(12C) Base Conversion

[Thomas Klemm]

This program is a slightly shorter version of the 2nd solution that uses the function NPV:

Code:

01 -    44 13  STO PV
02 -       36  ENTER
03 -    45 15  RCL FV
04 -       10  ÷
05 -    43 25  INTG
06 -    43 13  CFo
07 -    45 15  RCL FV
08 -       10  ÷
09 -    43 25  INTG
10 -    43 35  x=0
11 - 43,33 14  GTO 14
12 -    43 14  CFj
13 - 43,33 07  GTO 07
14 -    45 14  RCL PMT
15 -        1  1
16 -       24  Δ%
17 -    44 12  STO i
18 -    45 13  RCL PV
19 -    42 13  NPV
20 -    45 14  RCL PMT
21 -    45 15  RCL FV
22 -       30  −
23 -       20  ×
24 -       40  +

Example: 2014₁₀ → 3736₈
10 STO PMT
8 STO FV
2014 R/S

Formulas Used

Net Present Value

NPV = net present value of a discounted cash flow.
CF_j = cash flow at period j.

\(NPV=CF_0+\frac{CF_1}{(1+i)^1}+\frac{CF_2}{(1+i)^2}+\cdots+\frac{CF_n}{(1+i)^n}​\)

Percentage

\(\Delta\%=100\ \frac{x-y}{y}\)

Explanation

We can calculate the remainders by dividing the number continuously by the base we want to transform to (here stored in FV):

Code:

2014 ÷ 8 = 251  →  6 = 2014 - 8 × 251
 251 ÷ 8 =  31  →  3 =  251 - 8 ×  31
  31 ÷ 8 =   3  →  7 =   31 - 8 ×   3
   3 ÷ 8 =   0  →  3 =    3 - 8 ×   0

These remainders have to be multiplied by powers of the base we transform from (here stored in PMT):

Code:

   6 = 1 × 2014 - 8 × 1 ×  251
  30 = 10 × 251 - 8 × 10 ×  31
 700 = 100 × 31 - 8 × 100 ×  3
3000 = 1000 × 3 - 8 × 1000 × 0

And then when we add up all the terms we get:

Code:

3736 = 2014 + 2 × (251 + 10 × 31 + 100 × 3)

To calculate the polynomial we use NPV.

For this to work we have to solve for \(i\) in:

\(1+\frac{i}{100}=\frac{1}{b}\)

where \(b\) is the base we transform from.

Thus

\(\begin{align*}
i &= 100\ (\frac{1}{b}-1)\\
&= 100\ \frac{1-b}{b}
\end{align*}\)

That's why we can use the \(\Delta\%\) function to calculate \(i\):

Code:

14 -    45 14  RCL PMT
15 -        1  1
16 -       24  Δ%
17 -    44 12  STO i

I was a bit surprised that ENTER is needed here:

Code:

01 -    44 13  STO PV
02 -       36  ENTER
03 -    45 15  RCL FV

But it appears that STO PV disables the stack lift.

From the manual:

Quote:In addition, the stack does not lift when a number is entered if the last operation performed was storing a number into a financial register.