7

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? $$\sqrt[\sqrt{2^3}]{2}$$

Ali Caglayan
  • 5,726
hit
  • 549
  • All those notation are correct, if a bit confusing. But there exist incorrect notations, of course. – Javier Jul 08 '13 at 18:23
  • @JavierBadia powers are not correct notation. It's a very bad convetion since powering is not associative. – Git Gud Jul 08 '13 at 18:27
  • 9
    @GitGud $$\left(a^b\right)^c=a^{bc}$$ so by convention $$a^{b^c}=a^\left(b^c\right)$$ – user5402 Jul 08 '13 at 18:37
  • @metacompactness I'm giving you ${a^b}^c$ wich version do I mean? (Ignore the fact that $\LaTeX$ makes it easy to guess due to the size of the characters). – Git Gud Jul 08 '13 at 18:39
  • 3
    @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). – user5402 Jul 08 '13 at 18:43
  • 6
    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. – OR. Jul 08 '13 at 18:43
  • @metacompactness: Be careful about $(a^b)^c=a^{bc}$. While this is true for numbers, it may not be true if $a,b,c$ denote some objects in a category. For example in Top the notion $a^b$ refers to the space of continuous maps from space $b$ to space $a$, equipped with a suitable topology. Then this identity need not hold, where $bc$ can be read as $b\times c$, the topological product. I still agree with $a^{b^c}=a^{(b^c)}$, since the $c$ would be lower if it were the power for $a^b$. – Stefan Hamcke Jul 08 '13 at 18:45
  • @metacompactness: Not an ideal example given that $(a\times b)\div c = a\times (b\div c)$. On the other hand, writing $a\div b\times c$ is bad (because of ambiguity), whatever someone says is the convention. Better notation and parentheses exist. (I agree about the standard meaning of $a^{b^c}$.) – Jonas Meyer Jul 08 '13 at 19:44
  • @StefanH. You're right in that this notation is ambiguous in more general settings but the OP is about (real) numbers. – user5402 Jul 08 '13 at 19:52
  • 1
    @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-Álvarez Jul 08 '13 at 19:56
  • 2
    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 08 '13 at 20:04
  • @metacompactness: I know that the OP is (most likely) about numbers. I just thought that someone who reads your comment might misunderstand it as saying that in all fields of maths the expression $a^{b^c}$ can never mean $(a^b)^c$ because then it would be written as $a^{bc}$. At least I did. But then again, I'm totally into category theory lately :-) – Stefan Hamcke Jul 08 '13 at 21:25
  • The notation that I found most problematic is the use of $\sin^{-1}(x)$ to mean $\arcsin (x)$ (and the other trig ratios, as well.) – DJohnM Jul 08 '13 at 21:17

4 Answers4

25

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.

MJD
  • 65,394
  • 39
  • 298
  • 580
  • 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
  • 1
    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 01 '15 at 18:06
18

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

Mariano Suárez-Álvarez
  • 135,076
3

$a^{\large b^{\Large c}}$ 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.

Akiva Weinberger
  • 23,062
1

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

amWhy
  • 209,954
John Joy
  • 7,790