Egyptian Pi

Hello reader!

I am here to explain another incredibly important aspect of our lives, that has affected our society for centuries:

No no no, not that type of pie (although it is quite pleasing to the stomach).

Ah, yes. One of the most famous irrational numbers in mathematics, pi.

So, what IS pi? What sentence would define pi instead of a number? People say pi is the ratio between the circumference of the circle and the diameter (this can be easily found with knowing that C= π * d, and proceeding to find π = C/d)

This lesson will be mostly a history lesson about mathematics. How have people in history calculated the exact value of pi if it cannot be expressed as a common fraction (that is, an integer numerator AND denominator)?

SIDE NOTE: In your circle geometry homework, the questions may ask you to express pi as 22/7, but if you put that exact fraction in the calculator, you won’t get exactly 3.14159… however, it is a close approximation of the irrational number.

Pi has some interesting properties: first, it cannot be expressed as a sum of any n-root of any rational numbers. Second, if using a compass/straight-edge, the area of a square can never equal the area of a given circle.

So, easily found approximations of pi. Greeks and Egyptians were some of the ancient civilizations that had found a fairly close estimation of pi, and both involved some sort of polygon approximation. We’ll look at the Egyptians’ method right now, because it’s a fairly simple method.

First of all, what DID the Egyptians estimate pi to be? 256/81, or approximately 3.16. This is fairly close to pi, being about 3.14.

How did historians find out what the Egyptians did exactly? Well, they found a manuscript from ancient history and within some hieroglyphs they found this diagram:

The Egyptians approximated pi by first drawing a 9x9 square, and inscribed a circle within it (therefore having a diameter of 9): (please excuse my horrid drawing skills, these diagrams are NOT to scale and are meant for a helpful representation)

Then they put points to divide the side length of the square into 3 parts and connected some of these points to each other to form an OCTAGON:

The Egyptians then hypothesized that this area of the octagon is roughly about the area of the circle (just by observing, it could be quite possibly true). To find the area of the Octagon, you simply took the entire area of the square (9 x 9 = 81) and subtracted the isosceles triangles on each corner of the square (which we know has a side length of 3). Having four of these triangles, the total area is 4.5 x 4 = 18.

81-18= 63 (which is the area of the octagon).

The Egyptians then proceeded to say that by taking one row and column away from the square (9+9=18) then this would also be equal to that of the square:

HOWEVER…

You see how in the bottom right corner of the square, there is one part being overlapped, and is counted twice, which makes the real value only 17 squares being taken away, leaving an 8x8 square behind. This may have caused the Egyptians to over-calculate their approximation of pi (just goes to show how a little mistake can immediately drive you off the right path!). Still, their result was only about 0.02 off, and this was several hundred (thousand?) years ago, which is impressive if you ask me.

Knowing the area of a circle is πr², the Egyptians then wrote:

π(9/2)² = 8²

And therefore:

π = 8² / (9/2)² OR π = 4 x 64/81= 256/81.

VOILA! An Egyptians interpretation of pi.

Of course, now that technology has advanced, we have calculated pi to about 10 TRILLION digits, and people have been memorizing the digits all the way up to 67000 digits. The human brain is certainly accelerating, which gives all the more opportunities to observe and question the world around you!

So long for now!

~The Octopi

Here’s a selection of pies to reward you for actually reading this:

The Number Devil~ Introductory on "Vroooom!"

Greetings fellow pupils!

As you may or may not know already, I am the Octopi, an aspiring mathematician. This blog is mainly to enlighten audiences with simple mathematical concepts that can be easy to comprehend. As an introductory to my posts, I will expose you to a fun, child-friendly book that is (surprisingly…) based on mathematics!

Meet the Number Devil:

As a young octopus, I came across this book when my older sibling had advised me to read this book, and I was highly likely to learn a thing or two about math along the way. I’ve re-read it over the years, and the way the author (Hans Magnus Enzenburger) portrays commonly boring topics, such as factorials, Fibonacci numbers, exponentiation, and permutation.

(Not to spoil anything) This book is primarily based on a young boy, Robert, who is being increasingly bored about mathematics in school, leading to anxiety. He experiences recurring dreams until he meets a devil-like creature in his dreams, called the Number Devil. During his sleep, the Number Devil educates him about various mathematical principles centered on numbers, and eventually gets Robert more and more interested into mathematics.

Boring, hey?

Just kidding. Imagine having a mentor that could visit you during the life and teach you some of some amazing adventures!

1! 2! 3!

4! 5!

6!

…

Ah. You might be wondering why I’ve been shouting consecutive natural numbers in exclamation, but the typically used exclamation mark is also representing the factorial (!) sign.

For example, 4! = 4 * 3 * 2 * 1. (NOTE: “*” means Multiplication)

Or, as expressed by a variable, x! = x * (x-1) * (x-2) * (x-3)…*1

Let’s just go over some simple factorials, starting from 1.

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

…

How did the result of these factorials move from 1, all the way to 720? This concept of how quickly the factorials accelerated were called “Vroom!” by the Number Devil. It has given children a simple way to remember the overall concept of these factorials by the sound of a vehicle.

Of course, factorials aren’t JUST the sound of a car racing down the road: in fact, probability problems can be commonly solved by factorials, in a situation called permutation. What is it, you may ask?

Here’s a simple mathematics problem: If you were given 3 oranges and picked two at random to eat later, how many combinations would you have (assuming order does NOT count)?

…

…..

…

Get it yet?

Simply, your answer should be 3. For questions like these, many people find a diagram incredibly helpful.

Okay. NOW let’s say you were given 15 oranges and you could pick 10 at random.

So, I have included several samples. The grey triangles also represent a possible combination of 10 oranges, but there’s much too many to portray with a couple of markers. It’s also incredibly tricking listing each method, because the number of oranges you’re choosing is so large. It’s hard to keep track of all these combos by just a few lines.

This is permutation. For larger probability questions such as these, using the “Choose” formula can get your answer within half a minute (provided you have a calculator for the final step).

This Choose formula is denoted by nCx, where n is the total number of items, and x is the number of items you wish to take out of the set of items.

The formula is:

nCx= n! / x! * (n-x)!

Or, using the example above:

15C10= 15! / 10! * (15-10)!

You may think this made the situation quite a little harder, but using a simple crossing out method the result will simplify it to a quite small multiplication statement:

So, there are a total of 3003 possibilities, which is quite a considerably large number, and would’ve taken a lot of time to reach that answer by counting each individual way.

There are many other applications of factorials in the mathematical world, such as factors in algebra and calculus.

This is all I shall explain today. Be sure to read the Number Devil sometime, and find out the Number Devil’s actual name… without using Wikipedia.

~The Octopi