.. i thought it should be (2 ^^ 1) ^^ 2 = 4

On 22.10.2017 16:20, Ilya Yaroshenko wrote:
> .. i thought it should be (2 ^^ 1) ^^ 2 = 4

2 ^^ (1 ^^ 2) == 2

It is standard for ^/**/^^ to be right-associative. (This is also the standard convention in mathematics.)

On Sunday, 22 October 2017 at 14:44:04 UTC, Timon Gehr wrote:
> On 22.10.2017 16:20, Ilya Yaroshenko wrote:
>> .. i thought it should be (2 ^^ 1) ^^ 2 = 4
>
> 2 ^^ (1 ^^ 2) == 2
>
> It is standard for ^/**/^^ to be right-associative. (This is also the standard convention in mathematics.)

true

On Sunday, 22 October 2017 at 14:44:04 UTC, Timon Gehr wrote:
> On 22.10.2017 16:20, Ilya Yaroshenko wrote:
>> .. i thought it should be (2 ^^ 1) ^^ 2 = 4
>
> 2 ^^ (1 ^^ 2) == 2
>
> It is standard for ^/**/^^ to be right-associative. (This is also the standard convention in mathematics.)

Yeah, and a height-3 tower $a^{b^c}$ (TEX notation) actually means "a to the power of (b to the power of c)", not the other way around.  Otherwise, it can be written as $a^{b \cdot c}$, which is only a height-2 tower.

The convention also makes at least the following sense.  An expression like
(((a ^^ b) ^^ c) ^^ d) ^^ e
already has an almost bracket-free notation as
a ^^ (b * c * d * e).
So it is useful to have a bracket-free way to write the other-way-associative variant,
a ^^ (b ^^ (c ^^ (d ^^ e))).

Ivan Kazmenko.

On Sunday, 22 October 2017 at 22:28:48 UTC, Ivan Kazmenko wrote:
> Yeah, and a height-3 tower $a^{b^c}$ (TEX notation) actually means "a to the power of (b to the power of c)", not the other way around.

Because you have explicit braces there.

Math doesn't have precedence for exponentiation operator because it's written as a superscript, which is always unambiguous.

On Sunday, 22 October 2017 at 22:28:48 UTC, Ivan Kazmenko wrote:
> Yeah, and a height-3 tower $a^{b^c}$ (TEX notation)

Is $a^{b^c}$ the same as ${a^b}^c$ ? They are drawn slightly differently, so I suppose it's ambiguous indeed.

On Thursday, 26 October 2017 at 10:02:54 UTC, Kagamin wrote:
> On Sunday, 22 October 2017 at 22:28:48 UTC, Ivan Kazmenko wrote:
>> Yeah, and a height-3 tower $a^{b^c}$ (TEX notation)
>
> Is $a^{b^c}$ the same as ${a^b}^c$ ? They are drawn slightly differently, so I suppose it's ambiguous indeed.

Surely not the same.

"3 to the power of (3 to the power of 3)" is "3 to the power of 27", or 7,625,597,484,987.
"(3 to the power of 3) to the power of 3" is "27 to the power of 3", or 2187.

For an argument, the TEX command "^" accepts either a single character or a bracket-enclosed string of arbitrary length.  So $3^3^3$ indeed transforms to ${3^3}^3$, but not for some deeper reason this time.

Ivan Kazmenko.

On Saturday, 28 October 2017 at 00:14:15 UTC, Ivan Kazmenko wrote:
> For an argument, the TEX command "^" accepts either a single character or a bracket-enclosed string of arbitrary length.  So $3^3^3$ indeed transforms to ${3^3}^3$, but not for some deeper reason this time.

On my TeX compiler, $3^3^3$ makes it give a warning/error.

On Sunday, 22 October 2017 at 14:20:20 UTC, Ilya Yaroshenko wrote:
> .. i thought it should be (2 ^^ 1) ^^ 2 = 4

Imagine 2^^10^^10^^7. It's a big number, isn't? (up-up-and up) Where would you start from?