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Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to be simplified with parentheses? Do people write maths like this?

a radical expression with the root being a radical expression? $$\sqrt[\sqrt{2^3}]{2}$$

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@GitGud $$\left(a^b\right)^c=a^{bc}$$ so by convention $$a^{b^c}=a^\left(b^c\right)$$ – whatever Jul 8 '13 at 18:37
@GitGud It'a a matter of convention, like $a\times b\div c$ which anyone does it (by convention) from left to right. The convention for $a^b^c$ is from right to left (up to bottom). – whatever Jul 8 '13 at 18:43
There is a Cuban writer, Onelio Jorge Cardoso, that in a tale made a humming bird say: "Things are not how they are called, but how we name them along the way." He said that to mama-bird who was complaining that her chicks were calling their nest a ship after throwing it into the river. Everyone uses your tower of exponentials and understands them as being $2^{(2^{(2^{3})})}$. So, it is not wrong. There is no wrong language, as long as it is understood. That is the ultimate purpose of language. – Mlazhinka Shung Gronzalez LeWy Jul 8 '13 at 18:43
@StefanH. introducing in the context of the particuar questions qualms about what formal properties the notation might or not have in category theory seems rather disingenuous! – Mariano Suárez-Alvarez Jul 8 '13 at 19:56
Another example is $a+b\cdot c$ which by convention is $a+(b\cdot c)$ but that's just a convention; there's no inherent reason why it could not be $(a+b)\cdot c$. And BTW, that we use "+" for addition and $\cdot$ (or $\times$) for multiplication is pure convention, too. There's nothing "additive" in "$+$" and nothing "multiplicative" in "$\cdot$" or "$\times$". – celtschk Jul 8 '13 at 20:04

5 Answers 5

up vote 21 down vote accepted

Your teacher is mistaken. There is a well-established and universal convention about the meaning of an expression like $$2^{2^{2^3}}$$it is always understood to mean $$2^{\left(2^\left(2^3\right)\right)} =2^{2^8} = 2^{256}$$ People can and do write expressions like these. For example this paper, "Analog of the Skewes Number for Twin Primes", by Marek Wolf, contains the expressions $$10^{10^{10^{10^3}}}\qquad\text{and}\qquad 10^{10^{529.7}}$$on the first page, with no further explanation. Similarly "Some Rapidly Growing Functions" by Craig Smoryński has $$10^{10^{10^{34}}} < e^{e^{e^{e^{4.369}}}}$$ and similar expressions. (I picked these two papers arbitrarily; they were the first two hits in Google Scholar for "Skewes' Number".)

There is a good reason for the convention about what $a^{b^c}$ means: $a^{b^c}$ could be understood as either $a^\left({b^c}\right)$ or as $\left(a^b\right)^c$. But if it were understood as $\left(a^b\right)^c$, one would never need to write $a^{b^c}$, since it would be equal to $a^{bc}$. So it is always understood as $a^\left({b^c}\right)$.

Nobody ever writes $$\sqrt[\sqrt{2^3}]2$$ even though its meaning is clear. Partly this is because it would have been difficult to typeset with old-fashioned metal type, so there is a tradition of expressing this differently. And partly it is because it looks bad.

Since by definition, $$\sqrt[a]b = b^{1/a},$$ one would almost always write something like $$(2^{1/2})^{1/2^{3/2}}$$ instead, at which point it would become clear that the expression could be simplified to $$2^{(1/2)(1/2^{3/2})} = 2^{1/2^{5/2}} = 2^{2^{-5/2}}.$$ Good notation enables and encourages this sort of simplification; bad notation obscures and impedes it.

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Isn't it $-3/2$ in the exponent? – Javier Jul 8 '13 at 19:03
@JavierBadia Thanks; I think I have it correct now. – MJD Jul 8 '13 at 19:09
Isn't it just $2^{1/2^{3/2}}$? Because $\sqrt[3]2=2^{1/3}\ne(2^{1/2})^{1/3}$. – Akiva Weinberger Oct 24 '14 at 15:51
Sorry, I don't follow your reasoning. There is no $\sqrt[3]2$ in my post. – MJD Oct 24 '14 at 15:56
Not sure where @columbus8myhw got the $\sqrt[3]{2}$, but I agree with her/him about $2^{1/2^{3/2}}$. By application of $\sqrt[a]{b}=b^{1/a}$, we have $\sqrt[\sqrt{2^3}]{2} = 2^{1/\sqrt{2^3}} = 2^{1/\left(2^{3/2}\right)}=2^{1/2^{3/2}}$. – Ben Blum-Smith Jul 1 at 18:06

Towers of exponents have been standard for ages. Cajori, in his book History of mathematical notations, §313, tells the story. He reproduces an image from a book by Waring published in 1785:

enter image description here

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It is sad that we don't find &c used like that anymore! $x^{x^{x^{\&c}}}$. – Mariano Suárez-Alvarez Jul 8 '13 at 19:58
A translation might be helpful. – Alraxite Jul 8 '13 at 20:42
@Alraxite, it is just an explanation about how to compute derivatives of such beasts: the purpose of the image here is just pictorical. – Mariano Suárez-Alvarez Jul 8 '13 at 20:45
@Alraxite it is Latin: enjoy its power and elegance even without translation :) – Avitus Jul 9 '13 at 10:20
+1 for the brave Latin scan! ("Onus probandi incumbit ei qui dicit") – Avitus Jul 9 '13 at 10:25

@Git Gud I think I'm starting to see the problem; exponentiation is right associative. Perhaps a more sensible notation (for $a^{b^c}$) would be $${}^{{}^c} {}^b a$$

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Like so: ${}^{{}^c} {}^b a$ – Daniel McLaury Oct 24 '14 at 21:55
Thank you Daniel, I just made the correction. – John Joy Oct 26 '14 at 11:24

$a^{\large b^{\Large c}}$ means $a^{\large(b^{\Large c}\large)}$. This is not in dispute.

Note that $\left(a^{\large b}\right)^{\large c}=a^{\large bc}$, so there would be no reason for it to mean this.

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The first thing you need to realize is that mathematics doesn't have notations on by itself, it's people who use notation to represent ideas.

Having said this, some common notations are incorrect, yes.

Your example ${{2^2}^2}^3$ is one such instance. Let us consider a simpler version: ${a^b}^c$. This can be read in two different ways, namely: ${(a^b)^c}$ or $a^{(b^c)}$. You can find examples in which the two symbols do not coincide and hence the notation is ambiguous (or incorrect if you prefer).

Another example of ambiguous notation is $2+ 3\div 5$, for the same reason: associativty fails, therefore you're not allowed to use the concatenation of the symbols $2,+,3,\div$ without parentheses.

As for $\sqrt[\large \sqrt{2^3}]{2}$, there's nothing wrong with it because, by definition $\sqrt[\large \sqrt{2^3}]{2}=\sqrt 2^{\left(1/\sqrt{2^3}\right)}$ and the RHS is well-defined.

Wise people will stick in enough parentheses to make it impossible for anyone to mistake the meaning - Gerry Myerson.


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I believe this is wrong. Specifically, $a^{b^c}$ is not ambiguous, any more than $2+3\cdot 4$ is, because there is a clear and well-established convention about what it means. Also, your example $2\cdot3\div5$ fails, because both $2\cdot(3\div 5)$ and $(2\cdot 3)\div5$ are equal. – MJD Jul 8 '13 at 18:51
There are people who have a descriptive view of syntax, and others who have a prescriptive view. Whatever direction one may lean, it is not reasonable to think of the other viewpoint as wrong. – André Nicolas Jul 8 '13 at 19:00
The notation $a^{b^c}$ has been standard pretty much from the time that the notation $a^b$ was invented... – Mariano Suárez-Alvarez Jul 8 '13 at 19:04
I would place your position on the prescriptive side of the spectrum. I am myself not prepared to put my body on the line for parentheses. – André Nicolas Jul 8 '13 at 19:08
I imagine the downvotes are due to the fact that your answer is wrong according to long established standards. – Mariano Suárez-Alvarez Jul 8 '13 at 19:10

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