Math Mutation 64: Exponents Squared
Back in school, you probably recall that one of the earliest
arithmetic operations you learned was addition. Then the concept was
extended to multiplication: take a number x, and add it to itself a
bunch of times. A logical further step was exponents: take a number
x, and then multiply it by itself a bunch of times. And you probably
stopped there-- while your math classes went off in various other
directions, you didn't really learn any additional operator on this
'ladder'. But why should we stop there? If multiple repeated
additions become multiplication, and multiple repeated multiplications
become exponents, then what do multiple repeated exponents become?
It might surprise you to learn that there are futher operations in
this series, though they usually don't bother to teach them in
school. After exponentiation comes what is known as "tetration".
Tetration is usually symbolized with a number drawn to the upper left
of another number, as opposed to the upper right used for exponents.
Let's look at an example: what is 2 tetrated to the 4th? We would
draw this as a 2 with a little 4 to the upper left side. This is
equivalent to 2 to the 2nd to the 2nd to the 2nd power. Be careful
here-- we need to start expanding at the uppermost exponent, otherwise
we are effectively just multiplying the exponents together. So 2
tetrated to the 4th becomes 2 to the 2 to the 4th, or 2 to the 16th,
or 65536.
Needless to say, on positive whole numbers, tetration causes
values to grow incredibly fast. This probably helps explain why it's
not too practical in real life: after all, the simple operation of
exponentiation allows us to concisely express the number of atoms in
the universe, approximately 10 to the 80th power, or a 1 with 80
zeroes. In contrast, if we look at the relatively simple tetration
value or 10 tetrated to the 3rd, this is 10 to the 10 to the 10,
which is 10 to the 10 billionth power, or a 1 with ten billion zeroes
after it. I couldn't even begin describing 10 tetrated to the 80th in
any comprehensible form, other than just saying "10 tetrated to the
80th". It's hard to imagine a real-life application that needs such
immense numbers.
But even without real-life applications, there are some
mathematicians that find tetration a very interesting topic to study.
On small numbers, it can have some bizarre properties. For example,
get out a calculator and try calculating some tetrations of the square
root of 2. root-2 tetrated to the 2nd, or root-2 to the root-2 power,
is approximately 1.63. root-2 tetrated to the 3rd is aproximately
1.76. And as you increase the tetration, you will find that
bizarrely, root-2 tetrated to higher and higher values always gets
closer and closer to 2! So in effect, root-2 tetrated to infinity is
just 2, instead of growing huge like you would expect. In addition to
converging on small numbers, there has also been extensive study into
the properties of tetration on complex numbers, which leads to lots of
pretty multicolored pictures that you can see linked in the show
notes. There is actually a registered web domain, "tetration.org",
devoted entirely to the study of this operation, as well as some other
sites centered on this topic.
And, as you probably suspected already, tetration is just the
beginning. Mathematicians with lots of time on their hands have
defined further operations in the series: just as tetration is a
chain of exponents, an operation called "pentation" is a chain of
tetrations, "hexation" is a chain of pentations, and so on. This
general series of definitions is known as the "hyper-operation
sequence". As with tetration, none of these have any real-life
applications, as far as I can tell with a little web searching. But
has that ever stopped mathematicians before? This whole class of
operations, however, is actually a subset of a general operation that
is known as the "Ackermann Function", defined by German mathematician
Wilhelm Ackermann in the 1920's, which does have some significance in
theoretical computer science. So, at some level, these
hyper-operations do have a tenuous connection to reality.
And this has been your math mutation for today.
References:
The "Home of
Tetration"
Tetration.org
Tetration at
Wikipedia