endless iteror Ψ #2
Frances K*, 2016
© Kreative commons
We will unravel how decimal notation works, see how large numbers are handled in history, and explore the boundaries of mathematics.
2.1 Radix notation
The numbers you use daily are in the decimal system.
This is a type of radix notation
with number base
Although in any radix its base is written as
Here we show numbers in decimal base on an orange background.
You've forgotten (internalized) how this actually works, it's complicated.
The base determines a set of digits in the range
to multiply the powers of the base with
and add that in series.
This way a word composed of digits equals a unique number
Show the popular radices, both in decimal notation (orange) and by simple counting of ones (yellow).
2 =: 11binary
8 =: 11111111octal
10 =: 1111111111decimal base
16 =: 1111111111111111hexadecimal
We use the sign
=: to change between formats.
Click @ to put the digits of each radix on the board.
After you agree on a base, apply repeated multiplication to it.
Index each position
by a radix power, starting right with
and put larger powers
10^p on the left
(Arabic writing direction).
On each numeral place you can write a digit, that multiplies its radix' power. Then add up that series to get your number, as below.
2016 = 1*6+
2015 = 1*7+
8^3*3 =: 3737in base
2020 = 1*0+
2^10*1 =: 11111100100
2018 = 1*2+
=: 7e2in base
Radix notation is optimal,
because we can uniquely express all natural numbers up to any
n within a minimum word space.
Any other system with a set of characters of radix size
will do worse.
This goes for all natural number bases,
but real bases are possible too. Try them in the above App:
10 in base
2 show increasing overlap
as they approach the lower limit of
A growing proportion of their digit series expresses the same numbers.
So radices in the range
could model the overlap produced by arithmetical systems
2.2 Trends in history
The mumbo-jumbo of radix notation soon became automated, else society would now be crippled. What luck that children in the ages before social media were able to learn their elementary operations in decimals. We stand on the shoulders of midgets, as well as giants!
The ancient Egyptians had eight numeral hieroglyphs
for the powers of ten up to
that were directly added in any position to depict numbers
and baffle the crowd.
The Romans some
years ago had just those seven letters to work with.
Yet the digit concept was already present in speech.
quingenti in latin,
octo milia equorum.
The left position of decimal digits is most important.
But powers of ten elude the human psyche after a few
Bankers who believe Reaganomics will last, choose to ignore the impact of compound interest and the large debt that produces.
Any growth must stop somewhere, but exponential growth rapidly becomes unsustainable.
For example. If a family of two keeps growing at a general annual rate of
it will be larger than the current world population.
Enter that on a calculator.
in scientific notation
means that you multiply the decimal factor on its left by
to get the number, or often an approximation.
A percentage sign
is equivalent to an exponent
on the factor.
Multiply the current quantity with that number to get the increase.
Add the increase to the quantity, to get the next total.
2.3 Physical limits
Radix notation seems more economical than
notation, but the extra digit signs employed
for number input/output
do not help the operations in the throughput.
If your goal is to generate extreme numbers,
without regard for the unnumbered gaps in between:
then using just the character
1 is most frugal,
an easy win.
10^^r1 for example.
A unary number
is written in exactly that many places.
In radix notation this space is a power of
10 smaller, that is
10^^r digit places.
This reduction in word size is sufficient to express only the simplest tetrations, given the resources of our physical universe.
3^^3 = 3^3^3 = 3^27
4^^3 = 4^4^4 = 4^256
≈ 1E154exponential notation
2^^5 = 2^2^2^2^2 = 2^65536
≈ E2E4double exponential notation
3^^4 = 3^7625597484987
≈ EEE1multi-exponential notation
Basically, a power of
10 less subtracts
from the tetration iterator.
But even in exponential notation, to approximate moderate input size tetrations, your output becomes too long. Worse is, by reducing your system's precision most numbers will escape the net.
Because we are physical beings we cannot uniquely express the majority of the numbers in the arithmetical sea between the scattered islands of tetration. Even with the help of computers we can only point out a small portion of these illusive numbers in the rather random choice of systems we can make.
the universe as a computer has at most
2^300 binary places.
Then the radix representation of tetrations like
by far defeats any known physical data capacity.
The human mind can envision even larger constructs: check out the
in buddhist poetry.
Moving higher, we can still contemplate the strength of our rules, but the big numbers they produce lie wholly beyond imagination…