An Old Babylonian Algorithm and Its Modern Applications

Abstract

In this paper, an ancient Babylonian algorithm for calculating the square root of 2 is unveiled, and the potential link between this primitive technique and an ancient Chinese method is explored. The iteration process is a symmetrical property, whereby the approximate root converges to the exact one through harmonious interactions between two approximate roots. Subsequently, the algorithm is extended in an ingenious manner to solve algebraic equations. To demonstrate the effectiveness of the modified algorithm, a transcendental equation that arises in MEMS systems is considered. Furthermore, the established algorithm is adeptly adapted to handle differential equations and fractal-fractional differential equations. Two illustrative examples are presented for consideration: the first is a nonlinear first-order differential equation, and the second is the renowned Duffing equation. The results demonstrate that this age-old Babylonian approach offers a novel and highly effective method for addressing contemporary problems with remarkable ease, presenting a promising solution to a diverse range of modern challenges.

1. Introduction

The old Babylonian algorithm is an ancient method for computing various mathematical values. One of its notable applications is in calculating the square root of a number.

This algorithm works by making an initial guess and then refining it through a series of iterations. For example, to find the square root of a number N, an initial guess x₀ is chosen. Then, in each iteration, a new approximation x_n+1 is calculated using the formula x_n+1 = (x_n + N/x_n)/2. As the iterations progress, the value of x_n gets closer and closer to the actual square root of N.

This algorithm is surprisingly effective and efficient. It shows the ingenuity of ancient mathematicians in developing practical methods for solving complex mathematical problems. Although developed thousands of years ago, the old Babylonian algorithm still has relevance today and can be used as an alternative approach to modern computational methods in certain situations.

The evolution of ancient mathematics unfolded in a multitude of ways across a diverse array of geographical regions, as evidenced by the extant evidence [1,2,3]. Prior to the publication of Euclid’s Elements [4], the mathematical tradition of ancient China exerted the greatest influence [5,6,7]. The most striking difference between the two is undoubtedly the axiomatic framework of Euclid’s work and its abstract, formal character, which stands in stark contrast to the ancient Chinese mathematics, which was developed for practical applications.

The contrast between these two mathematical traditions is quite remarkable. Euclid’s work is characterized by its axiomatic framework and highly abstract, formal nature. On the other hand, ancient Chinese mathematics was primarily developed for practical applications. This difference is evident in the way problems were approached and solved. In ancient China, mathematics was often used to address real-world issues, such as engineering, astronomy, and commerce. The focus was on finding solutions that could be directly applied to practical situations.

In contrast, Euclid’s approach aimed at establishing a logical and systematic foundation for mathematics. His axioms and theorems provided a framework for deductive reasoning and the development of abstract mathematical concepts. This formal approach laid the groundwork for much of modern mathematics. Despite their differences, both traditions have made invaluable contributions to the history and development of mathematics.

Prior to its widespread adoption in the West after 1202 AD, the double false position method [8], also known as the Ying Buzu Shu [9], had been utilized in ancient China for over a millennium. Newton’s 1670 iteration algorithm, which is the most widely used in modern numerical simulation, was already in use by ancient Chinese mathematicians in more sophisticated forms for over a millennium [10].

The double false position method is a clever approach that allows for the approximation of solutions to various mathematical problems. It involves making two initial guesses and then refining the solution through a series of calculations. This technique showcases the ingenuity and advanced thinking of ancient Chinese mathematicians.

On the other hand, Newton’s 1670 iteration algorithm, which is extremely popular and widely used in modern numerical simulation, is a more recent development in the Western mathematical tradition. However, it is astonishing to note that ancient Chinese mathematicians had been using similar if not more sophisticated forms of this algorithm for over a millennium. This highlights the remarkable achievements and foresight of ancient Chinese mathematics. It also emphasizes the fact that mathematical knowledge and techniques have developed and spread across different cultures and time periods, each contributing to the overall growth and evolution of the field.

The Nine Chapters [11], an ancient Chinese mathematical treatise, may be considered a Chinese counterpart to Euclid’s Elements, which exerted a dominant influence on Western mathematics for centuries. Similarly, the Nine Chapters came to be regarded as the foundational text of ancient Chinese mathematics for many millennia [12]. When ancient mathematics is re-examined from a modern mathematical perspective, it can provide a novel perspective on the applications of mathematics to practical problems [13].

The study of ancient Chinese mathematics is a well-established field of research. In a review article [14], some ancient Chinese algorithms were introduced, and their modern applications to solve nonlinear differential equations were elucidated. Additionally, a frequency formulation was proposed [14]. The formulation’s simplicity and effectiveness have led to its widespread use for rapidly and reliably obtaining insights into the periodic properties of nonlinear vibration systems. Elías-Zúñiga et al. used the concept to study a fractal vibration system [15] and gave a modified frequency formulation [16]; Zhang et al. extended the concept to nonlinear oscillators with generalized initial conditions [17]; Yang et al. extended it to fractional vibration systems [18]; and various modified frequency formulations have appeared [19,20]. Chun-Hui He further developed the ancient Chinese mathematical algorithm into a modern numerical method [21], which was subsequently named the Chun-Hui He algorithm or Chun-Hui He iteration method in the literature [22,23].

Another topic of considerable interest in the field of ancient Chinese mathematics is the He Chengtian inequality, which was extended to facilitate the solution of nonlinear oscillators [24]. This approach, which is known as the max–min approach [25,26], has been the subject of considerable research.

In this paper, we briefly introduce an ancient Babylonian algorithm [27,28]. The calculation of the square root has been a topic of considerable interest among scholars of ancient mathematics [29,30]; it was the focus of the ancient Chinese, Sanskrit mathematics, Mediterranean practical arithmetic, and the old Babylonian algorithm [31]. Here, we first introduce the Smyrna algorithm [29]; its iteration process is

$$\sqrt{N}={\mathrm{z}}_{i+1},x_{i+1}=x_i+N{y}_i,y_{i+1}=x_i+y_i,{\mathrm{z}}_{i+1}={\displaystyle \frac{x_{i+1}}{y_{i+1}}},x_0=y_0=1.$$

(1)

When N = 2, we have the following series converging to √2:

1/1, 3/2, 7/5, 17/13, 43/30, 103/73, 249/176, 601/425, 1451/1026, 3513/2487. This is a limiting process and not an exact equality for every iteration. The ancient Babylonian algorithm for finding square roots can be considered invalid in certain extreme cases, for example, when dealing with extremely small numbers close to zero. Due to the limitations of numerical precision and rounding errors, the Babylonian algorithm might not converge accurately to the expected square root. For instance, if attempting to find the square root of a number like 10⁻²⁰, the algorithm might encounter difficulties in accurately converging to the true square root value.

The old method exhibits a distinctive symmetrical quality, which is both intriguing and harmoniously pleasing from a mathematical perspective. It is noteworthy that the square root of 2 can be represented by a fraction with an arbitrarily high degree of precision. This phenomenon is even more intriguing when one considers that the irrational approximation can be expressed by a fraction of two prime numbers, specifically 17/13, 103/73, and 3513/2487.

With different initial guesses, we can obtain the following approximate root:

2/1, 4/3, 10/7, 24/17, 58/41, 140/99, 338/239, 816/577, 1970/1393

3/1, 5/4, 13/9, 31/22, 75/53, 181/128, 437/309, 1055/746, 2547/1801

5/3, 11/8, 27/19, 65/46, 157/111, 379/268, 1016/647, 2320/1673, 6313/3993

3/5, 13/8, 29/21, 71/50, 171/121, 463/292, 1047/755, 2555/1802, 6161/4359

There are additional primes that can be identified, e.g., 2547/1801.

The Smyrna algorithm is also called Theon’s Ladder [29]. The √2 ratio was also widely used in ancient architecture [32,33], especially in ancient China; the concept of a round Heaven and square Earth was used to design various architectures.

Now, we must first consider the following algebraic equation:

$$x^2=a.$$

(2)

Here, a is a positive constant. The old Babylonian algorithm is [27,28].

Quadratic equations occupied a significant place in Babylonian mathematics [34]. There are also some variants [35,36]. Calkin and others revealed the possible connection between the Newton iteration method and the ancient Babylonian algorithm [26]. Surprisingly, this ancient algorithm is still in use and has applications in modern times [37,38,39]. Here, we will show its connection to an ancient Chinese algorithm called the He Chengtian average [40]. This will be discussed in Section 2. After that, a general algebraic equation is considered in Section 3. As an example, a micro-electromechanical system (MEMS) is considered in Section 4. Then, the old method is extended to solve differential equations in Section 5. Finally, in the last section, the future research frontiers are pointed out.

2. He Chengtian Average

A simple sight into old Babylonian algorithm, we find that

$$x_n\le x\le {\displaystyle \frac{a}{x_n}} \mathrm{or}{\displaystyle \frac{a}{x_n}}\le x\le x_n$$

(3)

where $x=\sqrt{a}$; here, a is a real number, and x_n and a/x_n are two approximates.

The He Chengtian average [40] is

$$x={\displaystyle \frac{p{x}_n+qa/x_n}{p+q}}={\displaystyle \frac{p}{p+q}}{x}_n+(1-{\displaystyle \frac{p}{p+q}}){\displaystyle \frac{a}{x_n}},$$

(4)

where p and q are positive nature numbers. Equation (4) can be written in the form

$$x_{n+1}=\beta x_n+(1-\beta ){\displaystyle \frac{a}{x_n}}.$$

(5)

Here, β = p/(p + q). It is important to note that the initial guess must be different from x₁ = a. When β = 1/2, Equation (5) is Equation (2). This might have shown a possible link between the old Babylonian algorithm and the old He Chengtian average. A modified He Chengtian average was suggested in Ref. [41]. When β = (n − 1)/n, Equation (5) becomes the Newton method.

3. Cube Root and Beyond

In a modern mathematics view, the old Babylonian algorithm can be easily extended to solve a cube root:

$$x^3=a.$$

(6)

Similar to Equation (6), we have

$$x_{n+1}=\beta x_n+(1-\beta ){\displaystyle \frac{a}{x_n^2}}.$$

(7)

We recommend β = 2/3 for the cube root:

$$x_{n+1}={\displaystyle \frac{2}{3}}{x}_n+{\displaystyle \frac{1}{3}}{\displaystyle \frac{a}{x_n^2}}.$$

(8)

We consider a simple case where a = 2 and begin with the initial guess of x₀ = 1; three iterations lead to 1.33333333, 1.26388889, and 1.25993349, respectively, while the exact root is 1.25992105. The relative error is only 0.00098% for the third iteration result.

For a more general case,

$$x^m=a,$$

(9)

where m is a real number. We have the following iteration algorithm:

$$x_{n+1}=\beta x_n+(1-\beta ){\displaystyle \frac{a}{x_n^{m-1}}}.$$

(10)

Here, we recommend β = (m − 1)/m. We consider a simple case of a = 2 and m = 7; we begin with x₀ = 1, and the three iterations are 1.14285714, 1.10781907, and 1.10412697, respectively, while the exact one is 1.104089513673812, with only a relative error of 0.0034% for the third iteration result.

Now, we consider a general algebraic equation

$$f(x)=0.$$

(11)

We re-write Equation (11) in the form

$$x^m=F(x)$$

(12)

The iteration algorithm is

$$x_{n+1}=\beta x_n+(1-\beta ){\displaystyle \frac{F(x_n)}{x_n^{m-1}}}$$

(13)

We can construct other iteration formulations, which will be discussed in a forthcoming paper.

4. MEMS System

As a practical application, we consider a transcendental equation that arises in MEMS systems [42]:

$${(1+\sqrt{1-4x})}^2+8x\ln \left|1-{\displaystyle \frac{1+\sqrt{1-4x}}{2}}\right|=0$$

(14)

The root of Equation (14) is the threshold value for the applied voltage; when x is smaller than the root, a periodic solution [43] is predicted, while when x is larger than the root, the so-called pull-in instability [44,45] occurs. The MEMS system is now widely used as sensors with extremely high sensitivity [46]. Zhang et al. suggested a method for the estimation of the frequency of a MEMS system [47]. Lv studied the MEMS system in a fractal space [48]. The dynamic characteristics of MEMS systems were studied in Refs. [49,50].

We introduce a new variable, u, defined as

$$u=1+\sqrt{1-4x}$$

(15)

Equation (14) becomes

$$u^2=-2(1-{(u-1)}^2)\ln \left|1-{\displaystyle \frac{u}{2}}\right|$$

(16)

By Equation (13), we have

$$u_{n+1}={\displaystyle \frac{1}{2}}{u}_n+{\displaystyle \frac{1}{2}}{\displaystyle \frac{-2(1-{(u_n-1)}^2)\ln \left|1-\frac{u_n}{2}\right|}{u_n}}$$

(17)

We commence with u₀ = 1, and the initial two iterations yield the following outcomes: u₁ = 1.19314718 and u₂ = 1.47046999. The exact root is 1.43066373, with a relative error of 2.78% for the second iteration result. This level of accuracy is sufficient for practical applications, although it may be possible to achieve greater precision by continuing the iteration process.

5. Differential Equations

The above section shows that the modified old Babylonian algorithm is suitable for solving algebraic equations, but this is not enough. Most engineering problems are modelled by differential equations, so this section applies the old method to solve nonlinear differential equations.

Consider the following nonlinear differential equation:

$$u^{\prime }+u^2=0,u(0)=1.$$

(18)

Its exact solution is u(x) = 1/(1 + x). We have the following iteration formulation:

$$u_{n+1}=\beta u_n+(1-\beta ){\displaystyle \frac{-{u^{\prime }}_n}{u_n}},$$

(19)

where ${u^{\prime }}_n$ is an approximate value of the first-order derivative, and the initial guess should satisfy the initial condition.

Equation (18) shows $\underset{x\to \infty }{\lim }u(x)=0,$ so we choose β = 1/2 and begin with

$$u_0(x)={\displaystyle \frac{1}{1+bx}},$$

(20)

where b is an unknown constant. Equation (20) is the trial solution; its choice is similar to that in the variational-based methods, e.g., the Ritz method [51].

The approximation of the first-order derivative is

$${u^{\prime }}_0(x)=-{\displaystyle \frac{b}{{(1+bx)}^2}},$$

(21)

The first iteration gives

$$u_1(x)={\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{1+bx}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\frac{b}{{(1+bx)}^2}}{\frac{1}{1+bx}}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{1+b}{1+bx}}.$$

(22)

By the initial condition, we have

$$u_1(0)={\displaystyle \frac{1+b}{2}}=1.$$

(23)

So, we find b = 1, and we obtain the exact solution.

Another example is the following well-known Duffing equation arising in the nonlinear vibration system [52,53,54]:

$$u^{″}+u+\eta u^3=0,u(0)=A,u^{\prime }(0)=0.$$

(24)

We construct the following iteration algorithm:

$$u_{n+1}=\beta u_n+(1-\beta ){\displaystyle \frac{-({u^{″}}_n+u_n)}{\eta u_n^2}},$$

(25)

where ${u^{″}}_n$ is an approximate value of the second-order derivative. Note that the initial guess should satisfy the initial conditions.

We begin with

$$u_0(t)=A(1-{\displaystyle \frac{1}{2}}{\omega }^2{t}^2).$$

(26)

It comes from the approximate solution $u(t)=A\cos \omega t$; $\omega$ is the unknown frequency. According to Equation (25), the approximation of the second-order derivative can be obtained as

$${u^{″}}_0(t)=-A{\omega }^2.$$

(27)

The first iteration gives the following result:

$$u_1(t)=\beta A(1-{\displaystyle \frac{1}{2}}{\omega }^2{t}^2)+(1-\beta ){\displaystyle \frac{-(-A{\omega }^2+A(1-\frac{1}{2}{\omega }^2{t}^2))}{\eta A^2{(1-\frac{1}{2}{\omega }^2{t}^2)}^2}}.$$

(28)

It should satisfy the initial conditions, so we have

$$u_1(0)=\beta A+(1-\beta ){\displaystyle \frac{-(-A{\omega }^2+A)}{\eta A^2}}=A.$$

(29)

From Equation (29), we obtain the following frequency–amplitude relation:

$$\omega =\sqrt{1+\eta A^2}.$$

(30)

So, the approximate solution is

$$u_0(t)=A\cos \left[{(1+\eta A^2)}^{1/2}t\right].$$

(31)

Figure 1 illustrates that the agreement between the approximate solution and the exact solution for weak nonlinearity is indeed valid. As the value of the parameter η approaches infinity, the discrepancy between the approximate and exact frequencies increases, reaching a maximum of 18.03%.

To enhance the precision of the approximation, it is advisable to seek an approximate solution in the following form:

$$u_0(t)=a\cos (\omega t)+b\cos (3\omega t).$$

(32)

a and b are unknown constants satisfying the following relationship:

$$a+b=A.$$

(33)

According to Equation (32), we begin with

$$u_0(t)=A-{\displaystyle \frac{1}{2}}(a+9b){\omega }^2{t}^2,$$

(34)

According to Equation (34), the approximation of the second-order derivative can be obtained as

$${u^{″}}_0(t)=-(a+9b){\omega }^2,$$

(35)

The first iteration yields the following result:

$$\begin{array}{l}{u}_1(t)=\beta (A-{\displaystyle \frac{1}{2}}(a+9b){\omega }^2{t}^2)\\ +(1-\beta ){\displaystyle \frac{-(-(a+9b){\omega }^2+(A-\frac{1}{2}(a+9b){\omega }^2{t}^2))}{\eta A^2{(A-\frac{1}{2}(a+9b){\omega }^2{t}^2)}^2}}.\end{array}$$

(36)

According to the initial conditions and Equation (24), we have

$$\left\{\begin{array}{l}{u}_1(0)=A,\\ {u^{\prime }}_1(0)=0,\\ {u^{″}}_1(0)=-A(1+\eta A^2).\end{array}\right.$$

(37)

That means

$$u_1(0)=\beta A+(1-\beta ){\displaystyle \frac{-(-(a+9b){\omega }^2+A)}{\eta A^2}}=A,$$

(38)

$${u^{″}}_1(0)=\beta {\displaystyle \frac{(a+9b){\omega }^2}{\eta A^4}}-(1-\beta ){\displaystyle \frac{-2{(a+9b)}^2{\omega }^4}{\eta A^5}}=-A(1+\eta A^2)$$

(39)

By simultaneously solving Equations (33), (38), and (39), one can determine the values of a, b, and ω. It is recommended that β be set to 1/2.

6. Discussion and Conclusions

This brief article presents an overview of the Babylonian algorithm, originally developed to solve the equation x² = 2, and explores its potential applications to modern mathematical problems.

For a general algebraic equation

$$f(x)=0.$$

(40)

Its iteration algorithm can be expressed as

$$x_{n+1}=\beta x_n+(1-\beta ){\displaystyle \frac{f(x_n)+{(x_n)}^2}{x_n}}.$$

(41)

We recommend $\beta$ = 1/2. Consider the following example [40]:

$${0.1}^x+{0.2}^x+{0.3}^x=1.$$

(42)

Beginning with x₀ = 1, the first three iteration results are

$$x_1={\displaystyle \frac{1}{2}}\times 1+{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{0.1+0.2+0.3-1+1}{1}}=0.8,$$

(43)

$$x_2={\displaystyle \frac{1}{2}}\times 0.8+{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{{0.1}^{0.8}+{0.2}^{0.8}+{0.3}^{0.8}-1+{0.8}^2}{0.8}}=0.6280$$

(44)

$$\begin{array}{l}{x}_3={\displaystyle \frac{1}{2}}\times 0.6280+{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{{0.1}^{0.6280}+{0.2}^{0.6280}+{0.3}^{0.6280}-1+{0.6280}^2}{0.6280}}\\ =0.6829.\end{array}$$

(45)

The precise root is 0.67019, and the relative error of the third approximate solution is 1.89%. This is an acceptable level of precision given that only three iterations were performed.

For a general differential equation,

$$L(u)+N(u)=0.$$

(46)

where L and N are, respectively, the linear differential operator and the nonlinear operator. An iteration algorithm can be constructed as

$$u_{n+1}=\beta u_n+(1-\beta ){\displaystyle \frac{L(u_n)+N(u_n)+{(u_n)}^2}{u_n}},$$

(47)

where u₀ is such chosen so that it satisfies all boundary and initial conditions; it should involve an unknown parameter. For nonlinear oscillators, we can choose u₀ in the form

$$u_0=A\cos (\omega t+\sigma ),$$

(48)

where A and $\sigma$ are determined by the initial conditions; $\omega$ is determined by the following equation:

$$u_1(\overline{t})=\beta u_0(\overline{t})+(1-\beta ){\displaystyle \frac{L(u_0(\overline{t}))+N(u_0(\overline{t}))+{(u_0(\overline{t}))}^2}{u_0(\overline{t})}},$$

(49)

where $\overline{t}$ is a location point.

For (24), we can assume

$$u_0=A\cos (\omega t).$$

(50)

By Equation (47), we have

$$\begin{array}{l}{u}_1(t)=\beta A\cos (\omega t)\\ +(1-\beta ){\displaystyle \frac{-A{\omega }^2\cos (\omega t)+A\cos (\omega t)+\eta A^3{\cos }^3(\omega t)+A^2{\cos }^2(\omega t)}{A\cos (\omega t)}}.\end{array}$$

(51)

The location point is chosen as

$$\omega \overline{t}=\pi /6.$$

(52)

That means

$$u_1={\displaystyle \frac{\sqrt{3}}{2}}A.$$

(53)

From Equation (48), we have

$$\omega =\sqrt{1+{\displaystyle \frac{3}{4}}\eta A^2}.$$

(54)

This is exactly same as that obtained by the homotopy perturbation method [53].

For fractal oscillators [55,56,57], we consider the following example

$${\displaystyle \frac{d^{2\alpha }u}{d{t}^{2\alpha }}}+u+\beta u^3=0,u(0)=A,{\displaystyle \frac{d^{\alpha }u}{d{t}^{\alpha }}}(0)=0.$$

(55)

where ${\displaystyle \frac{d^{\alpha }u}{d{t}^{\alpha }}}$ is the two-scale fractal derivative [58,59,60], $\alpha$ is the two-scale fractal dimensions [61].

We begin with

$$u_0=A\cos (\omega t^{\alpha }).$$

(56)

By Equation (47), we have

$$\begin{array}{l}{u}_1(t)=\beta A\cos (\omega t^{\alpha })\\ +(1-\beta ){\displaystyle \frac{-A{\omega }^2\cos (\omega t^{\alpha })+A\cos (\omega t^{\alpha })+\eta A^3{\cos }^3(\omega t^{\alpha })+A^2{\cos }^2(\omega t^{\alpha })}{A\cos (\omega t^{\alpha })}}.\end{array}$$

(57)

Similar to the above example, the location point is chosen as

$$\omega {\overline{t}}^{\alpha }=\pi /6,$$

(58)

and

$$u_1={\displaystyle \frac{\sqrt{3}}{2}}A.$$

(59)

From Equation (48), we have

$$\omega =\sqrt{1+{\displaystyle \frac{3}{4}}\eta A^2}.$$

(60)

So, the approximate solution of Equation (55) is

$$\omega =\sqrt{1+{\displaystyle \frac{3}{4}}\eta A^2}.$$

(61)

$$u_0=A\cos \left[{(1+{\displaystyle \frac{3}{4}}\eta A^2)}^{1/2}{t}^{\alpha }\right].$$

(62)

It is exactly the same as that in Ref. [55].

Now, we re-study Equation (18); the iteration formulation is

$$u_{n+1}=\beta u_n+(1-\beta ){\displaystyle \frac{{u^{\prime }}_n+2{(u_n)}^2}{u_n}}.$$

(63)

We begin with u₀ = 1/(1 + bx) and obtain the following result:

$$u_1={\displaystyle \frac{1}{2(1+bx)}}+{\displaystyle \frac{-b/{(1+bx)}^2+2/{(1+bx)}^2}{2/(1+bx)}}.$$

(64)

Locating at x = 0, we have

$$u_1(0)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{-b+2}{2}}=1.$$

(65)

We can determine b = 1; this leads to the exact solution.

The old Babylonian algorithm is an amazingly effective way to solve the root of 2, even more so than the well-known Newton iteration method. When the old method is re-studied in view of modern mathematics, it can solve more complex problems. The ability to solve nonlinear differential equations offers a promising scenario for both mathematicians and engineers. Its convergence and reliability in practical applications require further research, and this work paves the way for scientists to use ancient achievements for modern applications. It is our expectation that this preliminary study on the old Babylonian algorithm will serve as a foundation for more advanced research into its iteration robustness and potential applications.

This article proposes a modified version of the old Babylonian algorithm that can be applied to modern problems, e.g., nonlinear vibration systems, fractal oscillators, and nonlinear differential equations. The modification is straightforward and effective in addressing the examples presented. However, the author acknowledges that, regardless of its effectiveness, the new iteration algorithm’s convergence should be rigorously demonstrated. In practical terms, this paper can be regarded as an exemplar of a successful paradigm.