# Zeno’s paradox: Achilles and the Turtle

## Introduction

Paradoxes are a huge interest I’ve had concerning ideas and the real world. The idea of something being so enigmatic to the human mind and yet still hard to visually understand. Paradoxes such as “Schrodinger's Cat” have left me for hours trying to make a visual reference for myself and still couldn’t get anywhere. I was later told about philosopher Zeno of Elea the great philosopher who proposed his famous paradox of famed mythological warrior Achilles not reaching the turtle and set into a state of infinite distance. The idea of distance being infinite when the turtle is right in front of achilles

This examination means to make sense of the Mathematics surrounding Zeno's paradox and to perceive any motivation behind why Achilles might be able to make up for lost time to the turtle, and on the off chance that it is conceivable, to what extent it would take for Achilles to make up for lost time to it. There are five areas to this examination; the presentation, clarifying the paradox, discrediting the paradox, conclusion, and the sources utilized. On the off chance that the oddity can be refuted as, a large portion of the examination will be spent discovering how to demonstrate that.

As far as the maths utilized, this examination will center around utilizing Infinite geometric arrangement and Sigma notation with the end goal to investigate the paradox in fine detail. The infinite geometric series is the whole sum of a geometric series, from an arrangement that would have no last term.

The basic function of a geometric sequence is the multiplication of an x variable. The x variable is then being multiplied by a common ratio then exceeding by its ratio infinitely.

These two mathematical concepts are very much integrated with in the idea of zeno's paradox, and through the use of the two concepts indeatial the examination could prove that the whole paradox is a sham based on the idea of how distance truly works in reality. This may prove to be basic in nature this investigation is propelled further.

## Clarifying the Paradox

Zeno’s paradox was created by the Greek philosopher of Elea, and the whole paradox run on this idea. Ancient mythological hero Achilles is in a foot race against all thing a slow Tortoise. The turtle then states that the hero Achilles will never catch up to him with the turtle having a head start. So the hero gave the Tortoise a 10 meter head start. But as soon as achilles made it to where the slow reptile would have been, but by the time the Toritse would have moved from his original position.

Now if the Tortoise was given a headstart notated as x meters, then it implies that Achilles would have to run the same x meters to get to where the reptile last was. The then by the time achilles runs to the last point of the Tortoise the Tortoise would have moved x/10 meters. Achilles would have to add those ten meters, but the tourist would have run x/100 by the time he reaches the Tortoise last position. This humorously displays the great greek Hero achilles who was known to be swift on his feet, was defeated by a slow heavy reptile. Now that the general premise of the paradox is clarified and the parameters drawn, there can be a conclusion to why the paradox is true or false and if the hero Achilles is truly being beaten by the hulking paper weight with legs?

## Debunking the Paradox

Although the mathematics say that Achilles will never beat the reptile in a race, real life logic and mechanics says a more realistic idea of how this paradox would operate. Through common knowledge it’s known that an oversized reptile with a weight of over hundred of grams would never reach the speeds of an average human, even if the toristie had the head start it would be surpassed by the speeds even an unathletic individual would have. This can be proven mathematically as well with an equation to show the time need for the greek hero to show his swiftness against the Tortoise.

First of all, Achilles and the Tortoise need to have their speed determined for when the hero catches the Tortoise. The speed will be noted as the variable s and time will be represented in form of t. The notation used to see how fast Achilles is will be s m/s (meters per second), this is done as the common ratio k as the Tortoise runs at ks m/s (meters per second). Knowing this connection an equation can be formulated to show at what time it takes for Achilles to catch up to the reptile.

This portion of the equation denotes the the speed at which it relates to the time and head start the Tortoise. which equates to st, and refers to Achilles’ speed in relation to time. This equation demonstrates the connection between the speed and time of Achilles to the speed and time of the Tortoise with a lead, additionally factoring in the basic proportions.

This equation will enable us to realize to what extent it will take for Achilles to get the turtle (within seconds). Realizing that Achilles speed in reference to the time is equivalent to st, the condition can be improved by isolating the condition by s. This simplification is written to be . With this final equation the time can be found for Achilles to overtake the Tortoise. Although there is an alternative way to calculate the time need for Achilles.

The use of this notation will also equate to the already established infinite series to . The use of these two method provides a more reliable representation of the time it took achilles to make up and catch the Tortoise. The infinite geometric series may give reason to believe that it would be impossible for Achilles to surpass the Tortoise but this proves that it is possible for Achilles to beat the impossibly swift turtle through the methods devised. The turtle won't have the capacity to endlessly keep separate from Achilles due to the fact that the distance being made is minuscule to the point that it is effectively outperformed. Despite the fact that there are no legitimate qualities that can be utilized, this still shows the natural inadequacies the paradox inherts.

## Conclusion

In conclusion, the long lived Zeno's paradox is really not so much a paradox but a clear representation of a mathematical concept. The paradox hides that it’s just an example of an infinite geometric series that can be solved by finding the sum. Where it was originally thought to be an infinite number it’s truly a finite number. The formulas provided indicate the time for Achilles needs to catch up to the Tortoise and proves that this paradox grounded in mathematics as an infinite geometric series. This investigation provided me with a new lease on how theoretical ideas interact with real world applications. Zeno’s paradox made a lack of clarity visually to me as i was perceiving it to be with in a real world setting, but this investigation brought the clarity require to visualize how the paradox operates. The application of mathematics into paradox and other similar concepts does intrigue me on the possibilities others can find. This investigation has satisfied my quench for explanations of the seemingly impossible, and lead me to give math a revitalization into my more inquisitive mind.

## Works Cited

1. Ray. 'How to Find the General Term of Sequences.' Owlcation. Owlcation, 12 July 2018. Web. 14 Nov. 2018. https://owlcation.com/stem/HowtoFindtheGeneralTerm-of-Geometric-Sequences
2. 'Zeno’s Paradox of the Tortoise and Achilles.' Zeno's Paradox of the Tortoise and Achilles. N.p., n.d. Web. 10 Nov. 2018. http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles
3. “Summation Notation.” Columbia, www.columbia.edu/itc/sipa/math/summation.html.
4. Stapel, Elizabeth. 'Geometric Series.' Purplemath. N.p., n.d. Web. 14 Nov. 2018. www.purplemath.com/modules/series5.htm.
5. 'Infinite Geometric Series.' Private Tutoring. Varsity Tutors, n.d. Web. 18 Nov. 2018. www.varsitytutors.com/hotmath/hotmath_help/topics/infinite-geometric-series.
07 April 2022
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