Mathematics & Computing — TNT Code Tutorial

Computing Digits of π — in Python and JavaScript

How to extract hundreds of decimal digits of π using two different approaches — one using Python’s mpmath library, the other using JavaScript’s BigInt and Machin’s formula from scratch.

Python Version ↓ JavaScript Version ↓

Why Not Just Use Math.PI?

Every programming language gives you π built in. Python has math.pi. JavaScript has Math.PI. So what’s the problem? The problem is precision — or rather, the lack of it.

Math.PI and math.pi both give you 15–16 significant decimal digits of π:

For our randomness test, we needed 600 digits — 40× more than what the built-in constant can provide.

Why Floating-Point Has a Ceiling

A 64-bit “double” (the standard in C, Java, JavaScript, Python’s float) stores a number as: 1 sign bit + 11 exponent bits + 52 mantissa bits. Those 52 bits give you about 15–16 significant decimal digits — no matter what number you store. π has infinitely many non-repeating decimal digits, so after position 16 the double just… stops. It has no more bits to spend.

Getting 600 digits requires a completely different approach: either a library that uses arbitrary-precision arithmetic (Python’s mpmath), or implementing a formula that works in pure integers and never touches floating-point at all (JavaScript’s BigInt).

The Math Behind It — Machin’s Formula

You’ve already seen the key ingredient in BC Calculus: the Taylor series for arctangent. Machin’s stroke of genius in 1706 was finding a clever way to make that series converge fast enough to be practical.

a) The arctangent Taylor Series

From BC Calc, you know that many functions can be written as infinite polynomials. The arctangent function has this Taylor series:

The special value we care about: arctan(1) = π/4, because tan(π/4) = 1. So π = 4 · arctan(1). The problem? Plugging x = 1 into the series means each term is only 1/3 smaller than the last — you’d need thousands of terms for 600 digits.

b) Machin’s Formula (1706)

John Machin discovered this algebraic identity:

Why Does Machin’s Formula Work?
It’s a proven algebraic identity — you can verify it by computing tan(4 · arctan(1/5) − arctan(1/239)) using angle-subtraction formulas and showing the result equals exactly 1 (so the expression equals π/4). Machin used it in 1706 to hand-compute 100 digits of π.
Why Does mpmath Use Something Else?
Python’s mpmath library uses the AGM (Arithmetic-Geometric Mean) algorithm, which converges quadratically — doubling the number of correct digits with each iteration. For millions of digits, AGM is far more efficient than Machin. But Machin is conceptually accessible and is exactly what the JavaScript version implements.

Python Version — mpmath

The Python approach is the “library” solution: tell mpmath how many decimal places you want, ask for mpmath.pi, and convert the result to a list of digits. The library handles all the advanced arithmetic internally.

Python piDigits.py
    Requires Python 3 and mpmath. Run locally, or paste into Google Colab (mpmath is pre-installed there).
    Try It In Your Browser
    No installation needed — paste the code above directly into a free cloud Python environment:
    • Google Colab — mpmath is already installed. Paste into a code cell and press Shift+Enter.
    • Repl.it — create a new Python repl, paste, run. Install mpmath once with pip install mpmath in the Shell tab.
    ℹ️ About the Copy Button: The Copy button copies from the raw JavaScript variable, not from the displayed panel. This ensures no HTML markup, keyword-highlight spans, or HTML entity characters are included — you get exactly the Python source text. Paste directly into your editor or Colab cell.

    JavaScript Version — BigInt + Machin’s Formula

    The JavaScript approach implements Machin’s formula directly, using BigInt for exact integer arithmetic. No library needed — this runs in any modern browser console or Node.js. It’s longer than the Python version, but every step maps directly to the calculus you already know.

    JavaScript piDigits.js
      Runs in any modern browser. No installation — see the “Try It” note below.
      Try It In Your Browser — Right Now
      Zero installation required. Open any browser, press F12 (or right-click → Inspect), click the Console tab, paste the entire code above, and press Enter. You should see Count: 600 and First 10: [1, 4, 1, 5, 9, 2, 6, 5, 3, 5] printed to the console, followed by the 600 comma-separated digits ready to paste into the Loaded Dice Analyzer.
      ℹ️ About BigInt: JavaScript’s BigInt type (introduced in ES2020) stores integers of arbitrary size — no upper limit. BigInt literals are written with an n suffix: 1n, 5n, 239n, 10n**620n. BigInt arithmetic is always exact; unlike floating-point, it never rounds. That’s the property we need to guarantee 600 correct digits.

      Two Approaches, One Answer

      Python asks a library to do the hard work. JavaScript builds the formula from scratch. Both produce identical digits. Which should you use? It depends on what you want to learn.

      Python — mpmath JavaScript — BigInt
      Approach Library does the work (AGM algorithm) You implement Machin’s formula
      Precision limit Unlimited — set mp.dps to any value Unlimited — BigInt integers grow as needed
      Lines of code ~20 (excluding comments) ~60 (excluding comments)
      Educational value See the big picture; trust the library Understand the math; every step is visible
      Where to run Python 3, Google Colab, IDLE Browser console (F12), Node.js

      The Sanity Check — Do Your Results Agree?

      Both functions should produce exactly the same 600-digit sequence. Run both and compare the first ten digits:

      If the Python and JavaScript outputs agree to all 600 digits, you have strong evidence that both implementations are correct — two completely different algorithms, two completely different languages, one answer. That kind of cross-verification is exactly how mathematicians and engineers build confidence in results that can’t be checked by hand.

      Once you have the 600-digit sequence, paste it into the Loaded Dice Analyzer (model 0–9, 600 rolls) and run all four randomness tests. Pi’s digits look random — but can they fool four statistical tests? That’s the question the Analyzing Random Integers guide explores.