Confused by this: What does 10^10^79 mean?

[Simon Peach]

I was reading a book on black holes by Kip Thorne the other day, well weeks really, and I came across this in one of the footnotes 10^10^79 (10 to the power of 10 to the power of 79) And I really don't know what it means. Does it just mean what it says? If it does can you raise a power to a power?

 

[fresh_42]

I was reading a book on black holes by Kip Thorne the other day, well weeks really, and I came across this in one of the footnotes 10^10^79 (10 to the power of 10 to the power of 79) And I really don't know what it means. Does it just mean what it says? If it does can you raise a power to a power?

It is possible, although a bit sloppy, because $2^{(3^2)} = 2^9 = 512 \neq (2^3)^2 = 8^2 = 64$, so there should better be brackets to avoid ambiguity. Without them, it probably means $10^{10^{79}}=10^{(10^{79})}$ a one with $10^{79}$ zeroes for otherwise one would have written $10^{790}$ instead.

 

[Simon Peach]

It is possible, although a bit sloppy, because $2^{(3^2)} = 2^9 = 512 \neq (2^3)^2 = 8^2 = 64$, so there should better be brackets to avoid ambiguity. Without them, it probably means $10^{10^{79}}=10^{(10^{79})}$ a one with $10^{79}$ zeroes for otherwise one would have written $10^{790}$ instead.

Thanks fresh_42, I sort of thought that but it's good to have it confirmed by someone that knows a bit more maths than I do

 

[jedishrfu]

If you think that’s bad you should look into Knuth arrow notation:

https://en.m.wikipedia.org/wiki/Knuth's_up-arrow_notation

 

[Simon Peach]

If you think that’s bad you should look into Knuth arrow notation:

https://en.m.wikipedia.org/wiki/Knuth's_up-arrow_notation

Just have wish I hadn't!

 

[Mark44]

With regard to operators, maybe someday mathematicians will get their acts together as well as the people who design programming languages. All of the languages based on C have tables of precedence and associativity, which enables one to unequivocally state how some expression will be evaluated. Although C, C++, C#, and Java don't have an exponentiation operator, Python does have such an operator -- ** -- the same as Fortran uses.

The Python docs say that this operator associates right-to-left, which means that 2 ** 3 ** 3 is the same as if written as 2 ** (3 ** 3). Most of the other operators associate left-to-right. Using a more mathematical notation, this would be $2^{3^{3}}$, which if evaluated using the Python rules would be the same as if written $2^{(3^3)}$.

We have PEDMAS, (or BEDMAS/BODMAS, for those who can't distinguish between parentheses, braces, brackets, and angle brackets) which gives the relative precedence of the different categories, with parentheses at the highest level, but this convention doesn't also provide information about how an expression with more than two operands associates, or groups.