Is 2=1 Possible?

Once my teacher gave a proof of 2=1 which is as follow:

$ { 1 }^{ 2 }\quad =\quad 1\\ { 2 }^{ 2 }\quad =\quad 2\quad +\quad 2\\ { 3 }^{ 2 }\quad =\quad 3\quad +\quad 3\quad +\quad 3\\ .\\ .\\ .\\ .\\ .\\ { x }^{ 2 }\quad =\quad x\quad +\quad x\quad +\quad x\quad +\quad ........\quad upto\quad x\quad terms.\\ \\ Now\quad differentiating\quad both\quad sides\quad w.r.t\quad x,\\ 2x\quad =\quad 1\quad +\quad 1\quad +\quad 1\quad +\quad ........\quad upto\quad x\quad terms\\ ie,\quad 2x\quad =\quad x\\ \quad \quad \boxed { { 2\quad =\quad 1 } } $

Now we were challenged to find the flaw in this proof ( Which we won )

Can you find the flaw in this proof ???

#Calculus

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$Since\quad x\quad can\quad take\quad only\quad positive\quad integral\quad values\quad \\ (as\quad { -1 }^{ th }\quad term,\quad { 0 }^{ th }\quad term,\quad { 0.5 }^{ th }\quad term\quad does\quad not\quad exists),\quad \\ the\quad function\quad { x }^{ 2 }\quad is\quad discontinuous\quad \\ and\quad thus\quad is\quad non\quad differentiable.$

Shravan Jain - 7 years, 3 months ago

Thank you for posting this question.

You're trying to use the equality:

${x}^{2} = x + x + ... + x$

You are trying to differentiate two functions. Let's see what we know about them.

Let $f(x)={x}^{2}$ . The domain for this function is clearly $R$ . And yeah when we differentiate we get $f'(x)=2x$ .

Let $g(x)= x + x + ... + x$ (for the equality to be true x repeats in the sum exactly x times), then since a number can't repeat itself $\sqrt { 2 }$ times the domain of this function is $N$ .

Now you differentiate. Tell me how you can calculate:

$g'\left( { x }_{ 0 } \right) =\lim _{ x\rightarrow { x }_{ 0 } }{ \frac { g\left( x \right) -g\left( { x }_{ 0 } \right) }{ x-{ x }_{ 0 } } }$

But you can't approach ${ x } _ { 0 }$ on $N$ . You can only differentiate on intervals or $R$ .

More than this the differentiation gives you the slope of the tangent in ${ x } _ { 0 }$ , but you can't have a slope to a function that's made of dots.

Even if we took $f$ defined on $N$ we still wouldn't have been able to differentiate it either.

So no, 2 isn't 1.

Adrian Neacșu - 7 years, 1 month ago

Well the you only differentiated the x's,the number of x's should also be differentiated as it is also a variable

Murlidhar Sharma - 7 years, 3 months ago

Since this function is non differentiable , there is no question of differentiating the no. of x's

Shravan Jain - 7 years, 3 months ago

Well,I dint even consider that lol

Murlidhar Sharma - 7 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$