In this article, we will use geometric figures to understand √2. In the first section titled “Geometric Proof of Irrationality of √2", we will prove that √2 is irrational by modifying the standard approach. In our approach, we will assume √2 is not irrational and use geometric arguments to arrive at a contradiction. This proof was discovered by Stanley Tennenbaum and is included in the paper “Irrationality From The Book"[1] by Steven J. Miller and David Montague, in which they generalize the idea to √n for n = 3, 5, 6, 10.The next section, titled “Continued Fractions, Pell's Equation and Visualizing √2", is inspired from a YouTube video “Visualising irrationality with triangular squares"[2] by Mathologer. In this section, we will learn how to: express square roots of every non-square whole number as continued fractions, generate a sequence of rational approximations to those irrationals and find non-trivial integer solutions to Pell's equation. This knowledge will allow us to create a visual representation of √2.

Geometric Proof of irrationality of√2

Before we begin, let’s remember how the standard proof of irrationality of √2 works. In a nutshell, we assume √2 is a rational number and then use a series of logical arguments to arrive at a contradiction. In this section, we will use a similar procedure, i.e., proof by contradiction with geometrical arguments to prove √2 is irrational.

We begin by assuming √2 is a rational number. Then, √2 can be expressed as the ratio of two whole numbers a and b. In other words,√2 = a/b which implies 2b2 = a2. So, according to our assumption, it is possible to find two whole numbers a and b such that 2b2 = a2. Since a and b are whole numbers, we can choose a pair of whole numbers a and b which satisfy the above-stated property and are smaller than every other possible pairs. Let m and n be those two numbers. Then, as seen in the figure 1, the sum of areas of the two green squares with sides n units must be equal to the area of the red square with sides m units.

Now take a look at figure 2. Here we have moved the two green squares onto the red square. The yellow region represents the overlap of the two green squares. Since the sum of areas of the two green squares is equal to the area of the red square, therefore the area of the overlapping region (yellow square) must be equal to the area of the of the omitted region (two smaller red squares). This fact is shown in figure 3 .

Geometrically, it is clear that we have found two identical squares with sides smaller than n (red squares in figure 3) such that the sum of their areas is equal to a square with sides smaller than m (yellow square in figure 3). But m and n were the smallest whole numbers which satised this property. So, we have arrived at a contradiction and therefore, the proof must be complete. But before we say our proof is complete, let us attempt to increase the rigorousness of this proof by proving the following claims algebraically:

Claim: m-n < n

Proof: Let’s assume m-n ≥ n. Then m ≥ 2n which implies m/n ≥ 2 > √2. This contradicts our assumption, i.e., m/n =√2. Therefore, m - n < n.

Claim: 2n - m < m

Proof: Since n < m, therefore m-n is a positive integer and n-(m-n) < m, i.e., 2n - m < m

Claim:(m - n)2 + (m - n)2 = (2n - m)2

Proof: We need to show(m - n)2 + (m - n)2 = (2n - m)2i.e. 2(m - n)2 - (2n - m)2 = 0.

2(m - n)2 - (2n - m) = (2m2 + 2n2 - 4mn) - (4n2 + m2 - 4mn) = m2 - 2n2 = (√2n)2 - 2n2 = 0

Now the proof of irrationality of √2 is complete.

Before we proceed further, I would like to draw your attention to the figures 1, 2 and 3, which have allowed us to visualize the proof. According to our proof, these figures should not exist. And before you question the mathematically rigorous and visually elegant proof, know the following. The numbers, i.e., m and n, have been selected in such a way that the difference in the area is one square unit.

m = 99; n = 70;m2 - 2n2 = 1, Since the difference is comparatively small, I was able to illustrate the impossible. At this point, we can ask the following question: "What other values of a and b can we take to mimic this behaviour?” And the answer is simple, i.e., all non-trivial integer solutions of the equation a2 - 2b2 = 1. In the next section, we will learn more about this equation and use that knowledge to create at a visual representation of √2.

Continued Fractions, Pell's Equation and Visualizing √2

This section is further divided into three subsections. The first subsection, titled "Continued Fractions and Convergents"[3][4], introduces the concept of continued fractions and convergents of a continued fraction. The aim of this subsection is to calculate the simple continued fraction expression of √2 and find the sequence of convergents of the simple continued fraction expression of √2. The second subsection, titled "Pell's Equation"[5], defines Pell's equation (x2 - ny2 = 1) and describes an algorithm which uses convergents of the simple continued fraction expression of √n to find non-trivial integer solutions. The aim of this subsection is to find the nontrivial positive integer solutions to n = 2 case of Pell's equation, i.e. x2 - 2y2 = 1. The third subsection, titled "Visual Representation of √2"[2], concludes this article with a visual representation of √2. In this subsection, the nontrivial positive integer solutions to n=2 case of Pell's equation are used to construct a geometric pattern, which when extended indefinitely, give rise to squares with sides x and y units such that y/x =√2.

Continued Fractions and Convergents

Continued Fraction

A continued fraction is an expression of a number as the sum of an integer and a quotient, the denominator of which is the sum of an integer and a quotient, and so on. In general,

Simple Continued Fraction

If bi = 1 for all i, then the expression is called a simple continued fraction. A simple continued fraction can be represented by the abbreviated notation [a0 ;a1; a2; a3;...]. All integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Finite and Infinite Continued Fraction

If the simple continued fraction expression of a number contains a finite number of terms, it is called a finite continued fraction. Similarly, if it contains an infinite number of terms, it is called an infinite continued fraction.

Continued Fraction Expression of a Rational Number

The simple continued fraction representation for a rational number is finite and only rational numbers have finite representations. Therefore, the number of terms ai in the simple continued fraction representation of a rational number of terms is finite; for example, 415/93 = [4, 2, 6, 7].

Continued Fraction Expression of an Irrational Number

Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite simplied continued fraction. Therefore, the number of terms ai in the simple continued fraction representation of a irrational number of terms is infinite; for example, √3 = [1; 1,2], in which the bar spans the sequence of terms that repeats indefinitely. Since this article is focused on √2, we will ignore transcendental numbers entirely and focus on square roots of positive nonsquare integers, which is a very small subset of algebraic irrational numbers.

Continued Fraction Expression of √n

A process which can be used to arrive at a continued fraction expression of square roots of positive non-square integers is given below.

Replacing √n in the RHS yields the following equality:

Repeating this process for infinite steps results in the following:

Simple Continued Fraction Expression of √2

Convergents

An infinite simple continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the simple continued fraction.

Convergents of √2

Pell's Equation

Definition

Pell's equation is any polynomial equation of the formx2 - ny2 = 1where n is a given positive non-square integer and integer solutions are sought for x and y.

Solutions

Pell's equation has infinitely many distinct integer solutions, including the trivial solution with x = 1 and y = 0. Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers x and y, and non-square n, if x2 - ny2 = 1, then x/y is a convergent of the simple continued fraction for √n.

Fundamental Solution via Continued Fractions

Let hi/ki, denote the sequence of convergents to the simple continued fraction representation for √n. Then the pair (x1, y1) solving Pell's equation (x2 - ny2 = 1) and minimizing x satises x1= hi and y1 = ki for some i. This pair is called the fundamental solution. Thus, the fundamental solution may be found by performing the simple continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

Additional Solutions from the Fundamental Solution

Once the fundamental solution is found, all remaining solutions (xk,yk) may be calculated algebraically from

This yields the recurrence relations

Solutions of x2 - 2y2 = 1, i.e., n = 2 case of Pell's Equation

Trivial Solution

x0 = 1; y0 = 0

Fundamental Solution

Additional Solution

These solutions can be used to accurately approximate √2 by rational numbers of the form x/y , i.e.,

Visual Representation of √2

A geometric process to generate squares with sides xk+1 and yk+1 units from squares with sides xk and yk units, where (xi; yi) are the non-trivial positive integer solutions of n = 2 case of Pell's equation (x2 - 2y2 = 1), is shown in figure 4.

Let us prove the claims made in figure 4 algebraically:

Claim: xk+1 = 3xk + 4yk

Proof: Since x1 = 3; y1 = 2, thereforexk+1 = x1xk + 2y1yk = 3xk + 4yk

Claim: yk+1 = 2xk + 3yk

Proof: Since x1 = 3; y1 = 2, therefore yk+1 = y1xk + x1yk = 2xk+ 3yk.

Claim: xk+12 - 2yk+12 = 1

Proof: Since xk2 - 2yk2 = 1, therefore

xk+12 - 2yk+12= (3xk + 4yk)2 - 2(2xk + 3yk)2= xk2 - 2yk2 = 1=> (xk+1, yk+1) is a solution of n = 2 case of Pell's equation.We know that the sequence of non-trivial positive integer solutions (xi, yi) of n = 2 case of Pell's equation (x2 - 2y2 = 1) can be used to accurately approximate √2 by rational numbers of the form xi/yi and limiti->∞xi/yi = √2.

Therefore, to create a visual representation of √2 (as shown in figure 5), we start from squares with sides x1 and y1, use the recurrence relations xk+1 = 3xk+ 4yk and yk+1 = 2xk + 3yk to generate squares with sides xi and yi and continue indenitely in order to obtain squares with sides x∞ and y∞ such that x∞/y∞= √2.

REFERENCES

[1] Steven J. Miller and David Montague. IrrationalityFrom The Book. 2009.arXiv: 0909.4913v3 [math.HO].

[2] Mathologer. Visualising irrationality with triangular squares. YouTube. accessed on Oct. 2, 2020. url: https://youtu.be/yk6wbvNPZW0.

[3] Encyclopaedia Britannica, Inc. accessed on Oct. 2 2020. url: https://www.britannica.com/science/continued-fraction.

[4] Continued fraction. Wikipedia. accessed on Oct. 2, 2020. url: https://en.wikipedia.org/w/index php?title=Continued_ fraction&oldid= 980449968.

[5] Pell's equation. Wikipedia. accessed on Oct. 2, 2020. url: https://en.wikipedia . org / w / index . php ? title = Pell % 27s _ equation & oldid =980884884.10