full transcript
#### From the Ted Talk by Colm Kelleher: What is Zeno's Dichotomy Paradox?

## Unscramble the Blue Letters

This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending ridelds have inspired mathematicians and philosophers to better udsearnntd the nature of infinity. One of the best known of Zeno's problems is clelad the doicmtohy paradox, which means, "the paradox of ctunitg in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his hosue to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his jrouney takes some finite aunomt of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this tekas a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, ddnviiig whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the pecies of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be innftiiy? This argument, by the way, is cpmleetloy general. It says that tavrinelg from any location to any other location should take an iinfinte amount of time. In other wrods, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno wklas at one mile per hour. Common ssnee tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. smiumng up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the eqioatun, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up ilnfiteniy many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the rinmnaeig half in half, and so on. While we're doing this, let's keep track of the araes of the pieces. The first slice makes two parts, each with an area of one-half The next sicle divides one of those halves in half, and so on. But, no maettr how many tiems we slice up the boxes, the ttaol area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've otenaibd the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that ftinie answer is the same one that coommn sense tells us is true. Zeno's journey takes one hour.
## Open Cloze

This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending **_______** have inspired mathematicians and philosophers to better **__________** the nature of infinity. One of the best known of Zeno's problems is **______** the **_________** paradox, which means, "the paradox of **_______** in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his **_____** to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his **_______** takes some finite **______** of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this **_____** a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, **________** whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the **______** of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be **________**? This argument, by the way, is **__________** general. It says that **_________** from any location to any other location should take an **________** amount of time. In other **_____**, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno **_____** at one mile per hour. Common **_____** tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. **_______** up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the **________**, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up **__________** many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the **_________** half in half, and so on. While we're doing this, let's keep track of the **_____** of the pieces. The first slice makes two parts, each with an area of one-half The next **_____** divides one of those halves in half, and so on. But, no **______** how many **_____** we slice up the boxes, the **_____** area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've **________** the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that **______** answer is the same one that **______** sense tells us is true. Zeno's journey takes one hour.
## Solution

- pieces
- summing
- infinite
- dichotomy
- common
- equation
- total
- house
- finite
- infinitely
- words
- walks
- remaining
- riddles
- obtained
- sense
- journey
- cutting
- amount
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- matter
- slice
- infinity
- understand
- areas
- called
- takes
- times
- dividing
- completely

## Original Text

This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending riddles have inspired mathematicians and philosophers to better understand the nature of infinity. One of the best known of Zeno's problems is called the dichotomy paradox, which means, "the paradox of cutting in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his house to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his journey takes some finite amount of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this takes a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, dividing whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the pieces of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be infinity? This argument, by the way, is completely general. It says that traveling from any location to any other location should take an infinite amount of time. In other words, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno walks at one mile per hour. Common sense tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. Summing up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the equation, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up infinitely many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the remaining half in half, and so on. While we're doing this, let's keep track of the areas of the pieces. The first slice makes two parts, each with an area of one-half The next slice divides one of those halves in half, and so on. But, no matter how many times we slice up the boxes, the total area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've obtained the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that finite answer is the same one that common sense tells us is true. Zeno's journey takes one hour.
## Frequently Occurring Word Combinations

### ngrams of length 2

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journey takes |
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finite amount |
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ancient greek |
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common sense |
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sense tells |
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part takes |
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finite answer |
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infinite series |
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### ngrams of length 3

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common sense tells |
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