\(2^{{2}^{1}}=2^2=4\)
I saw this question posed in Quora and it evoked hundreds of responses.
The term used in this context is tetration which Wikipedia explains as follows:
Addition
\(a + n = a + \underbrace{1 + 1 + \cdots + 1}_n\)
\(n\) copies of 1 added to \(a\).
Multiplication
\(a \times n = \underbrace{a + a + \cdots + a}_n\)
\(n\) copies of \(a\) combined by addition.
Exponentiation
\(a^n = \underbrace{a \times a \times \cdots \times a}_n\)
\(n\) copies of \(a\) combined by multiplication.
Tetration
\({^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n\)
\(n\) copies of \(a\) combined by exponentiation, right-to-left.
Here is a proper definition:
For any positive real number \(a>0\) and non-negative integer \(n \ge 0 \), we define \(\,\! {^{n}a} \) by:$${^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{if }n>0 \end{cases} $$As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). For example:$$\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left[2^{\left(2^2\right)}\right]} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536$$Note that exponentiation is not associative, so evaluating the expression in the other order will lead to a different answer:$$\,\! 2^{2^{2^2}} \ne \left[{\left(2^2\right)}^2\right]^2 = 2^{2 \cdot 2 \cdot2} = 256$$Exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as right-associative.
Source: https://en.wikipedia.org/wiki/Tetration
Note: there's nothing very original about this post. I've copied the rather involved LaTeX code from the Wikipedia editing page, modified it by getting rid of the maths arrow brackets and replaced them with \( signs. I was mainly interested to see how the LaTeX would convert and as it turned out the transfer from Wikipedia to Blogger was quite straightforward.
The term used in this context is tetration which Wikipedia explains as follows:
In mathematics, tetration (or hyper-4) is the next hyper-operation after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra-(four) and iteration. Tetration is used for the notation of very large numbers.
The notation \({^{n}a}\) means \({a^{a^{\cdot^{\cdot^{a}}}}}\), the application of exponentiation \(n-1\) times.Shown below are the first four hyperoperations, with tetration as the fourth:
Addition
\(a + n = a + \underbrace{1 + 1 + \cdots + 1}_n\)
\(n\) copies of 1 added to \(a\).
Multiplication
\(a \times n = \underbrace{a + a + \cdots + a}_n\)
\(n\) copies of \(a\) combined by addition.
Exponentiation
\(a^n = \underbrace{a \times a \times \cdots \times a}_n\)
\(n\) copies of \(a\) combined by multiplication.
Tetration
\({^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n\)
\(n\) copies of \(a\) combined by exponentiation, right-to-left.
Here is a proper definition:
For any positive real number \(a>0\) and non-negative integer \(n \ge 0 \), we define \(\,\! {^{n}a} \) by:$${^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{if }n>0 \end{cases} $$As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). For example:$$\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left[2^{\left(2^2\right)}\right]} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536$$Note that exponentiation is not associative, so evaluating the expression in the other order will lead to a different answer:$$\,\! 2^{2^{2^2}} \ne \left[{\left(2^2\right)}^2\right]^2 = 2^{2 \cdot 2 \cdot2} = 256$$Exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as right-associative.
Source: https://en.wikipedia.org/wiki/Tetration
Note: there's nothing very original about this post. I've copied the rather involved LaTeX code from the Wikipedia editing page, modified it by getting rid of the maths arrow brackets and replaced them with \( signs. I was mainly interested to see how the LaTeX would convert and as it turned out the transfer from Wikipedia to Blogger was quite straightforward.
No comments:
Post a Comment