(This can be a re-post of a 2017 video and submit, with a special video presentation)
I acquired an electronic mail I needed to share with you. Wolfgang got here up with a false proof that 2 = 0. Nobody in his class, not even his trainer, might determine the error. Are you able to?
I current the false proof and the error in a brand new video.
“Show” 2 = 0. Can You Discover The Mistake?
Right here is the “proof” in textual content.
2 = 1 + 1
2 = 1 + √(1)
2 = 1 + √(-1 × -1)
2 = 1 + √(-1)√(-1)
2 = 1 + i(i)
2 = 1 + i2
2 = 1 + (-1)
2 = 0
Or hold studying for a textual content rationalization.
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False Proof 2 = 0
(Because of Rolan for alerting me of a typo on this submit.)
2 = 1 + 1
2 = 1 + √(1)
2 = 1 + √(-1 × -1)
2 = 1 + √(-1)√(-1)
2 = 1 + i(i)
2 = 1 + i2
2 = 1 + (-1)
2 = 0
The error is between traces 3 and 4.
√(-1 × -1) ≠ √(-1)√(-1)
This can be a misapplication of the product rule for sq. roots. The product rule is assured to work solely when each values are optimistic.
If x, y ≥ 0, then
√(xy) = √(x)√(y)
When x = y = -1, the product rule could not apply, and as demonstrated, it isn’t a legitimate step as a result of it results in the conclusion that 2 = 0.
If you be taught a property in math class, make sure that to concentrate to the precise situations when it applies. If you happen to don’t you possibly can find yourself with an absurd end result like 2 = 0, which might break all of math!
So how are we speculated to simplify a sq. root of a detrimental quantity? It’s really a mistake to make use of the product rule (which KhanAcademy teaches):
√(-52) = √(-1)√(52) = i √(52)
This can be a mistake! You shouldn’t use the product rule until each phrases are optimistic–though on this case you do get the proper reply.
Khan Academy clarifies it is best to use this provided that each numbers are optimistic, or just one is detrimental. They do say utilizing it for each detrimental numbers is an issue. Nonetheless, I believe one ought to keep away from the product rule altogether, until each are non-negative.
What I used to be taught is we outline the sq. root of detrimental numbers as follows (see web page 529 in right here):
If b is an actual quantity higher than 0 , then
√(-b) = i √b
So the proper approach to discover the reply is by definition:
√(-52) = i √(52)
You would possibly suppose this can be a nit-picking distinction because the Khan Academy technique will get to the proper reply. However keep in mind that the method issues in math–it isn’t about getting the proper reply, it’s about getting the proper reply by the proper technique, and the way issues are taught do matter to college students being confused. With nice funding comes nice accountability.
References
Throwback to 2017 video/submit
https://youtu.be/1irvvZzbJkU
https://mindyourdecisions.com/weblog/2017/03/23/prove-2-0-can-you-find-the-mistake/
Making the rounds in 2025 on Reddit
https://www.reddit.com/r/theydidthemath/feedback/1ohee1e/requestwhere_is_the_math_wrong_here/
My sq. root guidelines are based mostly on this web site’s presentation
https://www.onlinemathlearning.com/exponents-roots-gre.html
Mistake
https://www.wolframalpha.com/enter?i=2+%3D+1+%2B+sqrtpercent28-1percent29+*+sqrtpercent28-1percent29
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