# Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)

Consider the following:

• $1 = 1^2$
• $2 + 2 = 2^2$
• $3 + 3 + 3 = 3^2$

Therefore,

• $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$

Take the derivative of lhs and rhs and we get:

• $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$

Which simplifies to:

• $x = 2x$

and hence

• $1 = 2$.

Clearly something is wrong but I am unable pinpoint my mistake.

• Right before taking the derivative, x was an integer, and what you did before that point only makes sense for x integer. Then you computed the derivative, and then that does not make sense. Jul 29, 2010 at 2:48
• @Kaestur I am not sure what you want me to do. Is there a specific name for this fake proof?
– user116
Jul 29, 2010 at 2:51
• @Srikant: To clarify, the reason I asked that is so that people about to ask a similar question in the future may see yours come up as a suggested duplicate. Something including "derivative of 1+...+1 (x times)" would be uniquely identifying, I think. If there is a canonical name for this one, chances are people about to ask it won't know it, but will recognize that line. Jul 29, 2010 at 3:04
• @Kaestur I actually think retaining the title as it is written now is better as most people are more likely to write 'proof of 1 = 2' rather than some description of their proof in the title.
– user116
Jul 29, 2010 at 4:22

I think the discrete/continuous issue is sort of a red herring. To me, the problem is forgetting to use the chain rule!

To un-discretize, think of the function $F(u,v) = uv$, which we could think of as $u + \dots + u$, $v$ times. Then $x^2 = F(x,x)$. Differentiating both sides gives $2x = F_u(x,x) + F_v(x,x)$, which is perfectly true. In the fallacious example, the problem is essentially that the $F_v$ term has been omitted. In some sense, one has forgotten to differentiate the operation "$x$ times" with respect to $x$! Of course, the notation makes this easier to do.

• This is a nice explanation. Aug 11, 2010 at 4:35
• I side with you on this one. I answered a similar thing here
– Pedro
Jul 27, 2012 at 2:41
• any one please could help me? what are $F_u$ and $F_v$ ? Sep 6, 2020 at 3:19
• @JoséMarín: Partial derivatives. The partial derivative of $F$ with respect to $u$ is $F_u(u,v) = v$, and likewise $F_v(u,v) = u$. Sep 6, 2020 at 3:26
• @NateEldredge Thank you very much! now I understand and agree. In the original argument ( fallacy) the problem is with the abuse of notation ( en.wikipedia.org/wiki/Abuse_of_notation ) with the sum operation under the derivative. Sep 6, 2020 at 3:36

You cannot take the derivative of $$\underbrace{x + x + x + \dots + x}_{\text{repeated x times}}$$ with respect to $$x$$ one term at a time because the number of terms depends on $$x$$.

Even beyond that, if you can express $$x^2$$ as $$\underbrace{x + x + x + \dots + x}_{\text{repeated x times}}$$, then $$x$$ must be an integer, and if the domain of the expression is the integers, (continuous) differentiation does not make sense and/or the derivatives do not exist.

(edit: I gave my first reason first because the second reason can be smoothed over by taking "repeated $$x$$ times" to mean something like $$\underset{\lfloor x\rfloor\mathrm{\ addends}}{\underbrace{x+x+\cdots+x}}+(x-\lfloor x\rfloor)\cdot x$$.)

• A simpler version: 1+1+...+1 repeated x times = x. Now you can see that the left hand side is obviously not constant with respect to x. Feb 22, 2011 at 19:44
• Of course, the derivative of $(x-\lfloor x\rfloor)\cdot x$ is $2x-\lfloor x\rfloor$ when the derivative is defined, so the value you'd get is $2x$ if you took the continuous version. May 16, 2017 at 20:30
• "You cannot take the derivative with respect to x of x + x + x + ... (repeated x times) one term at a time because the number of terms depends on x."-@issac i really loved this statement...but could you please explain about it a bit more? Mar 25, 2019 at 4:19

We can create the same "paradox" with finite differences over integers.

Given $f: \mathbb Z \to \mathbb Z$ define the "discrete derivative" $$\Delta f (n)=f(n+1)-f(n)$$ we have the following obvious "theorems":

• $\Delta(n)=n+1-n=1$
• $\Delta(n^2)=(n+1)^2-n^2=2n+1$
• $\Delta (f_1+\cdots +f_k)=\Delta f_1 + \cdots +\Delta f_k$
• $f(n)=g(n) \; \forall n \quad \implies \quad \Delta f(n)=\Delta g(n) \; \forall n$

$$\underbrace{n + n + n + \ldots + n}_{n \textrm{ times}}= n^2$$

and we apply $\Delta$ on both sides taking advantage from the "theorems" above: we get $$\underbrace{1 + 1 + 1 + \ldots + 1}_{n \textrm{ times}} = 2n+1$$ so we conclude $n=2n+1$ and we have the paradox.

Here maybe the mistake is more clear: the rule $\Delta (f_1+\cdots +f_k)=\Delta f_1 + \cdots +\Delta f_k$ doesn't work when $k$ is a function (of the same variable of the $f_i$), in fact it amounts to do a computation like this: $$\Delta(\underbrace{n + \ldots + n}_{n \textrm{ times}})= \underbrace{(n+1) + \ldots + (n+1)}_{\color{Red}{n \textrm{ times}}}-(\underbrace{n + \ldots + n}_{n \textrm{ times}})=n$$ that is wrong, the right way being this: $$\Delta(\underbrace{n + \ldots + n}_{n \textrm{ times}})= \underbrace{(n+1) + \ldots + (n+1)}_{\color{Green}{(n+1) \textrm{ times}}}-(\underbrace{n + \ldots + n}_{n \textrm{ times}})=n+(n+1).$$

Here's my explanation from an old sci.math post:

Zachary Turner wrote on 26 Jul 2002:

Let D = d/dx = derivative wrt x. Then

D[x^2] = D[x  +   x  + ... +   x  (x times)]
= D[x] + D[x] + ... + D[x] (x times)
=   1  +   1  + ... +   1  (x times)
=   x


An obvious analogous fallacious argument proves both

•  D[x f(x)] = Df(x) (x times) = x Df(x)

•  D[x f(x)] = Dx (f(x) times) = f(x), via Dx = 1

vs. the correct result: their sum $$\rm\:f(x) + x\, Df(x)\:$$ as given by the Leibniz product rule (= chain rule for times). The error arises from overlooking the dependence upon x in both arguments of the product $$\rm\: x \ f(x)\:$$ when applying the chain rule.

The source of the error becomes clearer if we consider a discrete analog. This will also eliminate any tangential concerns on the meaning of "(x times)" for non-integer x. Namely, we consider the shift operator $$\rm\ S:\, n \to n+1\$$ on polynomials $$\rm\:p(n)\:$$ with integer coefficients, where $$\rm\:S p(n) = p(n+1).\:$$ Here is a similar fallacy

  S[n^2] =  S[n  +   n  + ... +   n  (n times)]
=  S[n] + S[n] + ... + S[n] (n times)
=  1+n  + 1+n  + ... + 1+n  (n times)
= (1+n)n


But correct is $$\rm\ S[n^2] = (n+1)^2.\:$$ Here the "product rule" is $$\rm\ S[fg] = S[f]\, S[g],\$$ not $$\rm\: S[f] g\:$$ as above.

The fallacy actually boils down to operator noncommutativity. On the space of functions $$\rm\:f(x),\:$$ consider "x" as the linear operator of multiplication by x, so $$\rm\ x:\, f(x) \to x f(x).\:$$ Then the linear operators $$\rm\:D\:$$ and $$\rm\:x\:$$ generate an operator algebra of polynomials $$\rm\:p(x,D)\:$$ in NON-commutative indeterminates $$\rm\:x,D\:$$ since we have

  (Dx)[f] = D[xf] = xD[f] + f = (xD+1)[f], so  Dx = xD + 1 ≠ xD

(Sn)[f] = S[nf] = (n+1)S[f], so  Sn = (n+1)S ≠ nS


This view reveals the error as mistakenly assuming commutativity of the operators $$\rm\:x,D\:$$ or $$\rm\:n,S.$$

Perhaps something to ponder on boring commutes !

You cannot differentiate the LHS of your equation

$$x + x + x + \cdots$$ (repeated $$x$$ times) = $$x^2$$

This is because the LHS is not a continuous function; the number of terms depends on $$x$$ so the LHS is not well defined when $$x$$ is not an integer. We can only differentiate continuous functions, so this is not valid.

• The problem is not so much that the LHS is not continuous: it is defined on a discrete set (so in particular it is continuous!). Jul 29, 2010 at 3:00
• -1. I know it's been a long time, but I always write a note after a downvote. Here, the reason for downvoting is that, as Mariano remarked years ago, continuity is not the problem. Apr 6, 2023 at 9:20

Lets define what is x+x+x+... x times for x - real. Natural definition is x+x+x.. := x*x (note - just the same as Isaac has wrote in his edit).

Suppose we want left our initial definition as is. We don't know what is x+x+.. repeat x times for x - real (and note we don't have rule how to obtain derivative from such func). So lets use definition of derivative. f(x):=x+x+x.. repeat x times, Df(x)=(f(x+h)-f(x))/h, h->0. Df(x)=((x+h+x+h+x+h.. repeat x+h times) - (x+x+x.. repeat x times))/h, h->0. Suppose x+x+... repeat a+b times := (x+x+.. repeat a times) + (x+x+.. repeat b times) we have Df(x)=((x+h+x+h+x+h.. repeat x times) - (x+x+x.. repeat x times) + (x+h+x+h+x+h.. repeat h times))/h, h->0, Df(x)=((h+h+h.. repeat x times) + (x+h+x+h+x+h.. repeat h times))/h, h->0, or Df(x)=(1+1+1.. repeat x times) + (x+h+x+h+x+h.. repeat 1 times), h->0 and at last Df(x)=x + x+h, h->0 = 2x

Applying the sum rule in differentiation we get: $$\forall k \in \mathbb {N} \,(k\,f(x))' = k\,f'(x).$$ The symbol $$x$$ does not represent any value, it is simply a placeholder meaning that $$\forall x \,(kf)'(x)=(kf')(x).$$ I skip the mention that $$x\in X$$, where $$X \in dom(f)$$ to make it look more simple.

So we get $$\forall k \in \mathbb {N} \,\bigl( \forall x \,(kf)'(x)=(kf')(x)\bigr).$$

As we see now, we can't substitute $$k$$ with $$x$$, because $$x$$ is from the scope of the quantifier $$\forall x$$.

But we can substitute $$k$$ and $$x$$ with the same number, for example, with 4, and get the right state. We get $$(4f)'(4)=(4f')(4),$$ so $$(4id)'(4)=\textbf{4} (4)=4= (4\,\mathbf{1})(4)=(4\, id')(4),$$ where $$\textbf{1}$$ and $$\textbf{4}$$ are constant functions.

Here is a nice way to write this joke ...

$$x^2 = \underbrace{\color{red}{x} + \color{red}{x} + \color{red}{x} + \dots + \color{red}{x}}_{\text{repeated \color{blue}{x} times}}$$ To differentiate the right side, use the chain rule for partial derivatives. We get two terms. One where we differentiate the red $$x$$ in the top line, the other when we differentiate the blue $$x$$ in the bottom line: $$\underbrace{1 + 1 + 1 + \dots + 1}_{\text{repeated x times}}\quad +\quad \underbrace{x + x + x + \dots + x}_{\text{repeated 1 times}}$$ and this is of course the right answer $$\frac{d}{dx}(x^2) = x + x = 2x,$$

It has already been pointed out that “$$x$$ times” makes sense only for integer $$x$$, and several ways of salvaging the example for continuous $$x$$ have been given. Here is another informal interpretation that seems fitting for first-semester calculus. The continuous analog of a discrete sum is a definite integral. So let’s define adding up a constant quantity $$f(x)$$$$x$$ times” by the integral $$\int_0^x f(x) \; dt \,.$$ Differentiating* this yields $$\int_0^x f’(x) \; dt + f(x) \,.$$ For $$f(x)=x$$, we get $${d \over dx} \int_0^x x \; dt = \int_0^x 1 \; dt + x \,.$$ In words, the derivative equals $$1$$ added $$x$$ times plus $$x$$ added once, which is exactly what @GEdgar derived and similar to @arena-ru.

*There are two ways to explain the differentiation. One is through the product rule applied to $$\int_0^x f(x)\,dt=f(x) \int_0^x dt$$, and the other is through the multivariate chain rule applied to $$G(u,v)=\int_0^u v\,dt$$.

That differentiation step is invalid, because that equation is not actually differentiable, because its domain is comprised of isolated points, because $$x\in\mathbb Z_0^+;$$ to wit:

In general, if a theorem's condition(s) have not been fulfilled, then its consequence is not guaranteed.

P.S. A note on the above notation: What does $\mathbb{R}_0^+$ mean?

The problem is that the equation $x + \cdots + x = x^2$ only holds for two values of $x$, namely if you added $x$ with itself $n$ times, it holds at $x=0$ and $x=n$. Therefore your equation becomes $nx = x^2$, the differential is $n = 2x$ (the number of terms in the LHS of the equation should not depend on the real/complex parameter $x$) and the only thing you can deduce from this is that the first equation is equivalent to $x^2 - nx = x(x-n) = 0$ and the second equation is $x=n/2$, so when $x=0$ or $x=n$ then the sum of the $x$'s and $x^2$ are equal, and when $x=n/2$ the derivative of the sum and $x^2$ are equal. The deduction that $1=2$ is simply not true. The equality $nx = x^2$ is not comparable to an equality like $\sin x^2 = 1 - \cos x^2$ : the equation $nx = x^2$ holds for only two values of $x$, where as the trigonometric equation holds for any real/complex value of $x$.

Hope that helps,