The square root of 2…

The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. 

There are a number of algorithms for approximating √2 as a ratio of integers or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the Babylonian method for computing square roots, which is one of many methods of computing square roots. It goes as follows:

First, pick a guess, a₀ > 0; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:

\(a_{n+1}=\frac{a_{n}+\frac{2}{a_{n}}}{2}=\frac{a_{n}}{2}+\frac{1}{a_{n}}\)

In general, suppose we want to evaluate the square root of a number n. Let’s assume the \(n=a^2+b\), where \(a^2\) is the biggest square number that is less or equal to \(n\).

Then, we can write it as:

\(n-a^2=b\)

or:

\(\left( {\sqrt n + a} \right)\left( {\sqrt n – a} \right) = b \Leftrightarrow \sqrt n – a = \frac{b}{{\sqrt n + a}}\)

and then:

\(\sqrt n = a + \frac{b}{{\sqrt n + a}}\)

Recursively replacing the \(\sqrt{n}\):

\(\sqrt n = a + \frac{b}{{2a + \frac{b}{{\sqrt n + a}}}}\)

or:

For \(n=2\) we have \(a=1\) and \(b=2-1=1\). So:

In general, we will write the sequence:

\({a_{n + 1}} = 1 + \frac{1}{{{a_n} + 1}}\)

In Python:

import numpy as np
import decimal

def twoSqrt(EPS):
    decimal.getcontext().prec = 10000
    
    x = decimal.Decimal(1)
    t = x
    # while error is still big
    while abs(x*x - decimal.Decimal(2)) > EPS:
        x = decimal.Decimal(1) + decimal.Decimal(1)/(decimal.Decimal(1)+t)
        # converge the answer
        t = x
    return x

EPS = decimal.Decimal(1e-200)

r1 = twoSqrt(EPS) 
r2 = decimal.Decimal(2).sqrt()

print('%.19f' % r1)

---

def twoSqrt(EPS):
    decimal.getcontext().prec = 10000
    
    myList = []
    myCount = []
    
    x = decimal.Decimal(1)
    t = x
    i = 0
    myList.append(x)
    myCount.append(i)
    # while error is still big
    while abs(x*x - decimal.Decimal(2)) > EPS:
        x = decimal.Decimal(1) + decimal.Decimal(1)/(decimal.Decimal(1)+t)
        # converge the answer
        myList.append(x)
        i = i+1
        t = x
        myCount.append(i)
    plt.xlabel("iterations")
    plt.ylabel("$\sqrt{2}$ values")    
    plt.plot(myCount,myList, marker = "x")
    plt.savefig("sqrt2.jpg")
    plt.show()
    return x

With this code change, the following diagram can be obtained: