Real Rationals

• Compare iOS calculator to Android calculator.

The purpose of a calculator app is to give correct answers. Floating point numbers are inherently imprecise, so a calculator that relies entirely on floating point arithmetic is like a house built on sand.

Representing numbers

• Most numbers cannot be precisely represented by floating point values, and must be rounded.
• Simple operations like summation of floating point values requires careful algorithms to handle error

Big numbers

• JavaScript BigInt solves this problem for integers
• Represent a fraction as two BigInt values
• How about representing pi or sqrt(2)?

Algebraic numbers

• Represent numbers by polynomial, so sqrt(2) is x^2 - 2 = 0
• Math operations are a bit trickier (add polynomials)
• Multiply with polynomial composition and resultants
• Still only works for algebraic numbers, does not get us pi

Create a diagram here:

• Image of hierarchy (natural (N) - integer (Z) - rational (Q) - real algebraic (A_R) - real (R) )
• Highlight irrational (real algebraic + real) and transcendental (Real)
• Real examples: e, pi, -2pi, cos(theta)

Rational real numbers

• link to mark bridger book

Constructive real numbers are numbers which can be computed to an arbitrary degree of accuracy.

You can't give me every single digit of , but you can define a function which gives me the value of that is within of the true value.

code.js
1// Leibniz formula: pi/4 = 1 - 1/3 + 1/5 - 1/7
2function pi(tolerance) {
3    let sum = 0;
4    let i = 0;
5    let term = 1;
6
7    while (Math.abs(term) > tolerance) {
8        term = 1 / (2 * i + 1);
9        sum = i % 2 == 0 ? (sum + term) : (sum - term);
10        i++;
11    }
12
13    return sum * 4;
14}