On the Subject of Mathemetallics

Basic arithmetic just got metal!

For addition or subtraction do the following:

• Write the second number below the first so the digits at the same positions are aligned.
• For each position perform the operation on the pair of digits.
• Now use the fact that p^(n+1) = k*p^n + p^(n-1) to modify answer digits to get rid of negative digits and digits above k (in other words, if you add +/- 1 to a digit, add -/+ k to the digit to the right and -/+ 1 to digit 2 positions to the right respectively). Don't forget that each instance of k is supposed to be followed by 0.

In case the operation is multiplication or division, a TFC method is needed (which you can still use for addition/subtraction):

• Find p=(k+sqrt(k*k+4))/2
• Convert both displayed numbers to base 10 by adding up all monomials in the form of Cn*(p^n) where n is the power depending on the position of the displayed digit Cn (... 3rd 2nd 1st 0th . -1st -2nd -3rd ...) (in other words, this is the same as manually converting from any natural base to base 10, except the base is p (irrational), the maximum possible coefficient is k and negative powers are present (since the numbers are irrational)).
• Example: Numbers 50103.225 and 20310.22, k=5
  • p=5.19258240356725201562535524577
  • a = 5p^4 + p^2 + 3 + 2p^-1 + 2p^-2 + 5p^-3 = 3665.4510876609361165315349071071
  • b = 2p^4 + 3p^2 + p + 2p^-1 + 2p^-2 = 1540.5379081916960603438487583239
• Perform the operation on the received numbers to get the answer in base 10.
  • c = a + b = 5205.988995852632176875383665431
• To get the digits Cn to submit, follow the backwards process equivalent to converting base 10 to any natural base, except, as stated previously, p is irrational, maximum possible Cn is k and any such Cn equal to k must be followed by a 0.
• Keep in mind, while subtracting monomials in the backwards process, due to number irrationality, TFC will never show 0 when the desired coefficients are found, but instead will show a number with a magnitude of 10 zeroes after the period.
  • c - p^5 - p^4 - 5p^3 - 3 - 4p^-1 - 4p^-2 - 5p^-3 = -0.000000000000000000000000000028815
  • Therefore the answer is 115003.445