Monthly Archives: February 2015

Numbers Courtesy of Fermat and Mersenne

A portrait of Pierre de Fermat, lawyer and amateur mathematician. Image: Author and painter unknown, via Wikimedia Commons.

It is not often a person contributes to a field they do not even work in the way Pierre de Fermat has contributed to the field of mathematics. Born to a wealthy leather merchant, Fermat received a bachelor’s in civil law from the University of Orléans and went on to become a lawyer, while at the same time engraving his name into math history books, doing said math just for recreation. His importance in mathematics lead to many theorems named after him, as well as numbers. These numbers are known as Fermat numbers, which are positive integers, that take of the form Fn = 2(2n) + 1, when n is nonnegative and an integer. For example for F1, F1= 2(2)+1= 5. The first five Fermat numbers are 3, 5, 17, 257, and 65537, and these numbers continue to grow to incredibly large magnitudes. Fermat believed this form created an infinite number of prime numbers, which are known as Fermat primes.

Fermat numbers are occasionally written as 2n+1, but since when n is greater than zero and Fn prime, n must be a power of two, the form Fn = 2(2n) + 1 is the common form for Fermat numbers. One of the main problems with Fermat claiming all these numbers are prime is the fact that they soon become too large to calculate for even today’s computers, let alone a man with his pen and paper in the 17th century. Unfortunately for Fermat, by the time the 18th century rolled around, he was dead. In 1732, mathematician Leonhard Euler found that F5, which is 4,294,967,297, is actually divisible by 641, most likely figuring this out from having a large amount of time on his hands. While this showed that some Fermat numbers are not actually prime, excluding when n=0 in the form 2n+1, it does not discount the fact that the Fermat number equation could still make an infinite number of primes, since there are infinite amount of Fermat numbers. However as of now, the only Fermat primes that are known are F0 through F4.

Now initially I found the idea of an equation, the equation here being Fn = 2(2n) + 1, that finds only certain prime numbers, most of which are way too large to even be calculated even 400 years after the equation for them was created, the equivalent to a student doing extra credit when he has a 98% in his class. What I’m trying to say is, I found Fermat numbers pointless and to be the 17th century mathematician’s version of a braggadocio. However, I know nothing and Gauss managed to find a relationship between “Euclidean construction of regular polygons and Fermat primes,” where he showed a regular 17-gon could be constructed. It was also found a regular n-gon can be created if n is the product of any number of Fermat primes and the number 2. These regular n-gons take the property of being able to be constructed with a compass and straightedge. Who would have thought one of the greatest mathematicians to ever live could leave me feeling so inadequate, at least mathematically.

Similar to Fermat numbers are what are known Mersenne numbers, created by 17th century French mathematician and music theorist Marin Mersenne. Yet again a person being a jack of all trades, except instead of being a master of none they were a master of a few or at least of one. Mersenne numbers take of the form Mn = 2n -1, and Mersenne primes are numbers that take that form which are prime. Mersenne believed that for n<=257, Mn was prime for n= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and the rest are composite. While this belief turned out to incorrect, he still got the name for the primes. Just like Fermat primes, it is unknown whether there are an infinite number of Mersenne primes, but as of now 48 Mersenne primes are known, the largest being 257885161-1, which again makes me wonder how much time do some of these mathematicians have on their hands.

Mersenne numbers were originally studied because of their connection to perfect numbers, which are positive integers that are equal to the sum of their divisors. Euclid proved that if the number 2n-1 is prime, then 2n-1(2n-1) is a perfect number, which many years later led to Euler discovering that all even perfect numbers come in this form. Another interesting fact is that the ten largest known prime numbers are Mersenne numbers. I personally find number theory incredibly interesting, partly because I like numbers and partly because how mathematicians are able to come with these theorems and proofs baffle me. I ultimately wonder if they had any true goals when thinking about these primes, or if it was just for the pure fun and interest in it.


Why not Babylonian numerals?

Numbers and systems for writing them have a very long and varied history. Not every number system used the same base, in fact, some used base 5, base 12, base 20, or even base 60!

But wait, you might ask, what is a base?

Positional number systems (like the customary Arabic numerals) represent numbers as multiples of a base and powers of it. For example, in our base ten Arabic numerals, this is what we mean when we write the number 1559.37:

Number 1 5 5 9 .3 7
Power of 10 103 102 101 100 10-1 10-2
Meaning One group of 1000 Five groups of 100 Five groups of 10 Nine ones Three groups of one tenth Seven groups of one hundredth

Now, we don’t have to 10 as our base as we did there. We could have used any number other than zero or one. Actually, base one number systems exist: tally marks use base one. However, they are not truly positional number systems.

The Babylonians used 60. In their base 60 number system, this would have been the way they thought about 1559.37:

Number 25 59 .22 12
Groups of this power of 10 601
Meaning 25 groups of 60 59 groups of 1 22 groups of 1/60 12 groups of 1/3600

Of course, the Babylonians didn’t write it like that. They would have written this:

The Babylonian number system has separate symbols for each number from one to 59:

Table of Babylonian numerals. Image: Josell7 via Wikimedia Commons

I will not write out any more Babylonian numerals with their notation. Instead I will use parenthesis around normal base 10 numbers. For example, I will write 60 like (1)(0).

Babylonian numbers were written in a fashion similar to ours. The first numeral on the left was the most significant, or the one representing the largest value. The second one was the second most significant and so on. At first, a blank space was used to mean what we would use zero for. That would have sometimes been problematic, as it might not always be clear how many blank spaces had been left. Around 311 BC a placeholder was added, . It was not the same as our zero. It was only used between other numerals as a placeholder. It never was used at the end of a number in the way we use zero in numbers like 10 or 200. The Babylonian system did’t include a direct equivalent to our decimal point. That is, if you wrote the numeral for one it wasn’t totally clear if you meant 1 or 60. The reader had to know something about what you were writing to be able to figure that out. On the other hand, it is usually pretty clear if a number should be 1, 60, or 3600. You wouldn’t wonder whether you were looking at one ox or 60 oxen!

This was a significant advance compared to previous numeral systems such as the non-positional Egyptian one. Before Babylonian numerals, most systems had a different symbol for each power of the base: a symbol for 10, another for 100, and so on. That meant that it was not possible to write numbers larger than a certain amount in those systems.

Base 60 numbers have significant advantages over base 10 numbers. 60 is a very nice number. It can be divided evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, or 30. Ten, on the other hand can only be divided evenly by 2 or 5. This makes a difference in how easy it is to write fractions. In base 10, such a simple fraction as 1/3 does not terminate: 0.33333… In base 60, it is easy to write since 1/3 is 20/60. It would be written as 0.(20) where (20) is a single numeral. In general, it is much easier to write a fraction if it’s denominator is a factor of the base, hence more numbers can be expressed simply with base 60 numbers.

Fun facts:

  • One drawback to base 60 numbers is that if you wanted to memorize a base 60 multiplication table, you would have to memorize 1830 entries! In base 10 there are only 55.
  • The angle composing 1/360 of the circle is a Babylonian invention. In base 60 numbers, it made sense to define it this way as 360 in Babylonian numerals would be (4)(0). Furthermore, the degree is divided into 60 minutes and each minute is divided into 60 seconds.
  • Our 60 minute hour also came from the Babylonian preference for base 60 numbers.
  • It has been suggested that base 60 arose from a finger counting system. On one hand, each of the 12 finger bones represented a unit. To represent multiples of 12, the thumb or a finger from the other hand was placed between two different fingers.
  • The Babylonians knew enough astronomy to realize that there are 365 days in a year. In base 60, the 365 is (4)(5). Writing it that way seems a bit tidier.


NRICH by the University of Cambridge

Babylonian Numerals on Wikipedia

Mayan Mathematics

After we talked about Babylonian mathematics and how they used a base-60 system, it got me thinking about different ancient cultures and the numbering systems that they used. My little brother is currently on an LDS mission in Guatemala, and he sent me some pretty cool pictures of Mayan ruins. Also, I remember back to 2012 when many people were hysterical about the “end of the world” because the Mayan calendar had stopped on a specific day in December. But man, if I was part of a civilization from thousands of years ago, I would NOT have made a calendar that went even that far. Had I been in charge, people might have thought the world was ending around the time Columbus made it to the Americas. A lot of what we know about the Mayan people was lost when they were invaded by the Conquistadors from Spain. A Spanish missionary, Diego de Landa, had a great respect for the Mayan people, but despised their religious customs. He ordered most of their religious icons, texts, and other documents to be destroyed, but of those that survived remain the Dresden Codex, the Madrid Codex and the Paris Codex. (Side note: I think it’s interesting that these documents are named after those that found them rather than those that created them.) The Dresden Codex, which will be discussed hereafter, is probably the most well-known. The Mayan people were a city-building and innovative society. There were 15 large cities (some of 50,000 or more) in the Yucatan peninsula. The people were governed by “astronomer-priests” that manipulated others with their religious instructions. (True case of where math and knowledge is power!)

The Maya number system. Image from MacTutor History of Mathematics.

Image from MacTutor History of Mathematics.

The Mayans had one of the most advanced number systems in the world at its time. It was a base-20 system (kind of) that also relied quite heavily on the number five. Some think that this is due to five fingers and five toes on each hand and foot. Their system relied on three different symbols. A “pebble” (small black circle) was used to represent the ones place. A “stick” (straight black line) was used to represent the fives place, and a shell was used to represent the number zero. In a report about the Mayan numbering system, JJ O’Connor and EF Robertson explained why the system was not exactly a base twenty system, but one that had been slightly modified: “In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20’s up to 19, the next the number of 400’s up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20’s up to 19, it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 202, the next the number of 18 × 203 and so on.” (O’Connor & Robertson, 2000)

The Dresden Codex contains written evidence of the use of this numbering system. Ifrah claims that the system was used in astronomy and calendar calculations. He states, “Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.” (Ifrah, 1998)

One of the reasons the Mayan numbering system was not a true base-20 system was because it was partially a base-18 system as well. The reason for this seems to be the calendars that they maintained. Robertson and O’Connor state: “The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 “months” of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short “month” of only 5 days that was called the Wayeb…” (O’Connor & Robertson, 2000).

It amazes me what the Mayans were able to do with the knowledge that they had available to them. Much of what we know of the Mayans comes from the few documents that have been preserved and the ruins that have been interpreted, but I wish that more information was available on how they used these systems in their societies and the other ways in which they used mathematics.

Ifrah, G. (1998). A universal history of numbers: From prehistory to the invention of the computer. London.
O’Connor, J., & Robertson, E. (2000, November). Mayan mathematics. Retrieved from The MacTutor History of Mathematics Archive:

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

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.


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

Where do numbers come from, anyways?

A short history of imaginary numbers

Mathematicians first came up against imaginary numbers in the mid 16th century and it wasn’t until the mid 19th century that they saw how awesome complex numbers could be. Before we look at how imaginary numbers came to be, let’s look at some other familiar number systems.

Number Systems Solve Problems

The first, most obvious, number system is the integers, or counting numbers. We have integers to answer really useful question that we see all the time in day-to-day life like, how many grapes can I really stuff into my mouth at a time? (About 9)


The next number system we might think about is the rational numbers, or fractions. These also serve to answer really useful day-to-day questions like that involve division like, if I have 6 roommates but only 1 pint of ice cream, what portion of the tub can I eat?

This assumes that I’m a fair roommate who would never eat more than her share of the communal ice cream- which leads us to our next number system. Negative numbers are used to measure debt; like how much ice cream I might owe my other roommates.


With these two systems we can count and divide stuff, but we also might have other sorts of problems like how to measure things. Like, for instance, I might need to walk 1 block south and 2 blocks east around a park to get to school, but since I’m inherently lazy (a good quality for all mathematicians), I cut straight through the park, and find that I’ve walked √5 blocks to get to school, which is totally irrational. We have to deal with irrational numbers when we measure distances because it turns out (to the Greeks’ great sorrow) that not all distances can be measured with rational numbers.

So what about imaginary numbers?

Where did they come from, and what are they good for?

We’ve got Real Problems: Imaginary Numbers give Real Results

In the mid 16th century a mathematician named Tartaligia came up with a general solution for finding the roots of 3rd degree polynomial, but he held his method as a closely-guarded secret. Another mathematician named Cardano eventually managed to convinced the reluctant Tartaligia to tell him the method, on the condition that he would never ever tell anyone else. Well, I think they should make a soap opera about 16th century mathematics because in 1545 Cardano completely betrayed Tartaligia by publishing the solution in his book ‘Ars Magna’.

Tartaligia’s method is really important in the history of imaginary numbers because there are some perfectly good 3rd degree polynomials with perfectly good real roots that this method doesn’t make sense for. When you use Tartaligia’s method for these certain polynomials, you get a nonsense step in the middle of the calculation where you have to take the square root of a negative number.

Consider for example the equation:

x3 = 15x + 4.

 This cubic has a real root x = 4, but when we apply Cardano’s formula we get:

x = ∛[ 2 + √(-121) ] + ∛[ 2 – √(-121) ]

The real problem (pun intended) was that even though everyone knew that taking the square root of -121 was totally ridiculous, they also knew that the root x=4 was a totally reasonable real solution. There was this breakdown in what the equation was trying to communicate.

The first mathematician to really break through this mold was Rafael Bombelli, who got around this problem with the crazy proposition that, well let’s just imagine that there’s some number that’s negative when we square it. With this assumption he was able to manipulate Tartaligia’s equation, for instance the example above becomes:

 ∛[ 2 + (√-121) ] + ∛[ 2 – (√-121) ] = (2 +(√-1) ) + (2 –  (√-1))   (**!)

= 4 – 2(√-1) + 2(√-1) – (√-1)2

= 4 (!)

Conveniently, the ‘imaginary’ numbers cancel out, leaving good real roots! Way to take a leap of faith, Bombelli!

** Okay, hold on, what just happened there? Well it turns out (2 +(√-1))3 is :

(2 +(√-1))3 = (2 +(√-1))*(3 + 4i) = (2 + 11i) = 2 + (√-121)

 Same goes for ( 2 – (√-1)). Neat.

About a half-century later in 1637, Descartes coined the term “imaginary” when he wrote about roots of nth degree polynomials in his book ‘La Geometrie’. He wrote that these polynomials might have as many as n solutions, but sometimes they have fewer, as some of the solutions are ‘impossible’, ‘improbable’ and ‘imaginary’. He meant it in a demeaning way- like we should be doing ‘real math’, not ‘day-dream math’.

At this point in history mathematicians swept imaginary numbers under the rug; they cautiously imagined that they might exist but only for long enough to cancel out and yield real solutions. It wasn’t conceivable that they might be useful by themselves.

It’s sort of a complex story

 The complex number system was really first understood as the incredibly powerful mathematically tool that it is in the 19th century when Gauss took an interest in imaginary numbers. He came up with a geometric interpretation for complex numbers (which, to be fair, was also independently discovered by the Norweigan mathematician Wessel and the French bookstore manager and amateur mathematician Argand). Gauss’ interpretation was that the imaginary number line is just like the real number line, so a complex number (a number with a real and an imaginary part) is actually a coordinate in a plane, like in the image below. We just say that real numbers lay on the horizontal axis, while imaginary numbers lie of the vertical axis.


The really amazing and exciting thing about this description is that it’s totally consistent with operations we might like to do on complex numbers, like addition and multiplication. Consider what happens when we multiply by i, for instance 1*i. We rotate 90 degrees, from the coordinate (1, 0) to the coordinate (0, 1), so we can say that multiplying by i is the same as rotating by 90 degrees. Then consider i*i (which is i2): we rotate another 90 degrees and end up at -1! Neat!

Later in the 19th century complex numbers got a lot of traction because they turned out to be very good at describing waves. At this point in history, physicists were developing ways to describe electricity and magnetism, and complex numbers enabled them to really understand these phenomena.

The neat thing about complex numbers is they show up everywhere in our day-to-day lives. Anything you have that uses electricity only works because some engineer somewhere knew how to build it using their understanding of imaginary numbers. Can you even imagine your life if you couldn’t send your mom photos of other people’s dogs? Any time you snap a photo or make a phone call your phone does a Fast-Fourier-Transform, which is a method based on complex numbers, to compress the data into just tiny amounts of storage.

So do imaginary numbers really exist?

Complex numbers are great representations for lots of natural phenomena, like electricity. Remember how we used our other number systems- like how we used integers to count how many grapes I could fit in my mouth? In some sense, it’s just the grapes that exist, not that the integers. The integers exist mathematically- they’re only there to describe the real world, and this is true for every number system. In this sense, not only do imaginary numbers ‘exist’ mathematically, but they’re first-class citizens because they describe so many awesome things that we use every day.


The Beauty of The Elements


A stone statue by Joseph Durham depicting the famous mathematician Euclid. Image: Garrett Coakley via flickr.

When I was in high school, I eventually learned about the mathematical subject known as geometry. Unlike most schools though, instead of our teacher having us sit down and listen to them talk about the subject, our teacher had each and every one of us go to the library and rent a copy of Euclid’s Elements (Book 1). From that point on till the end of the first semester, each day we would separately read from Euclid’s Elements and then try to prove to our teacher each and every postulate using Euclid’s methods. It wasn’t until recently that I discovered that most children do not learn about geometry in this fashion and how unique of an experience I had. While I can see some of the possible advantages behind the new ways people learn about geometry, I still believe that Euclid’s The Elements has its own advantages that some of these other sources don’t.

One of the most noticeable things about The Elements is that each and every one of Euclid’s postulates build exceptionally well off of each other. While I see proofs building off of each other in most other texts books, there is just something about the way it is done in The Elements that feels much smoother. Perhaps the big advantage with a book like The Elements is that it was never meant to be a “text” book but rather a book for people who are interested in learning about geometry. Because of this, it doesn’t have to continually throw out real world examples or ask the reader to try to use this proof in specific scenarios. Instead, The Elements will just make a statements, go about proving that statement, and then go straight into making another statement and most likely prove it using the previously proven statement.

Another difference between The Elements and other geometry books which I believe makes it far superior is the general way in which it goes about solving proofs. Nowadays, most geometry books will use a popular form of algebra and a number system to solve equations. However, Euclid’s Elements is fully self-contained and takes nothing for granted. Because this book was created in a time where people didn’t necessarily have access to other sources, everything that is necessary to understand what is being stated in this book is there; including its own algebraic system. This self-contained version of algebra within The Elements uses simple comparisons between lines and shapes to each other which replaces constants and variables found in other forms of algebra (which is also explained in the book) to prove that the different statements that are being made are true. These comparisons in combination with previously proven statements allows The Elements to create proofs of all different kinds. While the algebraic like system Euclid’s Elements uses to solve equations may be a little difficult to get one’s mind around sometimes it makes the proofs within its pages much more difficult to refute than other geometry books.

So, why do we not use this book to teach students about geometry today? Perhaps the biggest reason and most obvious is that The Elements is a difficult book to read. Unlike most textbooks today, it doesn’t use numbers and doesn’t give examples. However, just because current day geometry books are easier to teach with and easier for students to understand does not mean that they are better books. Perhaps the final reason that I believe The Elements is such a great geometry book compared to others is that the reader must want to learn about geometry if they wish to get anywhere in Euclid’s Elements.  But, if they are able to get through Euclid’s Elements, they will have a much stronger fundamental idea of geometry than from other textbooks. While it is easy to state the fact that someone who survives being stranded in the wilderness will have a better idea of how to survive in the wild than someone who hasn’t, it doesn’t change the fact that it is true.

Going back to my classroom experience, I thoroughly enjoyed going through the proofs in The Elements and I would spend most of my lunch time going to my teacher and proofing more of Euclid’s Statements. After about 2 weeks of starting the book, I had finished it. After that point, I spent the rest of time in class helping other students understand The Elements. Unfortunately, most of the other students had a hard time getting through that semester and only a few other students were able to understand it in a similar fashion as myself. However, those among us who did understand Euclid’s The Elements had no troubles passing the second semester of class which was going back to the more common form of geometry. In conclusion, I believe that Euclid’s Elements is a fantastic book that does more for geometry than any other book out there and, if someone is really interested in geometry, they should do their best to read through and understand The Elements if they want the best foundation in geometry they can have.




Hypatia of Alexandria – The Unsung Mathematician

In our studies of math, we learn from hundreds of individuals who’ve greatly influenced our perspectives of mathematics. These famous individuals created various theorems and proofs, and left us with questions to resolve. Individuals such as Euclid brought us marvels that were passed on for generations. His book The Elements created a foundation for many mathematicians, and helped them find breakthroughs in science and math. Unquestionably, our society and technology may not be where it is if it weren’t for these individuals. As we look at our own development of math, we learn names such as Pythagorean and use it repeatedly throughout our math education. Although the works of these persons are marvelous, there are many mathematicians and scientists who are unknown. They may not have found a theorem that completely changed the world around us, but they contributed towards something equally great. Truly these are our unsung mathematicians that are unknown to the world but known to the few who’ve learned or specialized in the field of math or science. One individual that I’ve found and never heard of before was Hypatia. Her biography may not be found in a history book about Rome, but she was an influential person to those around her. She is considered to be the first female mathematician known. Personally, reading about her impresses me because the math profession at the time was highly male dominated and lacked women. I can imagine that she had to go through many challenging experiences to prove herself as a respected mathematician. Her life could be seen as a tragedy and it is even romanticized in the 2009 film Agora, but those who knew her revered her greatly.

Depiction of Hypatia. Image: Public domain, via Wikimedia Commons.

Who was she?
There is little information about where Hypatia was born or even a birth date. Scholars believe that she was born between 351 AD and 370 AD. Around this time period, Rome was in a slow decline, and there were civil disputes because of the influence of Christianity. Her father Theon was a well-known mathematician and astronomer who taught Hypatia the foundations of her knowledge in those fields. It’s significant to note that Theon was the last known survivor of the Museum of Alexandria and played a part in the preservation of Euclid’s The Elements. Hypatia and her father Theon would collaborate on commentaries on classical mathematical works, and she would eventually have students of her own. As she followed her father’s footsteps, she also developed knowledge in philosophy and became head of the Platonist school of Alexandria. She primarily taught the works of Plato and Aristotle and it was recorded that many people would come from different places to learn from her. She would wear the robes of a scholar and nobody questioned that she didn’t wear the traditional female clothes. Her work was well respected everywhere and it simply didn’t matter how she appeared to her audiences. The people of Alexandria revered her as a virtuous figure and an intelligent scholar. She believed in Neo-Platonism, a belief system that states everything emanates from the One. This belief system states that the One is considered God or the Good of all things. But what’s interesting is that the One is neither existent nor a being. This abstract thought provoked the teachings of Christianity and most likely made Hypatia a target for zealots. Although Hypatia’s beliefs conflicted with others of the Christian faith, she tutored Synesius of Cyrene who later became a bishop and incorporated Neo-Platonism into Christianity.

Depiction of Hypatia’s Death. Image: Public domain, via WIkimedia Commons.

Hypatia’s popularity to those around her eventually led to her own death. There are a couple of different accounts of her death but it basically resulted from a dispute between her friend Orestes, the Governor of Alexandria, and Cyril, the Bishop of Alexandria. Whatever conflict or feud these two individuals had, this led to Hypatia being brutally murdered by a mob of Chrstian zealots. There are two different documentations of this event and little is known how old she was when she died. Although she was murdered, her students fled to Athens and the school she taught in Alexandria continued her teachings.

Her Works and Legacy
Many of Hypatia’s works were destroyed around 652 AD when the Arabs invaded Alexandria. However, a couple of letters that she exchanged with her students were still available. One of her students mentioned before, Synesius, exchanged a letter that talked about a hydrometer and astrolabe. These writings expanded on the concepts and structures of these two objects. Some other works that she has done, in collaboration with her father, were commentaries on Arthimetica by Diophantus and The Elements by Euclid.

Hypatia leaves a legacy that has been seen as an inspiration or tragedy. Many politicians, poets, and writers used her death as part of a cause for them and to influence others. I see her as one of our history’s unsung mathematicians who had so much more to give if she had lived longer. As time has progressed, we live in a day and age where there are more opportunities for unsung heroes such as Hypatia. Because of technology and access to colleges and universities, anyone can study mathematics. Although you and I may not be the next Euclid or even Hypatia, our small contributions and interest may eventually lead to something great and we can be known as the unsung heroes of our generation.


The Development of Zero

How many elephants are in the same room as you right now? Most people would answer zero to that question (if you answered something else, we should be friends). The concept of zero is familiar to us. Earlier today, my two-year-old cousin told me that his baby sister is zero years old. I filed sales taxes for my business and typed up countless zeros. Today, zero is part of daily life. Even a two year old understands the concept of zero.

Zero is nothingness — a void. If you think deeper, it’s fairly amazing that we throw around such a profound term. I can see, touch and count the number of teabags left in a box, but I can’t see, touch or count the number of elephants in my bedroom. There are also zero storm troopers, zero cookies and zero dinosaurs in my bedroom. In my bedroom, there are an infinite number of zeros. Our number zero, symbolized by “0,” enables us to do calculus, and it’s even half of the reason my computer works right now. In the early days of math, zero didn’t exist — there wasn’t even a word for it, which made even simple arithmetic a bit complicated. Thankfully, ancient Babylonian, Mayan and Indian mathematicians developed the concept of zero and paved the road for truckloads of discovery and innovation.

Just like ours, the Babylonian number system (2000 BC) was positional. In our base 10 system, having a positional number system simply means you have a position for ones, tens, hundreds, etc. Babylonians used the same concept except their ones position included the numbers 1-59 instead of 1-9. Regardless of base, the problem with having no zero is the numbers ‘11’ and ‘101’ suddenly both look like ‘11’. Most people can’t read minds, so that makes understanding other people’s writings a bit difficult. The Babylonians developed a place holding symbol to solve this dilemma. For example, if we used a period as a placeholder, those numbers would look like ‘11’ and ‘1.1’. It dispersed some confusion, but the placeholder could only be used between numbers, so ‘1’ and ‘100’ both looked like ‘1’. Without a zero, modern mathematics had no chance of developing.

Mayan placeholder symbol. Image: public domain via Wikimedia Commons.

Similarly to the Babylonians, the Mayans developed a placeholder symbol that stood for zero. They developed the notion completely independently of the Babylonians — after all, they were half way around the world and didn’t have texting. Their symbol for zero supposedly looks like a shell. To me, it looks more like a spaceship, but I digress. They had the concept of a placeholder, but like the Babylonians, they didn’t use the symbol on its own. Again, its a start, but you can’t add, subtract or multiply using a placeholder.

Brahmagupta, an Indian mathematician and astronomer. Image: public domain, via Wikimedia Commons.

A 19th century image of Brahmagupta. Image: public domain via. wikimedia commons.

The hero of this story is a Hindu astronomer by the name of Brahmagupta. Around 628 AD, Brahmagupta wrote down rules for getting to zero using addition and subtraction and the results of using zero in equations. There are earlier traces of zeros in Cambodia and various parts of India, but Brahmagupta’s account is primary because it gave the rules behind using zeros. Brahmagupta called zero ‘sunya’ or ‘kha’ which mean ‘empty’ and ‘place’ respectively. His rules included things like ‘the sum of two zeros is zero’, ‘the product of a zero and any other number is zero’, and ‘zero divided by a zero is zero’. These rules were revolutionary. As simple as they seem, this one list of rules effectively changed the entire human world. You may have noticed something wrong with one of those rules — our modern mathematics don’t allow you to divide by zero. Brahmagupta’s rules about dividing by zero may have been flawed, but that just means he left something for G.W. Leibniz and Isaac Newton to work on later!

After zero became a fully formed number, it spread like wildfire. Along with spices and other tradable goods, Arabian voyagers brought zero back from India. A hundred years after Brahmagupta discovered zero, it reached Baghdad. In the 9th century, a man named Mohammed ibn-Musa al-Khwarizmi started to develop algebra by working on equations that equaled zero. He called zero ‘sifr’ which turned directly into our word ‘cipher’ and eventually developed into our word ‘zero’. Come 879 AD, people wrote zero almost exactly like we do today; the only difference between our zero and theirs was size. They used an oval that was smaller than the other numbers — it became ‘1’, ‘1o’ and ‘1oo’. Finally, when the Moors invaded Spain they brought zero to Europe, and by the mid-1900s, Al-Khowarizmi’s work reached England at last.

Zero is universal; it transcends culture, space and time. It is part of our global language and is one of the most fundamental ideas in calculus, physics, engineering, computers, and a lot of financial and economic theory. Our lives are full of zeros. Plus, after traveling around the entire world and changing the course of human history, zero inspired this brilliant little video. Enjoy!


How Fermat’s theorem contributes to modern computer science

After reading several different sources of math history, I was attracted by Fermat’s history and his theorems about prime numbers. I realized that I am kind of interested in these numbers and anything related to these numbers after digging into the computer science major for few years. Why? I think the primary reason is because prime numbers are widely used in computer science for security. For example, the RSA algorithm, which is a public-key crypto system, uses prime numbers to generate public keys. If you are not familiar with the RSA algorithm, I will explain a little bit more here. Basically, the RSA algorithm generates a public key based on two large prime numbers. The prime numbers are secret. Anybody is able to use the public key to encrypt a message. When I first learned this RSA algorithm in my algorithm class, I felt like it was magic! After taking computer networking, the computer networking class, I realized that prime numbers were everywhere in my life. For example, you go to to buy some stuff, you need to login to your account and pay the bill online. Behind the scenes, there are prime numbers securing your account and transactions. This interest brought a new question to me, which was how to generate a big prime number? Another way to think about this question is how to determine whether a number is a prime?


FLT and modern computer science. Image by the author.

Because of my curiosity about prime numbers, I enjoyed reading and thinking about FLT. In class, FLT means Fermat’s Last Theorem, but in this blog post, it means Fermat’s Little Theorem. Basically, FLT says if p is a prime number, then for any integer a, a^p – p is a multiple of p. If a is not divisible by p, then a^(p-1) – 1 is an integer which is multiple of p (Wikipedia). How can this math theorem help a computer science student? I solved some programming questions related to prime numbers. For example, generate all prime numbers less than n. If n is 10, my program should return 2, 3, 5, 7 as a result. The other programming question was to write a program that runs a primality test, which is used to see whether a number is prime. For example, if the input number is 10, my program should return true if 10 is a prime number and false if 10 is not a prime number. I was able to solve such questions by using an inefficient algorithm. If a number was very big, it took more than an hour to get the final result. Obviously, I did not expect such a slow algorithm. By combining FLT into programming, I can write an efficiency algorithm to solve those questions.

Computer science majors might be wondering how a Fermat primality test works. First of all, we have an integer n and we need to choose some integers co-prime to n. Then we need to calculate a^(n-1) mod n. Let’s say the result is different from 1, then n is composite. If the result is 1, then we can say n may or may not be a prime number. For example, let’s use p = 341 and a = 2, then we have 2^340 = 1 (mod 341). Obviously, 341 is not a prime number because 341 = 11 * 31. So we call 341 a pseudoprime base 2. Such a pseudoprime number is also called a Carmichael number. This is a primary reason why we cannot directly use the FLT for a primality test: it will be fooled by some Carmichael numbers. That means we have to do something else to get it work on solving programming questions! There are only 21853 pseudoprimes base 2 for n from 1 to 25 * 10^9. How does this data help us? If 2^n – 1 mod n = 1, then n is a prime number, or n is one of those 21853 pseudoprimes. I think we can build a list of pseudoprimes before running our program for primality test. When we need to check if a number is a prime number, we can check if this number is a pseudoprimes in the list. If it is not one of the pseudoprimes, we can start running the program for primality test. The FLT was like the roots of a binary tree. People used this root to branch out its left children and right children. In our case, one of the children referred to is the Fermat primality test. At this moment, I cannot stop digging deeper in this tree. I want to traverse the whole tree just like running a breadth-first search in my brain.

Number theory is a mystery! After reading the articles, I asked myself a question “How many unknown theorems still exist?” It might be like the universe – people have only explored a tiny part of it. It may hide more secrets behind the scenes – people need time to reveal those secrets. Probably after a few years, scientists will discover many more theorems, just like FLT.

Egyptian Fractions: From Mythology to Computation


Horus and Set. Image: Soutekh67, via Wikimedia Commons.

While we in the United States often associate the All Seeing Eye with the Illuminati or the little eye on top of the pyramid on the dollar bill, in Egypt it is better known as the Eye of Horus – a symbol of healing, life, and resurrection. In Egyptian mythology, Horus was the son of Isis and Osiris, king of Egypt. Osiris had a brother named Set, who after years of watching his brother bring prosperity and joy to the people of Egypt became jealous of his brother and killed him to claim his throne. Once he had come of age, Horus sought to reclaim the throne that was rightfully his. In a battle between Horus and Set, Set gouged out Horus’s eye and ripped it into six pieces (the fight wasn’t too one sided – Set lost a testicle). Seeing the damage of the rightful heir to Egypt, the god Thoth collected the pieces of Horus’s eye and restored them using magic. Thus, the Eye of Horus represents health or wholeness.

The Egyptians ascribed a specific set of unit fractions to the six parts of the Eye of Horus. Each fraction is associated with one of the pieces that the eye was ripped into as well as each of the five senses (sight, smell, taste, hearing, touch) plus a sixth sense represented by the Egyptians as thought. These fractions, 1/2 for smell, 1/4 for sight, 1/8 for thought, 1/16 for hearing, 1/32 for taste, and 1/64 for touch, were often used to measure amounts of pigment, ingredients for medicines, and grain.

It is interesting to note that the Horus Eye fractions were the only unit fractions used in dividing grain and that they were written differently from other unit fractions when used in this context. The base unit of grain used by the Egyptians was the hekat which is approximately 1/8 of a modern day bushel. Rather than using standard unit fraction notation for hekat, each piece of the Horus Eye hieroglyphic became a symbol for each fraction (see table below). That way, if the symbol for a specific part of the Horus Eye was used, it was known to measure a fraction of hekat.

The only exception to this was ro. If the fraction required was less than 1/64, then a seventh unit fraction, ro, was used. Like the Horus Eye fractions, ro was not written as a typical unit fraction, but was symbolized by an open mouth hieroglyphic. One ro was somewhere between a teaspoon and a tablespoon and was used to measure 1/320 hekat. I was not able to find any writings about the origin of ro or why 1/320 might be a significant unit fraction other than 5 ro = 1/64 hekat.


Fractional representation of the Eye of Horus. Image:

From today’s perspective, this collection of numbers makes up the first six terms of a geometric series whose sum converges to one. It is interesting that the symbol for “whole” would only sum to 63/64 in Egyptian times. Some scholars think that the missing 1/64 represents the magic used to reassemble the eye to make it whole. Some consider it intentionally imperfect because mortals cannot achieve the perfection of the gods. Still others think it has value in relation to the Egyptian unit fraction ro. Whatever the reason may be, this collection of sacred unit fractions of the form 1/2n is just one of many forms of Egyptian mathematics that is derived from the fraction 1/2.

As a matter of fact, most of Egyptian mathematics is centered around halving or doubling quantities and manipulating those halved or doubled quantities to perform calculations. The only additional multiplier often seen in Egyptian mathematics is 2/3, but even this appears to be a result of manipulations of 1/2 (discussed below). The fraction 3/4 is also seen on rare occasion, but I couldn’t find any records or calculations based on this.

Archaeologists have found many Egyptian artifacts containing mathematical tables that were most likely used as reference materials by scribes during computation. Many examples of such tables can be found in the Rhind Mathematical Papyrus (RMP). The Recto of the RMP contains a table of unit fraction sums where the sum is equal to 2/n and n is an odd numbers between 3 and 101. Another table in the RMP lists the results of finding 2/3 of unit fractions with odd denominators. Problem 61 of the RMP begins with a table of five unit fractions and the results of halving those unit fractions. Elsewhere in the RMP, there is a table of doubled odd unit fractions. The lack of tables to describe operations with fractions other than 1/2 and 2/3 further supports the hypothesis that Egyptians strictly used these fractions to obtain results for other fractional quantities. Also, the lack of scratch work or steps along with the ordering of the problems suggests that these were reference tables rather than independent problems. This is particularly clear in Problem 61 of the RMP because the table is given before calculations were made in the first and second parts of the problem that clearly used the values in the table in their computations.

The Egyptians appear to have developed various algorithms for solving specific types of problems. For example, Problems 32 and 34 of the RMP show two sets of equations that exhibit the characteristic, if a*x=b, then (1/b)*x=(1/a). No additional work is provided around the problems, indicating that either scratch work was done elsewhere or this was a general rule known to Egyptian scribes.

In calculating halves of unit fractions described as a sum of other unit fractions, the Egyptians knew to double the denominator of each unit fraction in the sum. An example of this is found in problem 1 the Egyptian Mathematical Leather Roll (EMLR). In this problem, 1/4 is equated to 1/7+1/14+1/28. By doubling, the result is 1/8 is equal to 1/14+1/28+1/56.

A similar process is used to calculate 1/3 of a known unit fraction sum. Interestingly, Egyptians did not simply multiply their denominators through by three, likely because they were so dependent on their tables of halves and two-thirds. Instead multiple problems in the RMP show that scribes would first calculate 2/3 of each unit fraction and then calculate 1/2 of each of the resultant fractions to get to 1/3 of each of the original unit fractions in the sum.

The use of 2/3 instead of 1/3 is generally thought to be a result of manipulations of 1/2. There are many problems in the RMP that show the process of finding 2/3 of any unit fraction as a sum of other unit fractions. Problem 61B of the RMP show the Egyptians understood that 2/3 of any unit fraction, 1/n can be written as (1/2n)+(1/6n). For the case of even unit fractions, this is always appears to be simplified to 1/(n+(n/2)). Both of these calculations simply rely on manipulating halves or wholes of the original fraction to calculate the desired fraction.

Another interesting relation known the Egyptian scribes is demonstrated by ten, mostly successive lines in the EMLR. In Mathematics in the Time of the Pharaohs, Richard J. Gillings calls this the G Rule: “If one unit fraction is double another then their sum is a different unit fraction if and only if the larger denominator is divisible by 3. The quotient of the division is the unit fraction of the sum.” Many examples of this rule are written with no scratch work or intermediate steps shown such as line 11 where 1/9+1/18=1/6. But amongst those problems are problems of a slightly different form. Lines 1, 2, and 3 of the EMLR show that 1/10+1/40=1/8, 1/5+1/20=1/4, and 1/4+1/12=1/3 respectively. These seem to follow the same principal of the G Rule, but for cases when one fraction is triple or quadruple another and the larger fraction is divisible by four or five respectively. According the Gillings, this suggests that the Egyptians had a more inductive understanding of the G Rule: “If one of two unit fractions is K times the other, then their sum is found by dividing (K-1) into the larger number, providing the answer is an integer.”

From all these translations and analyses of original sources, I found it interesting that the groupings of problems on different papyri suggest mathematical rules that Egyptian scribes were familiar with but are not explicitly proven. This suggests to me that the Egyptians discovered rules of mathematics by discovering patterns through repeated computation and if those patterns were regularly helpful in further computations they were organized and made into reference tables. It was strange to me that many of these similar problems are not listed in any particular order or group all together unless they appear in a designated table. To me, this would suggest that these problems were done for practice, but so little information is known about the origins of these sources that it is difficult for even experts to deduce their original purpose.



Gillings, Richard J. Mathematics in the Time of the Pharaohs. Cambridge, Mass. MIT, 1972. Print