02/12/2012, 10:16 PM
When I first tried to write down the tetrations of 2, I noticed how fast the numbers grew. It's quite easy to write down 2^^5 in scientific notation: ~2.0035*10^19728
Or like calculators do it: 2.0035e+19728
But the next number (2^^6) would be like: 10^(6.0312*10^19727)
And then you get more and more 10^(10^(10^... which gets quite tedious.
I know now there are other methods for writing down numbers beyond the scope of scientific notation but I made up my own notation method. I basically wanted to have something that extends upon scientific notation.
There are computer systems that use the "e+" or a (back)slash as a shorthand of the usual "*10^" and since the exponent is more significant than the mantissa I decided to write the aforementioned 2^^5 as follows with the exponent first:
19728\20035
Notice that I omit the dot. The exponent already implies the "2" in this number has a value of 2*10^19728. Unnormalized form is not allowed in my notation.
When the number gets bigger you can add a secondary exponent. For example:
3^^4 ~= 12\3638334640024\12580
And now to show you the real power of my backslash notation we can write the complete exponent stack, of which the top should be a single digit number, and add the super exponent:
3\\1\12\3638334640024\12580
The super exponent (3 in this case) tells us the height of the exponent stack. This number becomes necessary when the exponent is so big that you can't write it down fully, like with 2^^6:
3\\4\19727\6031226062630295
Note that the mantissa isn't displayed here because the primary exponent can't even be fully displayed, unless you have lots of space for it but even then calculating such a number to that high precision takes a long time.
If the super exponent ever gets too large you can extend it with its own exponents and super exponents. Make sure you also add a higher level hyper exponent with 3 backslashes. I even managed to write down G1 (3^^^^3, the first step in reaching Graham's number) in backslash notation:
3^^^^3 ~= 2\\\\2\\\2\\1\12\7625597484986\\1\12\3638334640023\60023
Graham's number is still too big for my backslash notation, though.
Or like calculators do it: 2.0035e+19728
But the next number (2^^6) would be like: 10^(6.0312*10^19727)
And then you get more and more 10^(10^(10^... which gets quite tedious.
I know now there are other methods for writing down numbers beyond the scope of scientific notation but I made up my own notation method. I basically wanted to have something that extends upon scientific notation.
There are computer systems that use the "e+" or a (back)slash as a shorthand of the usual "*10^" and since the exponent is more significant than the mantissa I decided to write the aforementioned 2^^5 as follows with the exponent first:
19728\20035
Notice that I omit the dot. The exponent already implies the "2" in this number has a value of 2*10^19728. Unnormalized form is not allowed in my notation.
When the number gets bigger you can add a secondary exponent. For example:
3^^4 ~= 12\3638334640024\12580
And now to show you the real power of my backslash notation we can write the complete exponent stack, of which the top should be a single digit number, and add the super exponent:
3\\1\12\3638334640024\12580
The super exponent (3 in this case) tells us the height of the exponent stack. This number becomes necessary when the exponent is so big that you can't write it down fully, like with 2^^6:
3\\4\19727\6031226062630295
Note that the mantissa isn't displayed here because the primary exponent can't even be fully displayed, unless you have lots of space for it but even then calculating such a number to that high precision takes a long time.
If the super exponent ever gets too large you can extend it with its own exponents and super exponents. Make sure you also add a higher level hyper exponent with 3 backslashes. I even managed to write down G1 (3^^^^3, the first step in reaching Graham's number) in backslash notation:
3^^^^3 ~= 2\\\\2\\\2\\1\12\7625597484986\\1\12\3638334640023\60023
Graham's number is still too big for my backslash notation, though.
