Monthly Archives: September 2014

Why Do I Need to Learn This?

Why Do I Need to Learn This? Its an older than old question every student asks, any and every one, when stuck in a subject they have no interest in. Though as most of us can probably vouch for, no one ever hears this quite as much as in a mathematics class. Because really, when am I ever going to need to know how to find the area of a circle? Why should I care what pi is?  And who the heck ever uses calculus?

If we as a species were not mathematically inclined, then our society would never have developed the way it did. Even without the obvious technical professions that require a more advanced understanding, something as simple as bartering would have never existed without the concept of a fair exchange. Math does not just imply exact calculations; that feeling we get when the exchange “looks fair” is just our own inexact mental math giving us an idea of what is equal.

But that was years ago. We have money for that now, right? Well yes, but how do you think the system was outlined? What about credit and debit cards? The cards and the behind-the-scene workers that figure out your loans and credit are using all kinds of calculus so you don’t have to worry about it, trusting someone else to do it for you. You don’t need to know the background math as long as you have money right? Even if you don’t add or subtract the numbers yourself, how do you know whether the amount you have is great or small? This again returns us to our ability to gage an inexact mental measurement.

But why does it have to exist at all? Surely a time existed before math right?

Daniel Everett and Pirahã native on the cover of Don't Sleep, There Are Snakes: Life and Language in the Amazonian Jungle by Daniel Everett. Image: Robert Burdock, via Flickr.

Daniel Everett and a Piraha native on the cover of Don’t Sleep, There Are Snakes: Life and Language in the Amazonian Jungle by Daniel Everett. Image: Robert Burdock, via Flickr.

Well, not exactly. Mathematics, despite popular belief, was not something a bunch of smart people sat around and developed purely for the sake of driving their students to madness. It did not simply begin as the arithmetic we think of when hearing the word “math”, but simplicities such as numbers and magnitude. Even animal behaviorists are beginning to realize that they too have always had an active recognition of such things. Prehistoric evidence has given us a glimpse of our own ancestors’ concepts of just “one”, “two” and “many” that has not yet disappeared from this world. Though modern mathematicians have developed so very far beyond this humble start, there are still people today who do not have a counting system that extends any further. The Pirahã tribe, native to the Brazilian Amazons, is one such example. Their language does not contain numbers greater than two, though there is still debate whether this means some other part of their grammar does not imply counters. This tribe is an excellent example of a people with no advanced mathematic development, and yet a perfect example of how unquantified numbers are still a part of their everyday lives. Every day the hunters and gatherers are still faced with the question of how much food is enough for everyone, and though their numbers may not have a specific name or assigned symbol, their interpretation of “many” reaches greater estimation than mere numbers tell them.

Even if you’re never formally taught advanced mathematics, you still unconsciously uses math to some degree every day. Galileo was once quoted:

“The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics.”

Really, what is math but a tool? Just our own formally outlined way of understanding what goes on around us. There can’t be such a thing as a “world without math” because that world simply would not exist. Time would still pass, scientific and mathematical properties and functions would carry on even without our knowledge of what was happening. At least with even a slight mathematical background, we as humans are free to knowingly interact with our universe in a way no species has before. We do not simply trust the world around us, we understand it.

http://www.purplemath.com/modules/why_math.htm

http://www.wyzant.com/resources/lessons/math/calculus/introduction/applications_of_calculus

http://en.wikipedia.org/wiki/History_of_mathematics#CITEREFBoyer1991

http://edge.org/conversation/recursion-and-human-thought

Marveling at the Indian Genius: Ramanujan

Hardy left the cab. He was visiting an ill Ramanujan at Putney. Ramanujan asked about his trip. Hardy remarked the ride was dull; even the taxi number, 1729, was dull to the number theorist. He hoped the number’s dullness wasn’t an omen predicting Ramanujan’s declining health. “No Hardy,” Ramanujan replied, “it is a very interesting number.” The integer 1729 is actually the smallest number expressible as the sum of two positive cubes in two different ways. Indeed 1729 = 13 + 123 = 93 + 103.

leaf graphic
Ramanujan. Picture courtesy of Konrad Jacobs. From Wikimedia Commons

The mathematicians Srinivasa Iyengar Ramanujan and G.H. Hardy comprise the characters of this story. Ramanujan’s casual discovery of the smallest so-called taxicab number was no fluke. Ramanujan and Hardy’s most famous result was an asymptotic formula for the number of partitions of a positive integer. A partition of a number n is a way of writing n as the sum of positive integers. Reordering terms doesn’t change a partition. Thus the partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1; hence 5 has 7 seven partitions. Counting the partitions becomes difficult as n grows. Ramanujan had made several conjectures based on numerical evidence, and Hardy credits many of the needed insights to Ramanujan. The formula is complicated and counter-intuitive. It involves values of √3, π, and e, all very strange numbers for a counting formula. Near his death, Ramanujan discovered mock theta functions, which mathematicians are still “rediscovering” today.

These are only the achievements of later Ramanujan. The early life of Ramanujan is even more surprising. He spent his childhood in the poor south Indian town, Kumbakonam. In school, he scored top marks. He probed college students for mathematical knowledge by age 11. When he was 13, he comprehended S.L. Loney’s advanced trigonometry book. He could solve cubic and quartic equations—the latter method he found himself—and finished math tests in half the allotted time. If asked, he could recite the digits of π and e to any number of digits. He borrowed G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics by age 16 and worked through its many theorems. By 17, he independently investigated the Bernoulli numbers, a set of numbers intimately connected to number theory.

So How did Ramanujan do it? No brilliant teacher—excluding Hardy and J.E. Littlewood when Ramanujan was already an adult—taught Ramanujan. The odds were against Ramanujan from the start. Ramanujan often used slate instead of paper because paper was expensive; he even erased slate with his elbows, since finding a rag would take too long. He lost his first college scholarship by neglecting his every subject that wasn’t math. The first two English mathematicians he sent letters to request publication did not respond; was it luck that Hardy did?

Ramanujan credits Namagiri, a family deity, for his mathematics. He claimed the goddess would write mathematics on his tongue, that dreams and visions would reveal the secrets of math to him.

Ramanujan was undoubtedly religious. Before leaving India, he respected all the holy customs of his caste: he shaved his forehead, tied his hair into a knot, wore a red U with a white slash on his forehead, and refused to eat any meat. In the Sarangapani temple, Ramanujan would work on his math in his tattered notebook.

Hardy, the ardent atheist, didn’t believe that gods communicated with Ramanujan. He thought flashes of insight were just more common in Ramanujan than in most other mathematicians. Sure, Ramanujan had his religious quirks, but they were just quirks.

Looking at Ramanujan’s work, I find it hard to deny that some god aided the man. I cannot even imagine Ramanujan’s impact had he lived longer or had better teachers.


Sources and cool stuff:

biographical stuff: http://www-history.mcs.st-andrews.ac.uk/Biographies/Ramanujan.html, Christopher Syke’s documentary Letter’s from and Indian Clerk, Robert Kanigel’s The Man Who Knew Infinity

Ramanujan and Hardy’s original paper on partitions: http://plms.oxfordjournals.org/content/s2-17/1/75.full.pdf

Numberphile on taxicab numbers: https://www.youtube.com/watch?v=LzjaDKVC4iY

Wolfram on taxicab numbers: http://mathworld.wolfram.com/TaxicabNumber.html

Wolfram on Hardy-Ramanujan number: http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

Wolfram on Bernoulli numbers: http://mathworld.wolfram.com/BernoulliNumber.html

Ramanujan’s papers and notebooks: http://www.imsc.res.in/~rao/ramanujan/contentindex.html

mock theta functions: http://www.ams.org/notices/201011/rtx101101410p.pdf

G.S. Carr’s book: http://books.google.com/books?id=JLmCAAAAIAAJ&hl=en

S.L. Loney’s book: https://archive.org/details/planetrigonomet00lonegoog

Numberphile on Ramanujan constant: https://www.youtube.com/watch?v=DRxAVA6gYMM

Egyptian Numbers: Counting By Ten

When most people think of early mathematics, the first thing that comes to mind is probably Archimedes or Pythagoras of ancient Greece; or perhaps ancient Mesopotamia and its thousands of clay tablets, many of them containing math problems.  Few people think of ancient Egypt. This is mainly due to the fact that little is known about ancient Egyptian math in comparison to these other civilizations. However, by looking at what we do know about Egyptian math and the feats of engineering they accomplished with it, we can recognize how complex and sophisticated it was.

Rhind Mathematical Papyrus

A fragment from the Rhind Mathematical Papyrus. Image: Public domain, via Wikimedia Commons.

The majority of our knowledge of ancient Egyptian math comes from two papyri: the Rhind Mathematical Papyrus, which was originally written about 1985-1975 BCE; and the Moscow Mathematical papyrus, dating from around 1850 BCE. Math problems were not written on the walls of temples or great monuments in Egypt, so all math was probably done on papyrus or other perishable media, meaning that many mathematical works have not come down to us. This in in contrast to the mathematical documents from Mesopotamia, which were primarily done on clay tablets that are not as susceptible to the elements and ravages of time.

The ancient Egyptians probably developed the first base ten numerical system in human history. It was fully in use around 2700 BCE and perhaps even earlier. They had different symbols representing the powers of ten. A straight vertical line represented one, 10 was shown by a drawing of a hobble for cattle, a picture of a coil of rope served as a symbol for 100, a lotus plant delineated 1,000, a bent finger for 10,000, a tadpole or a frog for 100,000 and the picture of a god, perhaps Heh, represented 1 million. Although the Egyptian numerals were in a decimal system, it was not a positional place value system like the decimal system we use, or the sexagesimal (base 60) system developed by the Mesopotamians. Multiples of the powers of ten were written by repeating the symbol as many times as needed, and although they had a symbol for 1 million, other large numbers would have been very tedious to write. For instance, the number 987,654 would have required 39 characters. This Egyptian system possibly had an influence on the later Greek numerical system, but the Greeks improved upon it, creating a different symbol for each number 1-9, and other symbols for 20, 30 and so on.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

The Egyptians also had notation for fractions, although all Egyptian fractions were unit fractions (meaning they always had one in the numerator), with only two exceptions, 2/3 and 3/4. Fractions were marked by the hieroglyph for ‘R’ which is a long skinny oval, very similar in shape to the Eye of Horus. In fact, in an ancient Egyptian myth, the evil god Seth attacked his brother the god Horus and in the fight Seth gouged out the eye of Horus and tore it into pieces, fortunately the god Throth was able to put Horus’s eye back together and heal it. Thus, to honor the gods and this myth, whenever the Egyptians used fractions in relation to their measurement of volume, which is the hekat, the commonly used fractional parts were transcribed by using the corresponding parts of the Eye of Horus.

The great feats of engineering that the ancient Egyptians accomplished would have required an enormous amount of sophisticated math. The pyramids, for instance, are considered a marvel of mathematics and engineering. The base of the pyramids are almost perfect squares which the Egyptians would have achieved by using trigonometry, like the 3-4-5 trick. It was known in Egypt that a triangle with 3, 4 and 5 unit sides would always be a perfect right triangle. This Pythagorean triple, observed by the Egyptians long before Pythagoras, is sometimes called an “Egyptian triangle.” The Egyptians would have utilized this rule of geometry when laying out the base of a pyramid by tying knots in rope at 3, 4 and 5 unit intervals. Other tricks of trigonometry were known as well. The Rhind Mathematical Papyrus contains an equation for calculating the slope of a pyramid’s face, which is the same thing as finding the cotangent. It is also likely that they knew how to find a pyramid’s volume.

It is unfortunate that more is not known about the mathematics that were used by the ancient Egyptians, because they were obviously very skillful and innovative, and they must have been among the first people to develop important mathematical principals.

Sources:

http://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/

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

http://www.britishmuseum.org/explore/highlights/highlight_objects/aes/r/rhind_mathematical_papyrus.aspx

http://www.touregypt.net/featurestories/numbers.htm

Cryptography: A modern use for modular arithmetic

The common analogy used to describe modular arithmetic is fairly simple. All one has to do is look at an analog clock. For example, if it’s 11 AM and you want to know what time it will be in four hours, we instinctively know the answer is 3 PM. This is modular arithmetic, i.e. 11+4 = 3 mod 12. This is an important concept in the technology driven world we live in. Any time a product is purchased on the internet, cryptography comes into play. The remainder of this paper (pun most definitely intended) will describe how ancient modular arithmetic plays a very important role in today’s society.

History of modular arithmetic

The first known publication of modular arithmetic was in the 3rd century B.C.E, in the book Elements, written by Euclid. Within his book, he not only formalized the fundamentals of arithmetic, but also proved it. In what is known as Euclids Lemma, he states that if a prime number divides the product of two different numbers (x and y), then the prime number must also divide one of the numbers (either x or y), but it could also be both. Between the 3rd and 5th centuries a paper publish by Sun Tzu describes a modular arithmetic process known as the Chinese remainder theorem. This theorem is essentially the basis for modern RSA encryption schemes that are present on every banking/e-commerce website. It uses a congruent set of keys to produce the same numerical value. Imagine if there was a lock on a door that two differently cut keys could unlock and open, this is essentially how Chinese remainder theorem works.

Modern modular arithmetic

Oil painting of mathematician and philosopher Carl Friedrich Gauss by G. Biermann (1824-1908). Public Domain.

Oil painting of mathematician and philosopher Carl Friedrich Gauss by G. Biermann (1824-1908). Public Domain.

The modular arithmetic that we use today was discovered by Carl Friedrich Gauss in 1801.

Gauss is famous for numerous discoveries across a wide variety of fields in science and mathematics. Gauss’s proposition, from his book Disquisitiones Arithmeticae, defines modular arithmetic by saying that any integer N belongs to a single residue-class when divided by a number M. The residue-class is represented by the remainder, which can be from 0 to M-1. The remainder is obtained by dividing N by M. Given this fact, Gauss notices that two numbers that differ by a multiple of M are in the same residue-class. He then discusses the role of negative numbers in modular arithmetic. The following is an excerpt from his book:

“The modulus m is usually positive, but there’s no great difficulty in allowing negative moduli  (classes modulo m and -m are the same).  For a zero modulus, there would be infinitely many residue classes, each containing only one element.  [This need not be disallowed.]”

Modular Arithmetic’s Role Today

RSA encryption is named after those who invented it, Ron Rivest, Adi Shamir, and Leonard Adleman (RSA is obtained from their last names). RSA is the process by which information can be passed between two parties without another individual being able to intercept the message. Burt Kaliski has been one of the major contributors to RSA encryption since the 1980’s. I would like to start off with a passage from Burt Kaliski’s paper titled “The Mathematics of the RSA Public-Key Cryptosystem”:

“Sensitive data exchanged between a user and a Web site needs to be encrypted to prevent it from being disclosed to or modified by unauthorized parties. The encryption must be done in such a way that decryption is only possible with knowledge of a secret decryption key. The decryption key should only be known by authorized parties.”

This is a high level description of how RSA encryption works. It is also called public-key encryption, because anyone can obtain a copy of the encryption key it is publically available, but the decryption key cannot be obtained. This makes RSA encryption a secure way of passing data between an individual and a web site.

Simplified view of RSA encryption. Public Domain.

Simplified view of RSA encryption. Public Domain.

Performing this calculation (encrypting and decrypting text) is fairly simple. With a basic understanding of modular arithmetic it can be accomplished. First a public and private key must be produced by following the steps below:

  1. Generate large prime numbers, p and q (these should be hundreds of digits)
  2. Compute the modulus n, n = p×q
  3. Compute the totient, totient = (p-1)×(q-1)
  4. Choose an “e” > 1 that is co-prime to the totient
  5. Choose a “d” such that d×e = 1 mod totient

Once those steps have been completed, a public key (n, e) and a private key (n, d) have been generated. The public key can be distributed to anyone, but the private key must be kept safe. It’s easy to see that without the modular arithmetic this algorithm would be easy to discern. One could generate pairs of random numbers until a pair is found that when multiplied together, would equal the modulus n found in step two above. From there, it would be easy to find all numbers co-prime to the totient in step three. Modular arithmetic then comes into play, because it allows infinite pairs of numbers to satisfy the constraint listed in step five, but it would not allow the user to decrypt the message. In other words, 11+4 = 3 mod 12, but also 11+16 = 3 mod 12. This makes it impossible to determine what the original number was (it could be 4 or it could be 16, or any other multiple of 12).

Once the keys have been generated it is easy to encrypt and decrypt text. To encrypt a message “m,” given the public key (n,e) generated above:

C = me mod n

“C” is then the encrypted message that gets passed to the other party.

To decrypt the message “C” created above, all that is required is the inverse of the operation to encrypt:

M = cd mod n

Let’s do an example to illustrate the instructions listed above (note: we will be using small prime factors because the math is simpler).

  1. Select a p and q that are prime
    1. P = 11
    2. Q = 3
  2. The modulus n is then equal to P×Q = 11×3 = 33
  3. Computing the totient to be equal to (p-1)×(q-1) = (11-1)(3-1) = 20
  4. To select an “e” we must find a number that is coprime to 20
    1. The smallest value that is coprime to 20 is 3 because 3 is the smallest number that cannot divide 20 evenly, so “e” = 3
  5. Now we need to find “d”, d=e^(-1) mod [(p-1)×(q-1)]
    1. Using the Euclidian Algorithm we get d = 7

Now let’s say we want to encrypt the message “4.” To do this we need to know the public key, which in our case is (n=33, e=3).  All we have to do is compute:

 C = 43 mod 33 = 31

We can pass 31 (c=31) along to the website, which will then decrypt it using the private key (33, 7):

 M = 317 mod 33 = 4

Our message has been successfully “passed” from one place to another.

Thoughts

Without the work from previous mathematicians, this process would not be possible. Modular arithmetic plays a crucial role in our everyday lives and we don’t even notice it. I think it’s an amazing mathematical concept and provides a deep insight into the world of number theory. Even today there are computers constantly trying to figure out how to factor large prime numbers without success. I don’t know if RSA encryption will stand the test of time, but for now it’s the best we’ve got.

References

http://en.wikipedia.org/wiki/Cryptography#History_of_cryptography_and_cryptanalysis

http://www.britannica.com/EBchecked/topic/920687/modular-arithmetic

http://mathworld.wolfram.com/ChineseRemainderTheorem.html

http://www.mathaware.org/mam/06/Kaliski.pdf

http://www.numericana.com/answer/modular.htm#modulo

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://mathworld.wolfram.com/EuclidsLemma.html

Zero… A number

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

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

Think about our number system: now start counting out loud to five: 1…2…3…4…5. What happened to Zero? Most people do not think about using zero when they count. However, it is a very important and “worth-nothing” component of our number system. Without zero, we would not have placeholders and even doing simple math would be very difficult. Some may argue that the Babylonians were the first to utilize zero; but in fact, it was actually the Indians who initially made zero a number.

What the Babylonian and Indian number systems had in common was their use of “place values”. For example, this means the number 44 is broken down in the following way: the first four represents 40, while the second four represents the actual value of four. This is a concept that we still use in our in our number system today. Let’s think about the number 301. The three represents 300, the zero represents “nothing” in the ten’s place, and the one represents the actual value of one. Now imagine a mathematical world without zero. How would you explain 301?   The Babylonian’s created a symbol or marker to represent nothing in order to solve this “place holder” problem. That kind of sounds like the number zero right? Then what makes the Indian’s system so special as to deserve historical credit for number creation?

There are two different reasons why Indians deserves all the credit for our number system. First, they were the initial culture to represent zero as a circle, “0”, which is what we still use today. Second, zero was regarded as just another digit like 1 to 9; it was a number and not just seen a placeholder anymore. Brahmagupta an Indian mathematician and astronomer, developed the first set of rules regarding how zero would work as a number.

They are the following:

  • When zero is added or subtracted from a number, it remains the same.

Example: 4+0=4 and 4-0=4

  • When a number is multiplied by zero, it is zero.

Example: 5 x 0=0

Brahmagupta came up with these rules in the 7th century, and we are still using the exact same rules today in the 21st century.

However, when it comes to division, Brahmagupta did make a mistake. He said that any number divided by zero equals zero. We all know what is wrong with that statement; any number divided by zero is not zero. It wasn’t until the 12th century when another Indian mathematician, Bhaskara, proved that a fixed number divided zero was in fact infinity  and not zero. Once again, a very basic rule that we still use today, any fixed number that is divided by zero is infinity, came from an India.

I find it very interesting that our number system, as well as a lot of our basic mathematical skills, comes from India, and yet we are never taught this school. Our education system is very Eurocentric, meaning it is based on European principles that are centered on their history and their culture. After finding out how much India contributed to our current mathematics’ structure, I believe it is important for students today to learn about our system’s correct history and origin, as well.

As a future educator going into mathematics, I am going to teach my students where the math they are doing comes from. I encourage current and future educators to do the same thing so as to correctly teach students and future generations how greatly India has contributed to our mathematics system.

Works Cited

Bellos, Alex. “Nirvana by Numbers.” Theguardian.com. Guardian News and Media, 07 Oct. 2013. Web. 13 Sept. 2014. http://www.theguardian.com/science/alexs-adventures-in-numberland/2013/oct/07/mathematics1

“Brahmagupta – Indian Mathematics – The Story of Mathematics.” Brahmagupta – Indian Mathematics – The Story of Mathematics. N.p., n.d. Web. 13 Sept. 2014. http://www.storyofmathematics.com/indian_brahmagupta.html

Karmakar,Purnendu. “Brahmagupta.” Wikimedia Commons. JPG file.  http://commons.wikimedia.org/wiki/File:Brahmagupta.jpg

The Base of Money, and Other Adventures

Image: slgckgc, via Flickr.

Image: slgckgc, via Flickr.

The other day in class, there arose an interesting discussion about one of the most common topics in math classes. We were talking about bases. It is a very interesting topic, because there is so much room for creativity. One interesting thing about bases is that we don’t quite know why we are using a base ten number system. We have our theories, but nothing is quite concrete. (We just say that it’s the number of fingers we have, and um… yeah, we leave it there.) (I honestly think that’s what the ancients did.) In class it was mentioned that in the Bible the base ten number system is used, and it was theorized that maybe ten being the “holy base system” we cannot change it. Being in Utah, and with the predominant religion being Mormonism, Prof. Lamb asked what base they used in the Book of Mormon.

Bases Used in a Religious Text

In the Book of Mormon, while troop numbers are given as nice multiples of ten, indicating a base ten numbering system, in one part of the text (in Alma 11) the currency used by the culture is laid out, which has an interesting property of doubling from one unit to another.
Now, we know what doubling means, Binary! (only your computer is excited, sorry) In the text it lays out the following system:
Gold Coins
8= Limnah *(see bottom of entry)
4= Shum
2= Seon
1= Senine
Silver Coins
8= Onti *
4= Ezrom
2= Amnor
1= Senum
(and now for the interesting part where they go all fraction on us)
1/2= Shiblon
1/4= Shiblum
1/8= Leah
Now because we don’t know everything about this civilization, we don’t easily know how much any one of these coins actually bought, except for saying that either a “Senine” or “Senum” could be used to buy “a measure of barley” or any other kind of grain.
Let me be the first in saying that knowing that a “measure” of grain is equal to x is very little information, but if we decide that a “measure” is a useful quantity of grain that is enough to actually eat, we could say that 1 kilogram (or 2.2 lbs.) is a very useful amount of grain for a small family for a few days. To my knowledge the cost of that much wheat, or other grains is about $2. (I am assuming that there is a great deal of error involved in my guessing game.)

Now based on this, we might say that the system as it stands is a bit inflexible, unable to go to very high numbers, as it maxes out at roughly sixteen dollars, and hits a minimum at about 12.5 cents, but with a few tweaks on our part, applying the same pattern, we can achieve a wide array of numbers and a very intriguing property (at least to me). This most intriguing property, is that within this system any change given is rather simple to calculate, and give. This is based on the fact that if I used a theoretical 2^8 coin (256) for an item of value 102, I would get change in the form of I could get change as just a series of these coins (128), (16), (8), and (2) which is basically 10011010 (in binary)

Now I don’t know about you, but if I wanted to give you change in a way that I could just look at my coins and take out the biggest one that cuts down the difference with less of a need for calculations. This is called the greedy algorithm, where the largest coin possible is taken, and used until it can’t be used anymore, continuing until no more can be subtracted. In a base 2 coinage system, while performing this algorithm, or method for choosing, there is never a need for repeated coins.

The main drawback of this system is that it uses 14 different levels of currency to make it from 1/64 to 128 (roughly 1 cent to $100) while to go from $.01 to $100 we only need 12 (counting $.50, and $2, which are almost never used)

Basically the gist of the story is that within this religious text, The Book of Mormon, we find that they have a currency system with very interesting properties that come from being based on a binary system.

* In the text, it is stated that a Limnah, and an Onti are “as great as them all” which could mean that it is the sum of all the previous values, or in my interpretation, it is the greatest of them all, or worth more than all the others put together.

Sources

Original Text Where the Monetary system appears
https://www.lds.org/scriptures/bofm/alma/11.5-19

Why We use Base 10
http://ideonexus.com/2008/07/08/why-a-base-10-number-system/

Greedy Algorithm
http://math.stackexchange.com/questions/106317/for-what-coinage-systems-does-a-greedy-algorithm-not-work-in-providing-change
http://www.sciencedirect.com/science/article/pii/S0195669809001292

I thought we had him, Sophie

Portrait of Sophie Germain by Auguste Eugene Leray. Image: public domain, via Wikimedia Commons.

Portrait of Sophie Germain by Auguste Eugene Leray. Image: public domain, via Wikimedia Commons.

Like most people, I end up having dinner with my family every so often. Unlike most people, our family conversations always seem to include math or physics. Generally we end up trying to stump each other with various questions. It’s like a game to us.

A few nights ago we had one of those dinners. I found a lull in the conversation and I fired off, “Hey dad, do you know any significant female mathematicians?” His jaw went a little loose and he gave me a blank stare. It was the same look people give me when they find out I talk about math at the dinner table.

My crippling question had developed from a classroom discussion about an influential female mathematician, Sophie Germain. I’d realized in that discussion that I could name several influential female scientists. For whatever reason, I’d never heard of any women known for their mathematics.

As the feeling of triumph settled in, a smile developed across my face. Sophie and I were about to taste victory. I watched confidently as my father’s eyes slowly rolled back into his head. Gradually his eyes came back down and his smile met mine. I knew I was in trouble when he said, “Well, the oldest one I can think of is Hypatia”.

He went on to tell us the story of the daughter of Theon of Alexandria. Theon was a known scholar and professor of mathematics at the University of Alexandria. His daughter, Hypatia, was given all the best. In particular, her education was second to none. With such great influences, many historians believe she was able to eclipse her father’s knowledge at an early age. In time, people would come from distant cities to learn from her.

Sadly, none of her original work has survived to this day. As a mathematician we remember her largely for her editing and insightful comments on other great works of the time. Some of the more important works included the Arithmetica by Diophantus and Conics of Apollonius. The book on conics was particularly significant. It contained progressive information about cutting cones with planes that helped develop ellipses, hyperbolas and parabolas.

Unfortunately, Hypatia is possibly better known for the way she died than the way she lived. During her lifetime, the quality of life in Alexandria was on the decline. Fighting had developed among the different religious factions and it threatened to destroy the city. At one point, the Roman emperor ordered the destruction of all pagan temples. As an educated pagan that often spoke about non-Christian philosophy, Hypatia was a likely target. Ultimately her end came when a group of Christians pulled her from her carriage, drug her into a church, stripped her, beat her to death, tore her to pieces, burned the pieces and disgracefully scattered her remains.

At this point my mother had made it to the other side of the table. She began slapping my father’s arm while muttering something through clenched teeth. “Ok, Ok”, he conceded and continued, “My favorite was Emmy Noether anyway”. “She died of natural causes!” he taunted my mother as she wandered into the kitchen. By this time my smile was long gone. He’d already won the game and was just showing off.

Much like Hypatia, Noether was the daughter of a successful professor of mathematics. Her German family was quite wealthy and provided all her needs. Unfortunately, sometimes society isn’t as helpful.

Noether found herself suffering from restricted access to the University of Erlangen. Her father was on faculty there and I expect it felt like home. Fortunately, she was able to audit classes until they started accepting female students. I’m just speculating here, but I’d like to think she helped drive the decision. A brief four years later she was awarded her Ph.D. summa cum laude.

In spite of her obvious achievement, society still wasn’t quite ready for Noether. She spent the next several years working with David Hilbert and other prominent mathematicians at Göttingen University. Due to a lack of insight, much of the faculty refused to allow a female member. As a result, she worked for free.

As the years went by at Göttingen she worked on incredible theories and taught classes in Hilbert’s name. Eventually her situation improved when she was allowed to become a licensed lecturer. The university wasn’t paying her yet, but at least she could collect student fees. Years later she was finally granted a position as an adjunct teacher.

It’s worth mentioning she was considered a remarkable instructor. More than a few of her students went on to make significant contributions to mathematics. If all things were equal, it would be hard to imagine her without tenure.

While Noether may not be well known to the general public, great minds have given her enormous respect. Einstein referred to her as a “significant creative mathematical genius”. This may well be due to her contributions to general relativity.

Her major mathematical contribution was in a 1921. It was a groundbreaking paper on the theory of ideal rings. She was able to think of things in an extremely abstract way. Many consider her the mother of modern algebra.

One of the more easily understandable ideas she developed involved creating ascending or descending chains. Imagine a set A1 is contained in a set A2, which is contained in a set A3, etc. Sometimes you can see that after a certain point, the rest are the same. For example we might notice that A10, A11, A12, etc are all the same. On the surface it may not seem like a big deal. But techniques like these are frequently exploited in proofs.

Quite honestly, the scope of her contribution is breathtaking. It warms my heart to know people have been making such impressive advances in the last hundred years. It’s even more rewarding to see remarkable individuals like Emmy Noether push through an unjust barrier. In spite of never being treated equally, she emerge a champion with the landscape changed behind her.

By the time my father had finished his glowing praise of Emmy Noether, my face was in my palm. I had clearly lost my challenge. I thought we had him, Sophie.

http://www.smithsonianmag.com/womens-history/hypatia-ancient-alexandrias-great-female-scholar-10942888/

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

http://www.agnesscott.edu/lriddle/women/hypatia.htm

http://www.bu.edu/lernet/pathways/bios.html

http://www-history.mcs.st-andrews.ac.uk/Obits2/Noether_Emmy_Einstein.html

http://www.encyclopedia.com/topic/Emmy_Noether.aspx

http://en.wikipedia.org/wiki/Emmy_Noether#Ascending_and_descending_chain_conditions