## Journey into PI

### How to compute $$\pi$$ with a billion digits

Introduction

To achieve a given accuracy, all calculations must be made with precision at least equal to that of the final result.

That is, to obtain $$\pi$$ with a billion significant digits, all operations, multiplication, division, addition and subtraction, must be performed with arguments having at least a billion digits, so that the result of these operations have that precision.

Thus, the choice of the algorithm is very important: it must converge as quickly as possible to the result with the lowest possible number of operations.

Let us now consider a calculator that is only capable of handling numbers up to $$99$$. However, to make multiplications, this calculator can use two registers and thus, represent results up to $$9999$$.

With these limitations, how can it compute $$12345678\times87654321$$?

Using the features of our calculator we can organize the operations as follows:

$\begin{array}{rrrrrrrr} &&&& 12 & 34 & 56 & 78\\ &&&\times& 87 & 65 & 43 & 21\\ \hline &&&& 21 \times 12 & 21 \times 34 & 21 \times 56 & 21 \times 78\\ &&& 43 \times 12 & 43 \times 34 & 43 \times 56 & 43 \times 78\\ && 65 \times 12 & 65 \times 34 & 65 \times 56 & 65 \times 78\\ & 87 \times 12 & 87 \times 34 & 87 \times 56 & 87 \times 78\\ \hline &&&& 252 &714 &1176 &1638\\ &&&516 & 1462 & 2408 & 3354\\ && 780 & 2210 & 3640 & 5070\\ & 1044 & 2958 & 4872 & 6786\\ \hline & 1044 & 3738 & \raise 1.6ex {{\small 100} \atop 7598} & 2140 & 8192 & 4530 & 1638\\ \hline \raise 1.6ex {{\small 10} \atop 0} & \raise 1.6ex {{\small 38} \atop 1044} & \raise 1.6ex {{\small 77} \atop 3738} & \raise 1.6ex {{\small 22} \atop 7698} & \raise 1.6ex {{\small 82} \atop 2140} & \raise 1.6ex {{\small 45} \atop 8192} & \raise 1.6ex {{\small 16} \atop 4530} & 1638\\ \hline 10 & 82 & 15 & 20 & 22 & 37 & 46 & 38 \end{array}$

We managed to get $$12345678\times87654321= 1082152022374638$$, thus overcoming the limitations imposed by the calculator.