Author Archives: 24lsten

Chinese Mathematics: Not so Different from Western Mathematics

When we talk about mathematical discoveries certain names are mentioned.  These are names like Pascal, Euclid, Fermat, and Euler. These people become our mathematician Heroes. In our eyes, we often believe they pioneered the study. When we hear names like Mo Jing and Yang Hui in western society, most of us probably don’t even think anything of them.

But did you know that many of the great mathematical discoveries made in Western Mathematics were also made by Chinese mathematicians? In fact some mathematical discoveries we attribute to western mathematicians were even made by Chinese mathematicians far before they were discovered in the west.

I bring this up not necessarily to shame western culture, but because I find it fascinating.  We have two cultures that really didn’t intermix ideas and traditions, yet it seems that they have made many similar mathematical discoveries. In my opinion these similarities in a way show that two totally different cultures with cast differences still have profound similarities that can unite them.

Also in the great debate of whether math is manmade or discovered I personally believe the similarity between western and Chinese mathematics is a point for Team Discovered.  That might be only because I currently am on Team Discovered, though. I believe this is a point for Team Discovered because I feel if two separate cultures that are not trading ideas come up with the same mathematical truths then maybe they discovered them instead of just happened to share the same inventive thoughts. Still maybe this is the exact reason I should join Team Invention and I am just not thinking through my argument all the way.

Let’s talk about some of the similar discoveries in Chinese mathematics and western mathematics.  Let’s try to focus on the person behind a concept that both cultures discover/invented.  If feasible we should mention when the discovery/invention came about and how. Also how did it influence mathematics and the human race? I won’t focus on it, but you might even want to see if what we discuss puts you on team invention or team discovery for mathematics.

I guess the first Chinese work that I would like to point out is more of a compilation of Chinese works than the work of one individual, but did you know that of book very much like Euclid’s Elements existed in China?  This book was the canon of a group of people called Mo Jing. They were the followers of Mozi and the canon contained, among philosophical insights, works on geometry. Mozi was actually a Chinese philosopher, but his teachings inspired his followers to consider mathematics as well. In fact this book contained a definition for a point similar to Euclid’s. To be specific, “a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it.” (http://history.cultural-china.com/en/167H8715H13206.html)

Now, one of the most famous mathematical discoveries is “Pascal’s Triangle.” “Pascal’s Triangle,” is a fascinating work. To describe it you build it from the top down.  Put a one at the top.  Build the triangle down adding 1 more number in each row. The value of the number below is the sum of the two numbers directly above it. If it is an edge case the number is 1.

Image: Drini and Conrad.Irwin, via Wikimedia Commons.

This discovery, made by Pascal while, through letters, he was exploring probability with Fermat, was also discovered much earlier by a Chinese man named Yang Hui. Even before Yang Hui it was described earlier by Jia Xian in 1100.  Yang Hui in his book attributes the triangle to Jia and acknowledged that it was through this triangle that he found square roots as well as cubed roots.

I feel it is also very important that we discuss the book Zhou Bi Suan Jing.  This book, which is a collection of anonymous works, contains one of the first proofs of the well-known and widely used Pythagorean Theorem. As a refresher, this a2+b2=c2.  Controversy overshadows the actual date of the book which is assumed to be around 1046-256 BCE.

We can clearly see that mathematical ideas are not monopolized by western tradition.  In fact, in my studies of Chinese mathematics, I found references Pascal’s Triangle being found in India and Iran. Pascal was a genius, but clearly he was not the original discoverer of the triangle that bears his name. Mathematics is a global study, applied in many ways similarly by many cultures.

Take some time and identify a culture.  Make sure it is a culture that is so different from your own that, in a history class, this culture would study completely different things than what you studied. Now take what you know of your culture and the culture you chose and find similarities. Sometimes this can be hard. There are similarities such as in many cultures families eat together, but there are also many differences. What I am saying here is that in many ways math can be one of those similarities. This is neat! Math is as much a western study as it is an eastern study.

So next time you are learning about a western mathematician and how awesome he/she is, take some time and ask yourself if maybe the same ideas were explored by someone else in a different time in a different part of the world. Maybe even look it up. You might be surprised by what you find.

http://www.astro.umd.edu/~peel/CPSP118D_101/content/Zhou_bi_suan_jing.pdf

http://www-history.mcs.st-and.ac.uk/Biographies/Yang_Hui.html

http://history.cultural-china.com/en/167H8715H13206.html

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The Bridge of Asses

Image: Jenny Mealing, via Wikimedia Commons.

I don’t mean to be crude or inappropriate with my title.  After all a donkey used to be called an ass.  I don’t know what brought about taboo on the word, but in fact, it is by many considered a bad word.  I only used this title because it seemed a good attention grabber.  The phrase did, grab my attention after all.  But, what is “the Bridge of Asses,” and how does it have anything to do with Euclid or mathematics?  Why am I writing about “the bridge of Asses?” It may be because, “Ass” is the first swear word I ever used.  I was in a Shakespeare play called Much Ado about Nothing, and one of my character’s lines was, “You Are an ASS!”  Of course it was not so hard for me to accept the fact that I was swearing, as it was to accept the fact that my parents seemed to think it was funny and OK.  I was 12, but I digress. Let me tell you the real reason I am discussing, “The bridge of Asses.”

Sometimes we want to be able to tell if someone is really interested in something, or even if they are able to quickly grasp a concept.  Euclid’s fifth proposition in the first book of his elements was used to do just that.  Now before I proceed, lest I be accused of shaming people who have a hard time with math, I must say that I struggle very much with math and while reading about Euclid’s fifth proposition often felt like the “ass.” Don’t mock me! We all have our strengths and weaknesses. I am just trying to tell you about a something which I find interesting. Let’s talk about some history.

Around 1250 a man named Roger Bacon gave an alternate name to Euclid’s fifth proposition in the first book of his elements, which I will from here on out refer to as “the Bridge of Asses” or the fifth proposition.  The name he gave it was Elefuga, another word I will use freely to refer to the fifth proposition.  Elefuga, derived from Greek, means, “escape from misery.”  Medieval boys were presented with the Elefuga shortly before their “escape from misery.”  That is to say most medieval young men’s experience in geometry ended shortly after they encountered the fifth element, because it proved they simply did not want to go on or their mentor felt they should not.  They, like a donkey fears crossing a bridge, had a hard time grasping the fifth proposition or refused to grasp it. I personally believe they refused to try to grasp it or the mentor did not want to walk them through it well enough. This is because I think with time and patience people can overcome most barriers, but again I am digressing.

To better explain this, “the Bridge of Asses,” also known as the isosceles triangle theorem, is Proposition 5 of Book 1 of Euclid’s Elements.  But, also, pons asinorum, the Latin translation of “the Bridge of Asses,” became a metaphorical statement for a problem that will separate the confident from the unconfident. In other words it is a critical test, of the ability and understanding, of an individual. You see things like this all the time in movies. Usually someone has a sensei or master and they are trying to prove themselves. Eventually they come to the test that decides if they will continue with their training or not. For Bruce Wayne in Batman Begins it is, possibly, when he brings the flower to the League of Shadows high up in the mountain so that he can begin training with them.  Now we want to pass “the Bridge of Asses” for math, or proposition 5. Let’s see if you and I can manage to cross the bridge of elements together.

First, what is proposition 5?  Straight from Euclid’s elements, it is that, “In isosceles triangles the angles at the base equal one another, and if the equal straight lines are produced farther, then the angles under the base equal one another. Now, just hearing it makes sense, but to cross “the Bridge of asses” we must also prove proposition 5 and most importantly understand the proof.

Now I have read many blogs and articles proving the fifth proposition so I feel that I must make it clear that I am deriving this proof from an article, “the Bridge of Asses,” from www.britannica.com [1].  Also to make the proof more clear, I am going to list our proof in steps.

  1. We need to draw an isosceles triangle. It will have points ABC. For review, because I had forgotten, we must recognize that isosceles means that the sides AB and AC are equal.
    caption
  2. Now we want to extend past AB and AC indefinitely.

Step 2

  1. Now we want to add two more points D and E. The line AD will pass through the point B. The line AE will pass through the point C. AD and AE will be equal.

Step 3

  1. Now that we are past this point we must notice that the angle at DAC and the angle at EAB are equal. This is a simple to believe since they are the same angle.
  2. From step four we say that the triangle DAC and the triangle EAB have equal angles when all the corresponding side’s angles are compared. We can use the side-angle-side theorem to prove this.  It says that two triangles are equal when the triangles have two sides of the same length and the angle of those two sides is the same.

Step 4-5

  1. From step five we can conclude that the angles ADC and AEB are the same as well as that the lines DC and EB are equal.
  2. Now if we subtract AB from AD and AC from AE we can show that BD = CE.
  3. IT now holds by side-angle-side theorem that the triangles DBC and CEB are equal. If they are equal then so are the angles DBC and ECB
  4. We have now proven that the angels ABC and ACB are equal because the angle ABC = 180 degrees – the angle DBC and the angle ACB = 180 degrees minus the angle ECB when the two angles being subtracted are equal.

I hope that you found the proof I presented sufficient.  I don’t claim it as my own since I had to get help to cross this bridge. Hopefully I was able to help you across also, if you even needed help.  If you are still unsure, I suggest a pen and paper.  After all, that is really how it came to make sense to me.

Well now that we have crossed “the bridge of asses” together we are ready to further our careers in mathematics.  Really though I think the concept of “the Bridge of Asses” has a significant meaning. We will continually come across bridges in our education and careers. Sometimes we will feel that the bridge we are presented with is scary and hard to cross.  When I first saw the proof of proposition five that is what I thought. But if we take the time, think about it, and cross the bridge we will be that much better. Just like you and I crossed this bridge we can cross others. Don’t hold back, break a problem into steps, study it, think about it, and together we will cross “the Bridge of Asses.”

Sources

1) http://www.britannica.com/EBchecked/topic/724634/The-Bridge-of-Asses

2) http://aleph0.clarku.edu/~djoyce/elements/bookI/propI5.html

3) http://www.newworldencyclopedia.org/entry/Bridge_of_Asses

A New World of Thought in Fractions

I never thought about math as a fluid subject that would be approached differently depending on the needs of a culture.  We are taught from a young age how to do math and what it applies to, but I never thought that in different cultures, in a different time, people’s application of math could be so different from ours.

Let me explore my thought this way.  When I think of a math problem I need to solve on a daily basis, it is usually based around money, or computer memory.  I have five dollars and I want to buy a sandwich and fries.  Do I have enough money?  Or I have 3 GB of data for my phone.  Can I watch my favorite TV show on 4G or do I need to wait until I get home?  Many of the math problems we run into on a day-to-day basis support a monetary society where I have money, so I use that money to get more things.

This is starkly different from what we learned of different cultures in class.  The culture in particular I think of is Egyptian culture.  From the book, Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics, I learned that in ancient Egypt they had a very different fractional system than we do.  When we think of fractions we have a denominator and a numerator.  I like to think of the denominator as a bucket with n places for an object. The numerator is how many objects we have in the bucket, or how full it is.  In Egypt they had a very different system.  They represent fractions as a single number with a line over it.  The number represented an amount and the line indicated that it was a fraction.  It is a little foreign to us, but if the Egyptians wrote 6, with a line over the six, then we could think of it as 1/6.  It is important to note, and I will discuss later that this system can represent any fraction.  It would just take a bit more work in order to do that.

Why did they do it that way and why do we do things the way we do?  I think in our society our number system fulfills our desire to fill and make sure we have enough.  Our fractional system is very supportive of the money we use, because money is how we support ourselves. To elaborate on this I will use my bucket analogy for fractions. If you recall I said the denominator in our fractional system is like a bucket with room for n amount of objects.  The numerator tells us how many objects are in the bucket.  If the numerator and denominator are the same then we have filled one bucket. We use this all the time when we go shopping food.  When we shop the denominator is the price of what we want to buy.  The numerator is how much money we have.  If we have enough money, which represents the numerator, to fill the price, which represents the denominator, then we can get that item.  Or using my analogy you could say we filled the bucket.

In ancient Egypt their fraction system also could have supported their means of commerce.  They could have had more of a barter system were you worked for food and your master or boss gave you a portion of what you helped him produce.  For example if ten workers made 12 loaves of bread the boss possibly would have taken four for himself, leaving eight for the workers.  Using the Egyptian fractional system he could quickly and efficiently think, I have eight loaves for ten workers.  I will divide 5 loaves into halves giving me 2, with a line over the 2(1/2), which is, half a loaf for each worker.  Now I still have 3 loaves left.  If I divide those into fourths or 4, with a line over the 4(1/4), then I will have given each worker ½ a loaf and ¼ a loaf and there will be ½ a loaf left.  Know all I need to do is break that into tenths, but since it was a half loaf that would actually be twentieths.  That gives me 2 4 20, all with a with a line over them, which to us would be ½+1/4+1/20.  Now the boss had divided his loaves evenly and easily.

Learning this got me wondering.  How was it that other cultures did math?  Could I find another unique way fractions were used?  In the article, History of Fractions, Liz Pumfry talks about a Babylonian method of writing fractions.  Their method was also very simple and a little confusing.  To explain the Babylonian fraction system I first have to explain that the Ancient Babylonian number system was base 60.  The Babylonians wrote their fractions as demonstrated in the image below.

Image from nrich.maths.org.

The first grouping of symbols is for twelve and the second is for fifteen.  In Babylonian fractions this would come out to be 12 + (15/60).  That is, if this was a fraction.  Because the ancient Babylonians did not have a symbol for zero or a decimal point this grouping of numbers could have other meanings.

x60 Units Sixtieths Number
12 15 12+15/60=12
12 15 720+15

But why did the Babylonians use this method for fractions?  It seems to me that this method kept their number system simple, but in my quick analysis of this study I have found Babylonian fractions to be limited. In the article Babylonian Mathematics, it says, “Irregular fractions such as 1/7, 1/ 11, etc were not normally not used. There are some tablets that remark, ‘7 does not divide’, or ‘11 does not divide’, etc.” From this it seems we can deduce that the Babylonians disregarded some fractions.  It seems to me this could have limited them in some regards. For example, if they only used sixty as a denominator then the smallest number they could represent would be 1/60.  Let’s say they could increase this, though, using 602 or 60n.  If this is the case they could represent much smaller numbers, but it would still be difficult to represent values such as 1/7 or 1/13.  They might be able to do this by adding different values together, but that sounds very difficult and sloppy.   From this I conclude that they would lose accuracy with their fractional system.

Despite losing accuracy, addition and subtraction of fractions would be made much easier if the denominator was always sixty or a power of sixty.  This would make it easier for them to teach fractions as well as learn fractions.  They would not have to worry so much about finding a common denominator as we do.

It is very interesting to note that the Babylonian representation of fractions is very similar to our representation of decimals.  It is, in fact, so similar that I feel it is important to point out.  Our decimal system is base 10.  If we write 0.1 we are essentially writing 1/10.  This is essentially the exact same as the Babylonian system except they are using base sixty.  This is in fact so similar that it causes us to have the same problem representing some fractions.  If you want to write 1/7 in our decimal system you will quickly realize that this is impossible.  There is no concrete way to represent 1/7 in a base ten system.

It is very interesting to think about these different methods for representing fractions.  The Egyptians, if you think about it, actually had a very strong system that, while confusing, could represent all fractions very accurately.  It also suited their needs very well.  The Babylonians had a system very similar to our decimal system.  It seems foreign to us to use base sixty, but base ten is all we know and if they were confronted with our decimal system they would probably find it similarly confusing.  Our fractional system is, I believe, very strong, but in some ways I look at the Egyptian method and think it could have been very helpful when I wanted to share candy with friends or at the dinner table when everyone is fighting to make sure they get their equal portion of food.

In my studies I could not find any other Ancient Babylonian use of fractions.  I am also left to speculate on their reasons for their fractional system.  It is interesting to consider what a different cultures mathematical systems might have been and speculate as to why they used the method they did.

Sources

http://nrich.maths.org/2515

Count Like an Egyptian: A hand-on Introduction to Ancient Mathematics by David Reimer

http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf