Proof that square root of 2 is irrational | Algebra I | Khan Academy

Missed the previous lesson?

Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

Subscribe to Khan Academy’s Algebra channel:

[Skatinima]

The first guy who brought up the proof that sqrt(2) was irrational ended on a deserted island and was never heard of again.. Good times we live in. Sal didn't end up in a deserted island.

[rajsrivastava]

didn't understand after hours of practice...easily understood after watching this once.Thx man!!!

[cindywang]

There's probably a really simple answer to this, but I'm just gonna ask anyways: Why does a/b have to be irreducible? If it's reducible, couldn't you just simplify the answer?

[NoahAndABadger]

People in the comments either don't understand or are some kind of math major and can spot flaws in the proof 😓

[shainewinter]

Thank you! "No other common factors other than 1." I had a constant question about this where if we had 2/1, which is irreducible, then root-2 would be rational. But Khan to the rescue again =).

[amissanime]

Here's a simpler way.
✓2 = p/q
=> 2 = (p/q)^2
=> 2q = p^2/q

Since 2q is natural
And there's no common factor between p and q,
Therefore p/q or p^2/q can't be natural.

Hence, 2q = p^2/q can't be true.

[Yournature]

According to him -
Assume = Asoom 😂
Contradiction = khnantadiction
Co prime - Coh prime
Multiply = Multhifly
Two = theu
K = khay 🤭
Times - thaimes

[VoidHalo]

Thanks for this. I've taken up something of an obsession with root 2 lately, so I've been trying to find as many properties and neat tricks and such to do with it as I can. Really I'm watching this in preperation for the Mathologer video on this, which I assume goes more in depth, as they tend to do.

[kornelijekovac]

But this doesn't prove it is irrational number. It just shows that assumption of irreducibility is wrong.

[jyotiagarwal]

After listening to MIT professor I'm here. Understood really well now. Thanks

[HariCharann]

Hello sir I love the way you explained it. But I still have a doubt : if root 2 is expressed in decimal and a decimal number can be expressed in a fraction. Fractions are expressed in p/q form the how can we say root 2 is irrational number.

[fatimaalamien]

I really love this channel but please don’t make it in Hindi cause I can’t understand

[lampochka]

Thank you very much! I am really grateful for this lesson! You've helped me. Thanks again.

[scottcollege]

I don't get the reasoning. So what if it's reducible? 4/2 is reducible and it's a rational number.

[forrestberg]

Someone get Terrence Howard to watch this video haha

[خوارزمة]

can we generalize it for all √3 √5 for all

[HanSolo-dhrn]

It confuses me how a video from a channel with over a million subscribers has about 15 comments and 6000 views?

[Melomathics]

Is this the only non "overly complicated" proof? I keep seeing it everywhere.

[priya]

His voice reminds me of the rock (Dwayne Johnson)

[isaac]

to make it clear for others struggling with why both numbers being even proves the irrationality of root 2, A and B are DEFINED to be coprime (all rationals can be reduced to this form) and by assuming the inverse statement (root 2 is rational) and working your way to both A and B being even, have now shown that A and B have a common factor of 2. This is against the proposition made of coprime rationality. The important part now is that we have made a contradiction, and it is a fact of logic that a contradiction can ONLY be formed if a previous step was incorrect, but in this case the only step that was wrong was the assumption of rationality, so root 2 MUST be irrational.