What is Tetration?
Tetration is iterated (repeated) exponentiation. The exponent “b” tells you how many times to exponentiate the base. It is the fourth in a sequence of basic arithmetic operations:
- Multiplication (repeated addition),
- Exponentiation (repeated multiplication),
- Tetration (repeated exponentiation).
Pentation is the fifth operation (repeated tetration), and hexation (repeated pentation) is the sixth. The entire sequence is called the hyper-operation sequence.
Connection to Exponentiation
The idea is similar to exponents. For example, the exponent 34 is written as:
34 = 3 * 3 * 3 * 3 = 81.
The superscript (in this example, 4) tells you how many times to multiply out the base. The exponent in tetration (called the “height) tells you how many times to exponentiate the base. So, for 43 you take the base (3) and iteratively exponentiate it three times (giving a total of four “3”s in the equation):
43 = (3333) = 37.626 (to 3 decimal places).
Examples of How to Solve
To solve, simplify from the innermost nest first.
- 3[3(33)] =
- 3(327) =
- 3(327) = 37.626 (to 3 decimal places).
Make sure you start expanding at the uppermost (right) exponent, otherwise the result is just the multiplication of exponents, not exponentiation of exponents.
- 42 =
- 2222 =
- 224 =
- 216 =
Tetration Function Notation
The tetration function is a family of functions that undergo tetration. Tetration is denoted by ba, which is almost the same as the exponential function— except the exponent is to the left of the base.
The general form of the tetration function is:
tn(x) = n,x
Where n is the order of tetration.
Is there a Real Life Use?
For all practical purposes, no. Which is probably why it isn’t taught in school alongside addition and multiplication.
These calculations often result in trillions of zeros. Most calculators can’t handle that many digits, and so will give you an error.
To put this in perspective, 310 = 101010 which, if you wrote out all of the digits, results in 1 followed by ten billion zeros. Compare that to the number of atoms in the universe, which has a 1 with 80 zeros. Additionally, tetration has a few odd properties. For example, √2 tetrated to infinity equals 2 (Seligman, 2016).
Due to the problems with calculating the trillions of digits, you’ll probably only ever work with a second order (2x) or third order (3x) tetration function.
Lynch, P. The Fractal Boundary of the Power Tower Function.
Neyrinck, M. An Investigation of Arithmetic Operations. Retrieved November 26, 2019 from: http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf
Seligman, E. (2016). Math Mutation Classics: Exploring Interesting, Fun and Weird Corners of Mathematics. Apress.