but my expectation was wrong....😁 really it's so nice c video I have understood very well👍🏻 thnx so much
Excellent explanation i understood more than any other video I watched
Bahoot tez ho rhe 😂😂😂😂
Naughty baccha
Even me!
@@org_central 😀😂
Same
this helped me a lot...she teaches really nicely...thank you Miss Tori Matta😊
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Well put and easy to follow. Thank you.
You are from which country???
Well done Madam, Excellent methodology to explain. 🏅🏅🏅👌
You explain very smartly
then can it be written as a reducible fraction...?
Why do thry need to be irreducible? Please can you explain
Ma'am why a and b are taken as coprime ? Please reply
Can you explain root 16 is not a irrational no by this contradiction method ?
To all the people criticizing the assumption of an irreducible fraction, it's a non-issue because the canonical form of a fraction is equal to any multiple thereof. So, even if we assumed that sqrt(2) is equal to some fraction, c/d, that isn't in lowest terms, c/d can be reduced to a/b anyway. Therefore, sqrt(2) is still equal to a/b where a and b share no common factors.
This is basic part of the definition of a *rational number* that originated in ancient Greece during the time of Pythagoras in around 6 BCE. The "Pythagorean Order" believed that all numbers were perfect and divine and that any number could be expressed as a ratio two integers. Even an infinitely repeating decimal like 0.333... can be expressed as 1/3. If you couldn't calculate it down to a perfect ratio of just two integers, you just hadn't calculated enough. It was this very proof of sqrt(2) that demonstrated that there were, indeed, numbers that didn't follow this perfect structure; numbers that were irrational (cannot be expressed as a ratio of integers). And later, still, it was found that you can even go a step further. Even for irrational numbers like sqrt(2), they found that you could still describe them using an algebraic formula, using algebraic operations; exponents, addition, subtraction, multiplication, and division. These are now called Algebraic Irrational Numbers. But there are some that don't even obey that paradigm. Transcendental Irrational Numbers like _e_ and _pi_ won't even take an algebraic formula; the formula would keep going on and on with an infinite number of terms.
You don't even have to get that technical. By induction, if such a fraction were reducible, it would have to be infinitely reducible. This is obviously nonsense, so we still have a contradiction.
Anyways, even if a and b have common factors, the proof still holds!
+Jeremy, no rational can be infinitely irreducible. For a/b both a and b are natural numbers and are necessarily finite (as are all the real numbers). if g = gcd(a,b) then replace the a and b with A = a/g and B = b/g
-NB you have effectively claimed that g can be infinite.- Oops, I had misunderstood Jeremy.
+Thititoutou Wrong. If for a/b both had a factor of 2, then there wouldn't be a contradiction at the end of the step where we deduce that b must be even too.
+Jeremy, Ooops, I see that I misunderstood you. I had a brain fart. I'm sorry about that.
Nice Explanation. BTW if to Squared equivalence a squared & b squared equations ... could be 4b = a & 4k = b where; a/b = 1,,, 🤔
Thanks,hope this helps me 😊
If I had a teacher like her, I would go for Math classes everyday..
Why
@@themotivator1214 coz he tryina smash
Thanks mam i was also finding this that why a is even but other tutors were explaing only by prime factors vision.
Fantastic explaination 👍🏻👍🏻
Which standard maths are u teaching here¿
EXCELLENT MATH TEACHER PRAISE WORTHY WORK
Very super mam your a good teacher of youtube and all videos super explanation is very good and this video is very useful of irrartion numbers thank u mam bye...
AWESOME EXPLANATION....
❤
Nice explanation
I like your way to teach.
your explanation is awesome
Thanks mam this helps me lot in my examination
Best explaination. Being a ninth grader it's really helpful 👌👌
Breh
What about if this logic applied in a rational number? Will it be true?
I just tried it and it worked for me.. any help?
Good explanation.
Thank you tari matta g for this video ,overwhelming pretty video
th-cam.com/users/akdemyformathsbyawaneeshsir
Thanks for the video.
I tried this on √4 , and it still contradicted that the co primes have common factors
Root 4 is rational which can be reduced to 2x2 think
*3:10* Just because *(2n)² is even* doesn't mean that if *n² is even* than *n is even.*
E.g. *n² = 2 (so, n² is even)* but then *n = √2 (therefore, n is not even)*
Am I missing something here?
yes, n must be an integer - obviously. Otherwise you could sub in random decimals and ofc not come out with whole numbers let alone even whole numbers
Ma,am from where you are
So clear explanation...
Easily explained mam. Thank you so much
Mam please solve 1+root 3
Your video is really helpful for me
Thank you very much ma'am
Brilliant explanation madam❤️👍👍
I am able to understand it more than anyone else
Excellent Teaching
Hello madam
You teach very well
Love from India
That's a really good Explanation.
You have solved √2 is an irrational number can not be written in form of rational number such as fraction form p/q well explained by you by contradiction method
Saved me ❤️ thanx for the vid
how would the result be any different if you were to put a perfect square under the radical, because then it is rational and if you were to continue the proof there would still be a contradiction saying it couldnt be rational
Let's go through the argument with 4 instead of 2.
Suppose √4 is rational. Then √4 = a/b where a and b are integers and b is not 0.
Square both sides to get 4 = a^2/b^2.
Multiply both sides by b^2 to get 4b^2 = a^2
Now, the left hand side is divisible by 4. So the right hand side must also be divisible by 4.
This means a^2 is divisible by 4.
*Here's where things are different: we **_cannot_** conclude that a is divisible by 4 - the best we can do is conclude that a is divisible by 2* (I will explain why later)
So a = 2c for some integer c.
Then 4b^2 = (2c)^2 = 4c^2
Dividing both sides by 4, we get
b^2 = c^2.
Since b and c are both positive, we get b = c.
So √4 = a/b = (2c)/b = (2b)/b = 2.
And we get the actual answer instead of a contradiction.
Now, why can we not conclude that a^2 is divisible by 4? Well, let's look at some examples.
Suppose a = 2. Then a^2 = 4 is divisible by 4, but a = 2 is not divisible by 4.
Or suppose a = 6. Then a^2 = 36 is divisible by 4, but a = 6 is not divisible by 4.
So why does it work for 2 when it doesn't work in general?
One way is to notice that 2 is a prime number. If n^2 is a perfect square which is divisible by a prime number p, then n must be divisible by p as well. You can see this by taking a prime factorization of n, and then squaring all of the factors to obtain a prime factorization of n^2. Since p is a prime dividing n^2, it follows that one of the prime factors in the prime factorization of n^2 is p. But the prime factors in the prime factorization of n^2 are the same (but appearing twice as many times) as the prime factors in the prime factorization of n. Therefore, p is a prime factor in the prime factorization of n. So n is divisible by p.
More generally, by slightly modifying this argument, if m is a number which is not divisible by the square of any prime number and if n^2 is divisible by m, then n must also be divisible by m.
So since 2 is a prime (more specifically, since 2 is not divisible by the square of any prime), we know that if 2 divides a^2, 2 must also divide a. The same thing is not true for 4 since 4 _is_ divisible by the square of a prime - namely 2.
Great explanation
Why exactly can't a and b have any common factors? I tried the proof with n instead of 2 and it seems to work, "proving" that there are no rational numbers at all... My math is most certainly wrong, but please tell me how
Every rational number can be expressed in the form a/b where a and b have no common factors. You can always divide both the numerator and denominator by gcd(a,b), and you will have an equivalent fraction to the one you started with where the numerator and denominator are integers with no common factors.
So while rational numbers don't _have to_ be written in reduced form, they always _can_ be. Starting with the reduced form makes the argument cleaner.
Let's work through the argument with sqrt(n). Say sqrt(n) = a/b where a and b are positive integers which have no common factors. Squaring both sides and multiplying by b^2 gives us:
nb^2 = a^2.
Now here's where the argument breaks down for general n. Yes, a^2 is a multiple of n. However, this does not mean that a is a multiple of n. For example, let's say n = 4. It's possible for 4 to divide a square number without dividing the square root of that square number. For instance, 4 divides 36 = 6^2, but 4 does not divide 6.
The argument works fine for n = 2. The argument actually works just fine for n being _prime_ since prime numbers have the property that if they divide a product of integers, they must divide at least one of the factors. But not every number has that property.
@@MuffinsAPlenty Great explanation, thanks
Excellent teaching
I design the majority of my artwork in a root 2 rectangle. Phi is my favorite but 2 is easy.
th-cam.com/users/akdemyformathsbyawaneeshsir
You are from which country
can we prove that √4 isn't an irrational number with this method?
Let's use this method to prove that Square Root 3 is Irrational number.
Thanks a lot it really works for me
Thank you somuch sisters
what are ratoinal numbers
a rational number is any number that can be expressed as a fraction (p/q) of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number by definition.
Beautiful explation
1.41 what did I hear
I was not able to sleep so that’s why I am watching this vedio
🤭🤭🤭🤭😴
But now I am going to sleep
😂😂
Lol
But why 🤣?
*vedio*
HAHAHA GO SLEEP DUDE
I hate that number 2. It looks like a comic sans font of Windows 95 AND can be easily confused with symbols like alpha or the curved 'd' of the partial derivative. How would she solve the negative gradient of a potential [alpha]/(r^2)?? ^.^ =>
-(2/2x * 2/r^2)
-(2/2y * 2/r^2)
-(2/2z * 2/r^2)
A small slip would make it look like a 3.
Isn’t the proof hold for any irrational number which are square root of somthing
thank you
I don't understand something in your proof.
Why a and b haven't common factors?
(I'm not good at English . Sorry about it.)
Thanks Madam. very useful
hocam buyukluk saka mi
what about square root of 3 ? how to prove that?
You can prove it in the same way :)
The only difference is that instead of saying "a^2 must be even so a must also be even" you say that "a^2 must be divisible by 3 so a must also be divisible by 3."
Since 3 is a prime, we know this is true. You can check by taking a prime factorization of a, and then squaring each of the factors to get a prime factorization of a^2. Since 3 is a prime dividing a^2, it must be a prime factor of a^2. But the primes appearing in the prime factorization of a^2 are precisely the primes appearing in the prime factorization of a (since you squared a prime factorization of a to get a prime factorization of a^2). Therefore, 3 must be a prime factor of a. So a is divisible by 3.
how come just cause a and b are even they are irrational? 4/2 is rational?
I think it's because in the first she assumed a and b do not have any common factors (relatively prime),and in your case the common factor is 2
like it such types of teaching
this just cant be true,
so you are telling me that this is for college students ?
i am an indian ,14 year old and this is in my first chapter of maths book
when u assume root(2) is a rational number, why a/b must be irreducible? is this part of the definition of rational number?
Yes the definition of a rational number is a fraction in its simplest terms. Even if this wasn't the case consider getting the simplest form of the fraction a/b here. Through the same process used in the video it can be shown that the simplest fraction of a/b can be divided further which is clearly absurd and not possible
Beautiful maths N ma'am🥰 💕
Thanks for this video
Clear and awesome 👍
Why can't you reduce a/b when it's rational
Mam I have a problem _ what is the difference between a rational number and fraction? Please answer me mam
Hi Shalini,
A rational number is a type of fraction, although fractions also can describe things which are not rational numbers. Check out en.wikipedia.org/wiki/Fraction_(mathematics) for more information.
***** thank you very much mam😃😃😃😃
What is your hairstyle name Ma'am?
It is awesome
It's a bob cut
Let k=b/sqrt(2). In order to complete this process of proof successfully we HAVE TO assume that this expression for k is NOT an integer.
Whether it is or not, we do not know at this point, and we won't find out beyond this point. To my opinion the here presented proof of sqrt(2) is rational did NOT fail. A vicious circle.
hlw
So convoluted.
The "no common factor" statement was presented as a given, however it should be explained or proof why is that the case , then the proof will follow.
What is the definition of rational. It is that it can be expressed as a ratio. Which is a/b in this case. If they are co prime then we can simplify until they are not
@@magicbaboon6333 Your last sentence has got the definition the wrong way around.
Should read 'If they are _not_ coprime, then we can simplify until they are.'
more understandable, thanks
Ma'am, instead using the argument that a and b must be even, I think it will be better if we use The fundamental theory of arithmetic which is applicable for all primes.
Can u tell me how wolud that work? Tnq.
use the concept of co-primes
@@RzGyan98 thanks
@@jelenajonjic im late but there are two theorems.
Theorem 1: If a is a natural number and p is a prime number, then if p divides a^2 then p also divides a.
Theorem 2: If a and b are two natural numbers and p is a prime number, then if p divides ab then p divides a or p divides b or p divides both.
You can apply it in the equation so as to prove that a/b indeed has a common factor other than 1, hence proving its not a rational number
Made it simple to understand....thumbs up 😃😃
THANK U SO MUCH
I understood perfectly
It will come in board exam 100% garentee
Nice presentation. But I have to recommend to the lady that she unlearns to write the "2" as a delta
An actual delta looks like this: δ
I think you are referring to the partial derivative symbol, because, her 2, if looked at a certain way, does remind someone of ∂
Nice. Thanks.
An excellent ma'm
I enjoy her voice
I also
Thanks mam
Proved beautifully I have also proved it but in another way
Super tanks to teach
Very nice Hi Iam maths(lect) from India
So 3/6 is irrational? But Google says it isn't. Please help
Why would you think that 3/6 is irrational?
It is both rational (by definition) and reducible (3/6 = 1/3)
@@Grizzly01 becase 3/6 isn't in its simplest form
@@misan2002 You are getting 'irreducible' and 'irrational' mixed up, aren't you?
@@Grizzly01 yes, maybe.
Thanks