What should be the value of $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$?

Question

$\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$

Obviously, by using common-sense the answer is $0$. But I had thought of a different mathematical approach.

$Let:$

$x = \sqrt{0+\color{red}{\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}}$

Since the $\color{red}{red}$ part also symbolies/equals to '$x$', therefore we can rewrite this as:

$x = \sqrt{0 + x} \\ \text{Squaring, both the sides} \\ x^2 = 0 + x \\ x^2 - x = 0 \\ x(x-1) = 0 \\ \text{Hence, the solutions are 0 and 1}$

$1$ seems to be impossible, but it clearly satisfies my assuming 'x'. So is my assumption wrong ? or should is just ignore this fact.

Using color as a fundamental part of your notation is generally frowned upon, because of color-blind people and people using screen readers. — Thomas Andrews, Jul 16 '21 at 12:18
What you found was "If $x$ was a solution then $x=0$ or $x=1$" which is indeed a true statement. You are incorrectly reading the result however... it does not say that both $x=0$ is a solution and $x=1$ is a solution... it is saying that the solution happens to be among the numbers $0,1$ but it could yet be just a single one of those numbers... in particular the number zero. — JMoravitz, Jul 16 '21 at 12:20
"Let $a=0$. Then $a^2=a$, so $a=0$ and $a=1$ are the solutions. So $1=0$." — TheSilverDoe, Jul 16 '21 at 12:24
There are many examples (like this) where squaring an equation introduces “phantom solutions.” — Martin R, Jul 16 '21 at 12:24
As for proving it is zero... first, you must rigorously define the notation you are using. The most obvious and common way to define the infinitely nested radical notation is that $a_1 = 0,~a_2 = 0+\sqrt{a_1},~a_3=0+\sqrt{a_2},~\dots a_{n+1}=0+\sqrt{a_n},\dots$ However.... and here is an important realization... This is hugely depended on what we took $a_1$ to be equal to. If we were to have actually considered the sequence $\sqrt{1},\sqrt{0+\sqrt{1}},\sqrt{0+\sqrt{0+\sqrt{1}}},\dots,\sqrt{0+\sqrt{0+\dots+\sqrt{0+\sqrt{1}}}},\dots$ instead... we do get the limit equals $1$ — JMoravitz, Jul 16 '21 at 12:25
So, everytime, if i do 'Squaring both the sides' then I have to again verify if the both or any one actually satisfies my original equation ? — Saikat Das, Jul 16 '21 at 12:26
@JMoravitz: Your "most obvious" way is not the only way. To me at least, the obvious way to define $\sqrt{u_1+\sqrt{u_2+\sqrt{u_3+\cdots}}},,$ is by setting $a_1=\sqrt{u_1}, a_2=\sqrt{u_1+\sqrt{u_2}},a_3=\sqrt{u_1+\sqrt{u_2+\sqrt{u_3}}},\ldots$ and taking the limit if it exists. And this way you get unambiguously $0$, without any unwarranted assumptions about $a_1$. — TonyK, Jul 16 '21 at 12:45
...In fact this appears to be the usual definition. See here. — TonyK, Jul 17 '21 at 09:41

score 4 · Accepted Answer · answered Jul 16 '21 at 12:42

There are two useful responses that come to mind, both of which appeared in comments as I was typing this:

Firstly, not all implications go both ways. As @TheSilverDoe said in a comment, just because a number has some property does not mean it is the only number with that property. Squaring an equation is an example which can introduce alternative solutions as that comment shows: $x = 0 \implies x^2 = x$ which has solution $x=1$ too. Another classic error is multiplying by $0$; $x = 5 \implies 0 \times x = 0$ which is solved by $x = 7$ as well as $x=5$.
Secondly, as @JMorawitz emphasises, you have to make sure that mathematical expressions (with unique values you can work out) are well-defined objects, rather than being ways of defining equations which may have multiple solutions. (There's a hazy grey-area of multivalued functions, too, but let's not worry about handling these.) With infinite expressions, "things with dots in", this is particularly likely to be an issue. The usual mathematical formulation of infinite nested expressions (that avoids any confusion) is given by defining the result as the limit of a sequence. The above comment highlights one standard way to give a clear definition of what your expression means, namely we take a sequence $$a_{n+1} = 0 + \sqrt{a_n}$$ along with some initial value $a_0$, and ask what the limit of $a_n$ as $n\to\infty$ is. This depends on the initial value $a_0$. In fact, $$ \lim_{n\to\infty} a_n = \begin{cases}0 & \text{if }a_0 = 0 \\ 1 & \text{if }a_0 \neq 0\end{cases} $$ So you need to clearly define what you mean, perhaps by adopting a convention like $a_0 = 0$ since that's what you get if you 'stop writing roots' when 'filling in the dots'.

What should be the value of $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$?

1 Answers1