What are the p-adic numbers?

The p-adic numbers are a completion of rational numbers with respect to the p-adic norm and form a non-Archimedean field. They stand in contrast to the real numbers, which are a completion of the rationals with respect to absolute value and which do respect the Archimedean property. The p-adic numbers were first described in 1987 by Kurt Hensel and later generalized in the early 20th century by József Kürschák which paved the way for a myriad of mathematical work involving the P-adics in the just over a century since.

To start talking about the p-adics it makes sense to start with the Archimedean property, which as mentioned above the p-adics do not adhere to.  First we need an absolute value function or valuation, ||, which is basically a function that gives some concept of magnitude to an element of a field (so it’s a mapping from a field to the positive real numbers). Given this, the zero element should map to zero and the valuation of the product of two elements should be the same as the product of the valuation of each element individually. The last condition (which determines whether or not the valuation is Archimedean) is that if the norm of an element is less than or equal to 1, then the norm of 1 plus that element is less than some constant, C, which is independent of the choice of x. If the last condition holds for C equal to 1, then the valuation satisfies the ultrametric inequality: the valuation of 2 elements is less than or equal to the maximum of the valuation of each element. If the ultrametric inequality is satisfied then the valuation is non-Archimedean. Otherwise it is an Archimedean valuation.

While this is a bit opaque, it makes more sense now moving into defining Archimedean fields: a field is Archimedean if given a field with an associated valuation and a non-zero element, x, of that field, then there exists some natural number n such that |Σk=1nx|>1. Otherwise,  the field is non-Archimedean and the ultrametric inequality holds. Basically what this means is that if we can measure distance and we are living in a nice Archimedean world, if we walk forward we can go as far as we want. While if we were to live in a non-Archimedean world and we try to walk forward we would at best stay in place and possibly move backward.

Now that that’s out of the way and (hopefully) the weirdness of a non-Archimedean world has been established, it’s time to talk about the p-adics. Any non-zero rational number, x, may be expressed in the form x ,where a and b are relatively prime to some fixed prime p and r is an integer. Using this, the p-adic norm of x is defined as |x|=p-r , which is non-Archimedean. For example, when p=3, |6|=|2*3|=1/3, |9|=|32|=1/9 and |6+9|=|15|=|3*5|=1/3 or when p=5 , |4/75|=|4/(52*3)|= 25, |13/250|=|13/(2*53)|=125 while |4/75 + 13/250|=|17/325|=|17/(52*13)|=25. So now that we have this we can proceed identically as when constructing the real numbers using the absolute value and define p as the set of equivalence classes of Cauchy sequences with respect the p-adic norm. After some work it can be shown that every element in pcan be written uniquely as Σk=makpk,where am does not equal zero and m may be any integer.

The most common use of p-adics I found was in showing the existence (or lack thereof) of rational or integer solutions to problems. For example, the Hasse principle (also known as the local-global prinicipal ) was discovered by Helmut Hasse in the 1920’s and attempts to give a partial converse of the statement that if a polynomial of the form Σaijxiyj+Σbixi+c=0 has a rational solution then it has a solution for all expansions of Q. The Hasse principal asserts that if such a polynomial has a solution in R and every Qp then it has solution in Q. An example of this is x2-2=0, which has (irrational) solution square root of 2 in R. However, it does not have solution in Q5 , and so by the Hasse principal it does not have a solution in Q, which we know to be true. Another use of the P-adics which is fairly interesting is in transferring standard real or complex polynomial equations to their tropical (the semi ring of the reals with the addition of an infinity element under the laws of composition addition and min (or max)) polynomial counterpart, a process which runs into issues due to the necessity of the ultrametric inequality.

Sources:

http://www.cut-the-knot.org/blue/LocalGlobal.shtml

http://en.wikipedia.org/wiki/P-adic_number

http://mathworld.wolfram.com/p-adicNumber.html

http://mathworld.wolfram.com/p-adicNorm.html

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

http://www1.spms.ntu.edu.sg/~frederique/antchap5.pdf

http://www.math.ucla.edu/~i707107/HasseMinkowski.pdf

Isaac Newton and his Contributions to Mathematics

Sir Isaac Newton. Image: Arthur Shuster & Arthur E. Shipley, via Wikimedia Commons..

In class we discussed the Fundamental Theorem of Calculus and how Isaac Newton contributed to it, but what other discoveries did he make?

Sir Isacc Newton was born on January 4, 1643, but in England they used the Julian Calender at that time and his birthday was on Christmas Day 1642. He was born in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father had already passed prior to his birth and his mother remarried after his birth and left Isaac to live with her mother. He went to The King’s School, Grantham from the time he was twelve until he was seventeen. His mother removed him from school after the death of her second husband, but later allowed him to return by the encouragement of the school’s headmaster. He rose to be at the top in rankings in his school, mainly motivated to get revenge towards a bully. He began attending Trinity College in Cambridge in 1661. After receiving his degree he developed his theories on calculus over the span of two years during the plague [1].

Newton’s work in calculus intitially started as a way to find the slope at any point on a curve whose slope was constantly varying (the slope of a tangent line to the curve at any point). He calculated the derivative in order to find the slope. He called this the “method of fluxions” rather than differentiation. That is because he termed “fluxion” as the instantaneous rate of change at a point on the curve and “fluents” as the changing values of x and y. He then established that the opposite of differentiation is integration, which he called the “method of fluents”. This allowed him to create the First Fundamental Theorem of Calculus, which states that if a function is integrated and then differentiated the original function can be obtained because differentiation and integration are inverse functions [2].

Controversy later arose over who developed calculus. Newton didn’t publish anything about calculus until 1693, but German mathematician Leibniz published his own version of the theory in 1684. The Royal Society accused Leibniz of plagiarism in 1699 and the dispute caused a scandal to occur in 1711 when the Royal Society claimed Newton was the real discoverer of calculus. The scandal got worse when it was discovered that the accusations against Leibniz were actually written by Newton. The dispute between Newton and Leibniz went on until the death of Leibniz. It is now believed that both developed the theories of Calculus independently, both with very different notations. It should also be noted that Newton actually developed his Fundamental Theory of Calculus between 1665 and 1667, but waited to publish his works due to fear of being criticized and causing controversy [1].

Newton not only discovered calculus but he is also credited for the discovery of the generalised binomial theorem. This theorem describes the algebraic expansion of powers of a binomial. He also contributed to the theory of finite differences, he used fractional exponents and coordinate geometry to get solutions to Diophantine equations, he developed a method for finding better approximation to the zeroes or roots of a function, and he was the first to use infinite power series.

His work and discoveries were not limited to mathematics; he also developed theories in optics and gravitation. He observed that prisms refract different colors at different angles, which led him to conclude that color is a property intrinsic to light. He developed his theory of color by noting that regardless if colored light was reflected, scattered, or transmitted it remained the same color. Therefore color is the result of objects interacting with colored light and objects do not generate their own colors themselves [1].

Sir Isaac Newton was a truly amazing mathematician and scientist. He achieved so much in his lifetime and the amount of discoveries he made can seem almost impossible. He helped make huge advancements in mathematics and created theorems that we still use heavily to this day.

[1] http://en.wikipedia.org/wiki/Isaac_Newton

[2] http://www.storyofmathematics.com/17th_newton.html

Leonardo of Pisa – The Great Fibonacci

Figure 1-Fibonacci. Image: Public domain, via Wikimedia Commons.

Most mathematically inclined people are familiar with the famous and unique Fibonacci sequence. Defined by the recurrence relation (*) Fn=Fn-1+Fn-2 with initial values F1=1 and F2=1 and (or sometimes F0=1 and F1=1), the Fibonacci sequence is an integer sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …) with many remarkable mathematical and real world applications. However, it seems that few are as well informed on the man behind this sequence as they are on the sequence itself. Did you know that Fibonacci didn’t even discover the sequence? Of course not! Predating Fibonacci by almost a century, the so called “Fibonacci sequence” was actually the brainchild of Indian mathematicians interested in poetic forms and meter who, through studying the unique arithmetic properties of certain linguistic sequences and syllable counts, derived a great deal of insight into some of the most fascinating mathematical patterns known today. But with a little bit of time (few hundred years), some historical distortion, inaccurate accreditation[1], and a healthy dose of blind western ethnocentrism and voila! Every high school kid in America now thinks there is a connection between Fibonacci and pizza. Or is it Pisa? (That’s a pun, laugh.) While often given more credit than deserved for the “discovery” of the sequence, Fibonacci was nonetheless an instrumental player in the development of arithmetic sequences, the spread of emerging new ideas, and in the advancement of mathematics as a whole. We thus postpone discussion of Fibonacci’s sequence – don’t worry, we shall return – to examine some of the other significant and often overlooked contributions of the “greatest European mathematician of the middle ages.”[1]

Born around the year 1175 in Pisa, Italy, Leonardo of Pisa (more commonly known as Fibonacci) would have been 840 years old this year! (Can you guess the two indexing numbers between which Fibonacci’s age falls?[2]) The son of a customs officer, Fibonacci was raised in a North African education system under the influence of the Moors.[3] Fibonacci’s fortunate upbringing and educational experience allowed him the opportunity to visit many different places along the Mediterranean coast. It is during these travels that historians believe Fibonacci may have first developed an interest in mathematics and at some point come into contact with alternative arithmetic systems. Among these was the Hindu-Arabic number system – the positional number system most commonly used in mathematics today. It appears that we owe a great deal of respect to Fibonacci for, prior to introducing the Hindu-Arabic system to Europe, the predominant number system relied on the far more cumbersome use of roman numerals. It is interesting to note that while the Hindu-Arabic system may have been introduced to Europe as early as the 10th century in the book Codex Vigilanus, it was Fibonacci who, in conjunction with the invention of printing in 1482, helped to gain support for the new system. In his book Liber abbaci[4], Fibonacci explains how arithmetic operations (i.e., addition, subtraction, multiplication, and division) are to be carried out and the advantages that come with the adoption of such a system.

Figure 2-Golden spiral. Image: Weisstein, Eric W. "Golden Spiral." From MathWorld--A Wolfram Web Resource.

Figure 2-Golden spiral. Image: Weisstein, Eric W. “Golden Spiral.” From MathWorld–A Wolfram Web Resource.

Whereas the number system most familiar to us uses the relative position of numbers next to each other to represent variable quantities (i.e., the 1’s, 10’s, 100’s, 1000’s, … place), Roman numerals rely on a set of standard measurement symbols which, in combination with others, can be used to express any desired quantity. The obvious problem with this approach is that it severely limits the numbers that can be reasonably represented by the given set of symbols. For example, the concise representation of the number four hundred seventy eight in the Hindu-Arabic system is simply 478 in which “4” is in the hundreds place, “7” is in the tens place, and “8” is in the ones place. In the Roman numeral system, however, this same number takes on the form CDLXXVIII. As numbers increase arbitrarily so does the complexity of their Roman numeral representation. The adoption of the Hindu-Arabic number system was, in large part, the result of Fibonacci’s publications and public support for this new way of thinking. Can you imagine trying to do modern mathematical analysis with numbers as clunky as MMMDCCXXXVIII??? Me either. Thanks, Fibonacci!

Fibonacci’s other works include publications on surveying techniques, area and volume measurement, Diophantine equations, commercial bookkeeping, and various contributions to geometry.[4] But among these works nothing stands out more than that of Fibonacci’s sequence – yes, we have returned! Among the more interesting mathematical properties of Fibonacci’s sequence is undoubtedly its connection to the golden ratio (shall be defined shortly). To illustrate, we look momentarily at the ratios of several successive Fibonacci numbers. Beginning with F1=1 and F2=1 we see that the ratio F2/F1=1. Continuing in this manner using the recurrence relation (*) from above or any suitable Fibonacci table we find that F3/F2=2, F4/F3=3/2, F5/F4=5/3,F6/F5=8/5, F7/F6=13/8, F8/F7=21/13, … As the indexing number tends to infinity, the ratio of successive terms converge to the value 1.6180339887… (the golden ratio) denoted by the Greek letter phi. We may thus concisely represent this convergent value by the expression as the lim n–> infinity (Fn+1/Fn). Studied extensively, the golden ratio is a special value appearing in many areas of mathematics and in everyday life. Intimately connected to the concept of proportion, the golden ratio (sometimes called the golden proportion) is often viewed as the optimal aesthetic proportion of measurable quantities making it an important feature in fields including architecture, finance, geometry, and music. Perhaps surprisingly, the golden ratio has even been documented in nature with pine cones, shells, trees, ferns, crystal structures, and more all appearing to have physical properties related to the value of (e.g., the arrangement of branches around the stems of certain plants seem to follow the Fibonacci pattern). While an interesting number no doubt, we must not forget that mathematics is the business of patterns and all too often we draw conclusions and make big picture claims that are less supported by evidence and facts than we may believe. There is, in fact, a lot of “woo” behind the golden ratio and the informed reader is encouraged to be weary of unsubstantiated claims and grandiose connections to the universe. It is also worth mentioning that, using relatively basic linear algebra techniques, it is possible to derive a closed-form solution of the n-th Fibonacci number.

Figure 3-Computing the 18th Fibonacci Number in Mathematica.

Figure 3-Computing the 18th Fibonacci Number in Mathematica.

Omitting the details (see link for thorough derivation), the n-th Fibonacci number may be computed directly using the formula Fn=((φ)(n+1)+((-1)(n-1)/(φ)^(n-1))/((φ2)+1).[5] While initially clunky in appearance, this formula is incredibly useful in determining any desired Fibonacci number as a function of the indexing value n. For example, the 18-th Fibonacci number may be calculated using F18=((φ)(18+1)+((-1)(18-1)/(φ)^(18-1))/((φ2)+1)=2584. Comparing this value to a list of Fibonacci numbers and to a Mathematica calculation (see picture above), we see that the 18-th Fibonacci number is, indeed, 2584. Without having to determine all previous numbers in the sequence, the above formula allows us to calculate directly any desired value in the sequence saving substantial amounts of time and processing power.

From the study of syllables and poetic forms in 12th-century India to a closed-form solution for the n-th Fibonacci number via modern linear algebra techniques, our understanding of sequences and the important mathematical properties they possess is continuing to grow. Future study may reveal even greater mathematical truths whose applications we cannot yet conceive. It is thus the beauty of mathematics and the excitement of discovery that push us onward, compel us to dig deeper, and to learn more from the world we inhabit. Who knows, you might even be the next Leonardo of Pizza – errrrr Pisa. What patterns will you find?
[1] French mathematician Edouard Lucas (1842-1891) was the first to attribute Fibonacci’s name to the sequence. After which point little is ever mentioned of the Indian mathematicians who laid the groundwork for Fibonacci’s research.

[2] Answer: n=15 –> 610 and n=16 –> 987.

[3] Medieval Muslim inhabitants of the Maghreb, Iberian Peninsula, Sicily, and Malta.[2]

[4] Translation: Book of Calculation[3]


Bibliography

 [1] Knott, Ron. Who Was Fibonacci? N.p., 11 Mar. 1998. Web. 27 Apr. 2015.

[2] “Moors.” Wikipedia. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.

[3] Leonardo Pisano – page 3: “Contributions to number theory”. Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.

[4] “Famous Mathematicians.” The Greatest Mathematicians of All Time. N.p., n.d. Web. 28 Apr. 2015.

 [5] Grinfeld, Pavel. “Linear Algebra 18e: The Eigenvalue Decomposition and Fibonacci Numbers.” YouTube. YouTube, 2 Dec. 2014. Web. 28 Apr. 2015.

 Figure 1: Fibonacci. Digital image. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.

Figure 2: Golden Spiral. Digital image. Mathworld. Wolfram, n.d. Web. 1 May 2015.

Figure 3: Ross, Andrew Q. Closed-Form Computation of Fibonacci. Digital image. Mathematica, 28 Apr. 2015. Web. 28 Apr. 2015.

Convergence of a Divergent Series and other Tests

In class we have been toying with the idea of classifying diverging infinite series, such as the sum: Σ k (k = 1,) = 1 + 2 + 3 + … which, as we add it up, continues on to infinity. We also messed around with some series notations and came to a conclusion that the sum adds up to -1/12. Now, I have no intention to claim, nor prove, that it equals -1/12. In fact, I would like to do the opposite; I would like to show that it in no way converges. If an infinite series converges, that means it sums up to a real number s: Σ ak = s. Likewise, an infinite series diverges if the sum of the series equals ±∞ or it does not add to any value. There are a few methods I will use to prove that this series diverges, and these methods can also be used to determine whether any infinite series converges or diverges.

The first test I want to look at is the Root Test. In this test, we take the sequence ak raise it to the power 1/k, and take the limit of that as k→∞. If this limit is greater than 1, the series diverges; if the limit is less than 1, it converges. For the sum of all integers, that gives us limit as k→∞ of k(1/k) . As k→∞, limit of k(1/k) → 1. Unfortunately the limit equals 1, which means it is inconclusive; there is a chance that this does indeed sum up to -1/12. Let’s leave the score at 0-0.

I will now put the series through the Ratio Test. It is called the Ratio Test because we take a ratio of the k+1 term divided by the k-th term and take the limit of that as k→∞: the limit of ak+1/ak. Similar to the Root Test, if the limit is greater than one it diverges, and if the limit is less than 1 it converges. For the dubious series which is under examination, that limit (k+1)/k, and as k→∞ the limit equals 1. Once again, this test is inconclusive. The score is still 0-0.

Now it’s time to get serious; so far every test I’ve done has been disappointingly empty of an answer. In this test if a series bk > ak and bk converges, then ak must also converge and if bk < ak and bk diverges, then ak diverges as well. My claim is that Σ k (k = 1,∞) diverges and now I will compare it with a series Σ 1 (k = 1,n) = n. We can see that 1+1+1+… grows much more slowly than 1+2+3+… thus we can use it as a comparison. By another test, the Term Test, if the series converges then lim an = 0. The series must diverge then if the limit doesn’t equal 0. The limit of 1 as n→∞ = 1 ≠ 0, thus the sum of 1 on (k=1,∞) diverges. Since Σ 1 (k = 1,∞) < Σ k (k = 1,∞), the latter series must also diverge. Now the score is 1 for divergence, and 0 for -1/12.

While it can now be seen that Σ k (k = 1,∞) does indeed diverge, the comparison test relies on having knowledge that a similar infinite series will either converge or diverge. In this case, I have to know that Σ 1 (k = 1,∞) diverges before I can compare it to the original series. I used the Term Test to show that Σ 1 (k = 1,∞) diverged, but I can also use it to show that Σ k (k = 1,∞) diverges. The limit as n→∞ of k = ∞, and by the Term Test diverges. The problem with the Term Test is that the limit ak can be equal to zero, but the series can still diverge. Therefore, this test is only useful if the limit does not equal 0 or to insure that a converging series does indeed converge.

By both the Term and Comparison tests, I was able to show that Σ k (k = 1,∞) diverges and is not equal to -1/12. In class though, we determined that the infinite series itself doesn’t sum to a negative number, there is no possible way that adding large positive whole numbers together would result in a negative rational number, but rather -1/12 represented a value or categorization of the infinite sum Σ k (k = 1,∞). How did this number come about? The idea is definitely not of “fringe mathematics” and has some excellent arguments. The idea stems from Σ (-1)k (k = 0,∞) = 1 – 1 + 1 – 1 + … The mathematician Srinivasa Ramanujan gave this sum a “value” of ½, since it seems to jump between one and zero equally, with ½ as the average.

The second sum Σ k(-1)k-1 (k = 1,∞) = ¼ as this sum added to itself equals Σ (-1)k (k = 0,∞). That is 1 – (2 – 1) + (3 – 2) + … = 1 – 1 + 1 – 1 + … Finally, Σ k (k = 1,∞) – 4(Σ k (k = 1,∞)) = 1 + (2 – 4) + 3 – (4 – 8) + … equals 1 – 2 + 3 – 4 + … = Σ k(-1)k-1 (k = 1,∞), or rather -3(Σ k (k = 1,∞)) = ¼ Σ k (k = 1,∞) = -1/12.

Ramanujan discovering the value -1/12 for the the infinite series ak = k. Image: Srinivasa Ramanujan via Wikimedia Commons.

The specific stipulation given is that the series must go to its limit. The partial summations of any of these series will produce a number unlike the total summation. For the majority of mathematics, to say Σ k (k = 1,∞) = ∞ makes more sense and requires significantly less head-scratching than -1/12.

There is one more test for convergence that I did not talk about, as the infinite series I was examining did not apply, and that is the Integral Test. For Σ ak, the function f(k) = ak. The reason why I could not apply it is because it only works with positive, non-increasing functions bounded on [1,∞). If the integral of f(x) is a real number, then the series converges, whereas if that same integral diverges, then the series diverges. The first case I will look at is Σ 1/k (k = 1,∞), where f(x) = 1/x and is a positive, non-increasing function. The integral of 1/x = log(x) evaluated from 1 to ∞. This integral turns out to diverge, and therefore the series also diverges. The second case is similar: Σ 1/k2 (k = 1,∞), where f(x) = x-2. The integral of 1/x2 = -(1/x) evaluated over the same interval. The sum of these two bounds is 0 – (-1) = 0 + 1 = 1. Since the indefinite integral converges, the series also converges.

These tests help determine whether a series converges or not. I used them to prove with basic mathematics that Σ k (k = 1,) diverges, rather than converging on a negative rational number. While the values given to divergent indefinite series can provide an idea of how they relate to each other, they require a fair amount of assumption and a lot of counterintuitive work to calculate. It is far easier and more practical to state that Σ k (k = 1,) diverges.

Sources:

http://www.lemiller.net/media/classfiles/notes.pdf (Foundations of Analysis by Joseph L. Taylor)

https://www.math.hmc.edu/calculus/tutorials/convergence/

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

http://en.wikipedia.org/wiki/Series_%28mathematics%29

http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

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

Drop the Base

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A demonstration of exactly how a shift of base can change our perception of time. Image: Jeremy Keith via flickr.

As a kid, when we were first introduced to numbers, they were just something we memorized, learned to accept, and started using on a regular basis. While this seems almost second nature to most of us, there was a time where the idea of a number system was a new thing and, like all new things, it was discovered multiple times by different people who had different setups. One of the more interesting areas of variation between different number systems would be the base that different number systems used.

Before going into detail about what a base is, it is important to understand that base systems are primarily used by number systems which also use position to determine how large a number is. For example, the Arabic numeral system is positional because I can use the same symbol in a different position to change the value of the number. While 01 is only the value one in this system, by switching these numbers around to 10, I have changed the value now to ten. This is different from something like the Roman numeral system which, for the most part, wouldn’t be considered a positional system because in two different numbers, like X and XIII, the value of the symbol X doesn’t change.

Now, what does this have to do with the base of a number system? The thing is, the base of a positional number system is the number of different symbols you can have in any single position. For example, the Arabic numeral system is base 10 because we can have ten different symbols in a single position (1, 2, 3, 4, 5, 6, 7, 8, 9, 0). In addition to defining how many different symbols you can have in any one position, the value of the base will also affect how much of a change in value a symbol will have based on its position. As I had mentioned earlier, different number system have different bases. The primary reason why would most likely be just because they may have had a different system for counting which lead to that decision. Having a base 10 system is the more common one and a lot of people give credit to that due to the fact that the average number of fingers we have on our hands combined is ten and people like to count using their fingers. On the other hand, the Mayans had a numeral system which consisted of base 20. Unlike most people from Europe, the Mayans wouldn’t wear shoes which meant they could count using ten fingers and ten toes. Even the Babylonian’s had a numeral system with base 60. I honestly couldn’t say why but I am sure they had a good reason for doing so.

Even current day computers use a different base than 10. Instead, computers count using base 2 which means they can only have a 0 or 1 in any position. How can something like this work? The reason why different numeral systems can have different bases is because all positional systems use mathematics in combination with the base size to determine how important a certain symbol is based on its position. This means that it is easy to convert from any base system into a different one. For example, if I want to convert the binary number (100010) into a base 10 number, all I need to do is figure out the base 10 value at each position and add them together. Since this is base 2, every position will be multiplied by 2i with i being the current position. This means:

100010 -> 1*25 + 0*24 + 0*23 + 0*22 + 1*21 + 0*20 -> 1*32 + 0*16 + 0*8 + 0*4 + 1*2 + 0*1 -> 32 + 2 = 34

To make things even more bizarre, a base can be found in more than just numeral systems. Another great example of a system which has a base is the alphabet. In our case, the Roman alphabet has a 26 base system or 52 (if you include capitalization). In addition, a lot of the different measuring units we have also have their own setups for bases. There are 12 inches in a foot, 16 ounces in a pound, 60 seconds in a minuet, and even 12 months in a year. And yet, for all of these we use a base 10 counting system instead of creating our own symbols for each measuring units. Then again, imagine how confusing that would be. In most places, people realize how difficult it can be constantly converting from one base system to another which is why certain measuring systems like the metric system uses a constant base of 10 between unit sizes to make things easier.

In the end, the point is that different bases are used everywhere. Whether you are dealing with numbers or some other system entirely, you will usually be able to find a base of some kind connected to the system. While it may be difficult to have to constantly deal with different kinds of bases, bases are necessary for people to be able to have such a large variety with such a limited number of symbols. Bases are here and they are here to stay.

Sources:

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html http://mathcentral.uregina.ca/RR/database/RR.09.00/hubbard1/MayanNumerals.html

Female Mathematicians, the Unsung Super-heroines

During most of my schooling I’ve generally only heard about male mathematicians and how they changed the way we do math. On occasion I have heard a woman’s name come up, but she is usually brushed over and not too much detail is given. I did some research and there are quite a few women throughout history that have contributed to mathematics and made numerous discoveries of their own. Why don’t we hear about them more often? It’s somewhat the same as how we don’t see much about super-heroines, even though comics and superheroes are all the rage right now. Women tend to get over looked and I believe this should stop. Maybe if I write a comparison of how female mathematicians can be compared to superheroes it will bring about a different view and we can get behind the power of the female brain.

Portrait of Sophie Germain.

Sophie Germain

Black Widow.

Black Widow

Female superheroes have been around since the early 1940’s, but with some caveats. They were still portrayed as being controlled by outside forces. Also why do we have to put them in the clothes we do; it’s just another way to classify them as less than a man. Wonder Woman had to have her bracelets to keep from going crazy, and this symbolized control. This reminds me of how early female mathematicians had to hide who they were and only communicate with fellow colleagues through letters and under different names. Sophie Germain was one such mathematician. She started teaching herself, at the young age of 13, when she ran across a book in which the legend of Archimedes’ death was recounted in her dad’s library. The story fascinated her and began her love for math, but like most stories about our female heroes, she was told she was wasting her time and her parents tried everything to keep her from studying on her own. To no avail, they couldn’t keep her from the love that bloomed over a terrible tale of how “Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death” (Perl 64). When she was 18 Ecole Polytechnique, a school, was built, but being female she was unable to attend. How have we been so blind to the contributions that females can make?

Her inability to attend any type of formal classroom to advance her knowledge makes me wonder how we have made advances in science and math at all. It reminds me of how we highlight male superheroes to be better and why female fans are so rare to come by in the world of comics. Portraying women to be less or just objects to admire doesn’t seem to be the way to advance in any area, but it happens, still to this day. Germain’s view in different fields has won her recognition as a wonderful mind and a fresh breath of air to the field of math and science. She wrote to Lagrange, Gauss, and Legendre under a pseudonym, surprising each of them that she was in fact a she. They helped her grow and learn more than she could ever on her own, but it saddens me that she would even need a man to obtain documents and texts for her to even expand her knowledge in the first place. Being left out is something I am all too familiar with and I don’t know what I would do if I had to solely rely on one or two men to get me the proper supplies to further my education. I value my education and see myself going far.

I’m very happy to not live in the time where a man had to vouch for a woman to be accepted into the scientific community like Emmy Noether had to deal with. Having to see her mentioned in comments with the pretense that Einstein said she was good so therefore she must be is quite ridiculous (Angier). With her inability to teach under her own name for quite some time and then to receive a ridiculously small pay when they did decide she was worth pay is shameful (“Emmy Noether Lectures”). I believe that a person’s merit is there whether someone says it is or not, especially if you’ve done the same research and studying as the next guy. Just because you’re a woman doesn’t make you any less intelligent nor should you have to rely on a man vouching for your intelligence. How often have we watched a TV show or movie where there was a man vouching for a woman? Or where a man is trying to keep a woman from getting credit or keeping her under control as if she was a crazy person who has no self-control of her own?

Take Iron Man 2, Black Widow. At first he assumes she’s just a pretty face, until she has to come in and save him with her awesome butt kicking skills. Not only is he shocked, but he takes a bit to adjust to the fact that she is her own hero. Female mathematicians have had to face this same stigma throughout the years. Germain eventually won her own right to her work and proved she wasn’t just a pretty face. The saddest thing is she wasn’t the only female that had to prove her worth.

Throughout the years, women did have to prove themselves, but this was also not always the case. There were a few women who had respect for their minds from the get go. One such woman was Hypatia, the first women to make a substantial contribution to mathematics, or so believed or documented. She was taught by her father and I believe that this might have been why she was accepted so easily in the circle of great minds of that age. She was led in by a great man and had very little to prove besides to show that his words about her were correct. She was the head of a school in her time, leading discussions on math and science to many students. She was a great hero in her time and I only wish that the many other female mathematician super-heroines throughout the ages had an easy path to their greatness.

Perl, Teri. Math Equals: Biographies of Women Mathematicians + Related Activities. Menlo Park: Addison-Wesley Publishing, 1978.

Angier, Natalie. “The Mighty Mathematician You’ve Never Heard Of,” New York Times Science section, page D4, March 27, 2012 (print edition). March 26, 2012.

“Emmy Noether Lectures,” Association for Women in Mathematics.

Random walk

George Polya was a Hungarian mathematician who made contributions in many branches of mathematics, among which was probability theory. In this blog we will discuss the “Random Walk” problem in probability. What is interesting is that George Polya actually first theorized this problem by accident.

George Polya was at that time a professor at a university in Zurich, Switzerland. The beautiful landscape there helped him develop a habit: taking afternoon walks in local gardens. One day, when he took his afternoon walks, one strange thing happened: he met a young couple six times. Well, I don’t know how large the garden was and how many paths were there, but this coincidence did surprise our mathematician. He was wondering how could this be possible, considering that he was then taking a random walk. After he mentioned this to his wife, he decided to do some research on the probability regarding random walk. His research actually established a new topic in probability theory.

Now let’s begin our discussion about the random walk problem with the simplest case. Consider one object moving along a straight line. Our assumptions are that this object can only move leftward or rightward, and the distance it moves each time is just one unit. For each time, it has a 50% chance to move leftward and a 50% chance to move rightward. A random walk model is thus created. And now we can ask: if when the object return to the starting point, the movement will end, then what is the probability that the object could return to the starting point? Thanks to George Polya’s work, the answer is 100%. Although some people may think this is unlikely, that’s only because when we think about this problem, we have a pre-assumption that the time is limited, or finite, which is always true in our real life. However, this mathematical model sets the time as endless, so it means when there is enough time and the object keeps moving endlessly, at last it will almost always return to the starting point.

When combining this theoretical conclusion with real life experience, mathematicians created the so-called “Drunken Man Going Home” problem. Assume there is a drunken man. He comes out of a bar and walks randomly along the line connecting his home and the bar. Once he gets home he will stop. Then what is the probability that the drunken man could finally get home? The answer to this funny question is still 100%. The math model of this problem is equivalent to the previous one. Because when time approaches infinite, the different positions of fixed points on the line actually make no differences. In my own words, any finite number in front of infinite vanishes to zero. So in this case, we just move the ending point from the starting point in the first example, to a new fixed point which denotes the drunken man’s home in the second example. The result keeps unchanged.

But mathematical problems always go from simple cases to complex cases. In the original random walk along a straight line, if we change the condition from “along a straight line” to “in a two-dimensional plane”, the complexity of this problem will definitely increase. If we change the condition to “in a three-dimensional space”, the complexity of this problem will increase dramatically. And the answers to this problem also change. The law is that, when the dimension increases, the chances that the object could get back to its starting point decreases, with the assumption that other conditions keep unchanged. For example, for a three-dimensional space, the probability is 34%; for four-dimensional space, the probability is 19.3%; when the dimension reaches eight, the probability that the object can get back is only around 7.3%. This means that there is only a lucky “drunken man”, who could always find a way home; there is no lucky “bird”: in a three-dimensional space, if the bird flies randomly, its chance to get back to its nest is much smaller! (Even if it could fly endlessly.)

Some people may cannot help ask, what can the theorem on random walk be used for. Well, obviously it is not developed to help drunken men feel confident when they need find a way home. It does have very wide applications in various areas. I will broadly list its applications in three different areas.

In economics, the theorem on random walks generates a hypothesis called “random walk hypothesis”. With the help of this hypothesis, economists could construct math models to research factors affecting shares price and the change of shares prices. It is quite understandable because randomness is q characteristic of the stock market.

In physics, the famous Brownian motion can apply the theorem on random walk to achieve the purpose to simplify the system. Since molecules move randomly to every direction, if we make an assumption that the molecules will only move to a finite number N directions, then the problem can be converted to a random walk problem in finite dimensional space. We could estimate the case in real physical world through our approximation in theoretical random walk models.

In genetics, a random walk could be used to explain the change of genetic frequency — a phenomenon that finally leads to genetic drift. Because gene mutation is always random, when we view the original gene pool as a point in a multi-dimensional space, then each gene mutation will generate a new gene pool, which can be viewed as a new point that forms through the original point’s random walk. After countless random walks, we could use the computer to draw pictures that show the trajectory of these abstracted points. The trend and some important characteristics of a gene drift could be described by this.

Randomness is everywhere in nature. At first glance, randomness connotes lack of order, unpredictable and uncontrollable. However, after mathematician’s great job, we do gain many laws and theorems on randomness. Random walk now has very wide applications in various disciplines, and we must attribute this partially to the coincidence happened to George Polya.

Reference:

  1. http://www.math.wichita.edu/history/men/polya.html
  2. http://www.guokr.com/article/53059/
  3. http://en.wikipedia.org/wiki/Random_walk