Real Mathematics – Life vs. Maths #2

Matches

For thousands of years people tried find a precise value for the number π (3,1415192…). At first this special number was thought to be seen only when there is a circle around. Within time π started to appear in places where scientist didn’t expect it to be. One of them was an 18th century scientist Georges Buffon.

Buffon came up with a probability problem named “Buffon’s needle problem” in 1777 when he came across with the number π. As I didn’t possess that many needles, I modified the problem as “Serkan’s matches problem”.

Buffon’s Needle Problem: Take a piece of paper and draw perpendicular lines on it with specific amount of space between them. Buffon wondered if one can calculate the probability of a needle that will land on one of the lines.

To start Serkan’s matches problem you need at least 100 matches, a piece of empty paper, a ruler, pen/pencil and a calculator.

First of all, draw perpendicular lines with 2 matches-length spaces between them.

Then just throw the matches on the paper randomly.

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Start collecting the matches which land on a line. At last you should use your calculator to divide the total number of matches to the number of matches landed on a line.

In my experiment out of 100 matches, 32 of them landed on a line. That gave me 3,125 which is close to the magical number π.

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In fact, 100 matches are not enough for this experiment. In my second try 34 matches landed on one of the lines which gave 100/34=2,9411… Obviously this is not close to π. More matches we use, closer we will get to π.

In an experiment back in 1980 2000 needles were used to analyze Buffon’s needle problem. Result was 3,1430… which is seriously close to the number π.

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You could go to https://mste.illinois.edu/activity/buffon/ and use this simulator which uses 1000 needle. In my first try I got 3,1496… You should try and see the result yourself.

In the future I will be talking about why a needle (or a match) is connected to the number π.

One wonders…

Try to do your own experiment and repeat Buffon’s needle problem for five times. Take the arithmetic average of your solutions and see how close you are to π?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #9

Finding Pi

Humans discovered a connection between a circle’s circumference and its diameter around 4000 years ago. This connection, what we call the number pi now, affected everyone who was involved in mathematics and/or engineering in the early times. Countless people, including historical figures, worked in order to find what this number is, but all of them failed to find such answer.

Some of them tried to find an approximation and got really close. When it was their time to shine for geometry, one ancient Greek whom we are familiar with had taken approximation methods to a different level. This famous Greek dude was Archimedes.

Archimedes is believed to live between 287 BCE and 211 BCE. If you take a look at the history of mathematics it was common to see approximations for the number pi, especially in Babylon, Sumer, ancient Egypt and China civilizations.

Although, Archimedes was different in an important way: Before him, method to find the number pi was not that complicated. A certain circle would be inspected and compared to a similar regular polygon. Their areas or perimeters were thought to be same which would give an approximation for pi.

Let’s try to find an approximation for the number pi using this method.

Draw a circle and divide it into 12 equal parts, like a pizza.

Cut the 12th slice into two equal parts and lay down all the slices side to side.

Slices will seem like a rectangle which is a regular polygon. At this point circle and rectangle are believed to have same shapes.

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Circumference of circle is found with pi*R, where R is the diameter of the circle. It is obvious to the naked eye that the long sides of the rectangle would give the circumference of the circle. Hence, we can find an approximation for the number pi.

Try this for another circle that has different diameter length.

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And take the arithmetic mean of both results. There is the number pi… Well, it is just an approximation.

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Archimedes’ Method

Even though there were so many engineering marvels as well as important geometry knowledge, Archimedes was not happy with the methods used to find the number pi. He developed his own method that required inscribing and circumscribing the unit circle with regular polygons. Archimedes did something different than the methods which were used before him: He continued using polygons with more sides and compared those results until he found the best polygon.

His method today is known as “the method of exhaustion”. It is the earliest known version of calculus.

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You could draw a unit circle yourself. Try to inscribe and circumscribe a square. You’d get pi between 2,8 and 4, which is not bad for the first try. If you continue with polygons that have more sides, you’d get even better intervals.

The thing is, Archimedes didn’t try to find an exact value. Instead he found an interval which was 99,9% accurate!

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Archimedes found this interval with starting a hexagon, and he finished his method at a 96-sided polygon. You could also see that 22/7, which is given to students for pi in school, is the upper limit that Archimedes found nearly 2200 years ago!

M. Serkan Kalaycıoğlu