Category Archives: Practical Math

Who knew that an unlikely friendship and a few games of cricket with one of the greatest mathematicians in the early 20th Century could lead to a breakthrough in population genetics?

Today, it is almost commonplace for us in the scientific community to accept the influence natural selection and Mendelian genetics have on one another, however for the majority of human history this was not the case. Up until the early 1900s, many scientists believed that these concepts were nothing more than two opposing and unassociated positions on heredity. Scientists were torn between a theory of inheritance (a.k.a. Mendelian genetics) and a theory of evolution through natural selection. Although natural selection could account for variation, which inheritance could not, it offered no real explanation on how traits were passed on to the next generation. For the most part, scientists could not see how well Mendel’s theory of inheritance worked with Darwin’s theory of evolution because they did not have a way to quantify the relationship. It was not until the introduction of the theorem of genetic equilibrium that biologists acquired the necessary mathematical rigor to show how inheritance and natural selection interacted. One of the men who helped provide this framework was G.H. Hardy.

G. H. Hardy. Image: public domain, via Wikimedia Commons.

G. H. Hardy. Image: public domain, via Wikimedia Commons.

Godfrey Harold (G.H.) Hardy was a renowned English mathematician who lived between 1877-1947 and is best known for his accomplishments in number theory and for his work with the another great mathematician, Srinivasa Ramanujan. For a man who was such an outspoken supporter of pure mathematics and abhorred any practical application of his work[5], it is ironic that he should have such a powerful influence on a field of applied mathematics and help shape our very understanding of population genetics.

How did a pure mathematician come to work on population genetics? Well it all started with a few games of cricket. Whilst teaching at the University of Cambridge, Hardy would often interact with professors in other departments through friendly games of cricket and evening common meals [1]. It was through these interactions that Hardy came to know Reginald Punnett, cofounder of the genetics department at Cambridge and developer of Punnett Squares, which are named for him, and developed a close friendship with him[13].

Punnett, being one of the foremost experts in population genetics, was in the thick of the debate over inheritance vs. evolution. His interactions with contemporaries like G. Udny Yule, made him wonder why a population’s genotype, or the genes found in each person, did not eventually contain only variations, known as alleles, of a particular gene that are dominant. This was the question he posed to Hardy in 1908, and Hardy’s response was nigh on brilliant. The answer was so simple that it almost seemed obvious. Hardy even expressed that “I should have expected the very simple point which I wish to make to have been familiar to biologists’’ [4]. His solution was so simple in fact that unbeknownst to him, another scientist had reached the same conclusion around the same time in Germany [17]. In time, this concept would be known as Hardy-Weinberg Equilibrium (HWE).

In short, HWE asserts that when a population is not experiencing any genetic changes that would cause it to evolve, such as genetic drift, gene flow, selective mating, etc., then the allele (af) and genotypic frequencies (gf) will remain constant within a given population (P’). To calculate the gf for humans, a diploid species that receives two complete sets of chromosomes from their parents, we simply look at the proportion of genotypes in P’.

0 < gf < 1

To calculate the af, we look at the case where either the gene variation is homozygous and contains two copies of the alleles (dominant—AA || recessive—aa) or heterozygous and only has one copy of each allele (Aa). P’ achieves “equilibrium” when these frequencies do not change.

Hardy’s proof of these constant frequencies for humans, a diploid species that receives two complete sets of chromosomes from its parents, is as follows[1][4]:

If parental genotypic proportions are p AA: 2q Aa: r aa, then the offspring’s would be (p + q)2: 2(p + q)(q + r): (q + r)2. With four equations (the three genotype frequencies and p + 2q + r = 1) and three unknowns, there must be a relation among them. ‘‘It is easy to see that . . . this is q2 = pr” 

Which is then broken down as:

q =(p + q)(q + r) = q(p + r) + pr + q2

Then to:

q2 = q(1- p – r) – pr = 2q2 – pr     ——->   q2 = pr

In order to fully account for the population, the gf and af must sum to 1. And, since each subsequent generation will have the same number of genes, the frequencies remain constant and follows either a binomial or multinomial distribution.

One important thing to keep in mind, however, is that almost every population is experiencing some form of evolutionary change. So, while HWE shows that the frequencies don’t change or disappear, it is best used as a baseline model to test for changes or equilibrium.

When using the Hardy-Weinberg theorem to test for equilibrium, researchers divide the genotypic expressions into two homozygous events: HHο and hhο. The union of each event’s frequency ( f ), is then calculated to give the estimated number of alleles (Nf). In this case, the expression for HWE could read something like this:

Nf = f(HHο)  f(hhο)

However, another way to view this expression is to represent the frequency of each homozygous event as single variable, i.e. p and q. Using p to represent the frequency of one dominant homozygous event (H) and q to represent the frequency of one recessive homozygous event (h), gives the following: p = f(H) and q = f(h). It then follows that p² = f(HHο) and q² = f(hhο). By using the Rule of Addition and Associative Property to calculate the union of the two event’s frequencies, we are left with F = (p+q)². Given that the genotype frequencies must sum to one, the prevailing expression for HWE emerges when F is expanded:

Fp² +2pq + q² = 1

Using this formula, researchers can create a baseline model of P’ and then identify evolutionary pressures by comparing any subsequent frequencies of alleles and genotypes (F) to F. The data can then be visually represented as a change of allele frequency with respect to time.

HWE represents the curious situation that populations experience when their allele frequencies change. This situation is realized by first assuming complete dominance, then calculating the frequency of alleles, and then using the resultant number as a baseline with which to compare any subsequent values. Although there are some limitations on how we can use HWE—namely, identifying complete dominance, the model is very useful in identifying any evolutionary pressures a population may be experiencing and is one of the most important principles in population genetics. Developed, in part, by G.H. Hardy, it connected two key theories: the theory of inheritance and the theory of evolution. Although, mathematically speaking, his observation/discovery was almost trivial, Hardy provided the mathematical rigor the field sorely needed in order to see that the genotypes didn’t completely disappear and, in turn, forever changed the way we view the fields of biology and genetics.

Sources:

  1. Edwards, A. W. F. “GH Hardy (1908) and Hardy–Weinberg Equilibrium.”Genetics3 (2008): 1143-1150.
  2. Edwards, Anthony WF. Foundations of mathematical genetics. Cambridge University Press, 2000.
  3. Guo, Sun Wei, and Elizabeth A. Thompson. “Performing the exact test of Hardy-Weinberg proportion for multiple alleles.” Biometrics(1992): 361-372.
  4. Hardy, Godfrey H. “Mendelian proportions in a mixed population.” Science706 (1908): 49-50.
  5. Hardy, G. H., & Snow, C. P. (1967). A mathematician’s apology. Reprinted, with a foreword by CP Snow. Cambridge University Press.
  6. Pearson, Karl. “Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs.”Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character(1903): 1-66.
  7. Pearson, K., 1904. Mathematical contributions to the theory of evolution. XII. On a generalised theory of alternative inheritance, with special reference to Mendel’s laws. Philos. Trans. R. Soc. A 203 53–86.
  8. Punnett, R. C., 1908. Mendelism in relation to disease. Proc. R. Soc. Med. 1 135–168.[PMC free article] [PubMed]
  9. Punnett, R. C., 1911. Mendelism. Macmillan, London.
  10. Punnett, R. C., 1915. Mimicry in Butterflies. Cambridge University Press, Cambridge/London/New York.
  11. Punnett, R. C., 1917. Eliminating feeblemindedness. J. Hered. 8 464–465.
  12. Punnett, R. C., 1950. Early days of genetics. Heredity 4 1–10.
  13. Snow, C. P., 1967. G. H. Hardy. Macmillan, London.
  14. Stern, C., 1943. The Hardy–Weinberg law. Science 97 137–138. [PubMed]
  15. Sturtevant, A. H., 1965. A History of Genetics. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.
  16. Weinberg, Wilhelm. “Über vererbungsgesetze beim menschen.” Molecular and General Genetics MGG1 (1908): 440-460.
  17. Weinberg, W. “On the demonstration of heredity in man.” Boyer SH, trans (1963) Papers on human genetics. Prentice Hall, Englewood Cliffs, NJ(1908).

Figure: Wikimedia Commons

Map Projections

The world is not flat (citation needed). This is a very important aspect of our planet Earth; indeed, were our world flat it would not rotate on its axis the same way, would have to have an edge, and would probably crumble into a non-flat ball of rubble from its own sheer weight. We should all, therefore, be grateful that the world we live on is the 3D almost-perfect-sphere that it is. Cartographers tend to be a little less happy about our world’s roundness than others because it presents them with an irksome problem: how do we model a non-flat 3D world on a flat 2D surface? Initially this may seem like a straightforward issue. We could, for example, just draw the earth by what it looks like from space! Well this doesn’t quite work because, among other things, you would only have a 2D projection of the side of the earth facing you. Angles, sizes, and shapes get distorted, especially as we approach the edges of our disc. Perhaps we could imagine we have a globe that we cut a slice through from North Pole to South Pole, which we can spread out flat on a table! This method is unfortunately flawed as well, since the sphere will never lie flat regardless of how many cuts are made. Clearly this problem isn’t quite as straightforward as we initially hoped! Well luckily for us, mathematicians and cartographers love these types of problems, and many have offered many possible solutions, of which we will discuss a few.
The first and probably most familiar solution (called a projection) to this problem is the Mercator projection. This is probably the map that you had hanging on the wall of your elementary school classroom. Gerardus Mercator developed it in 1569. The goal of this particular projection is to maintain direction of rhumb lines (aka paths of constant bearing), which are lines that meet each meridian (lines between the two poles) at the same angle. It was particularly useful for navigation because of these lines. The Mercator projection’s biggest failure is generally that it distorts sizes more and more as we venture away from the equator, causing the poles to have infinite size. For example, Greenland and Africa take up roughly the same size on the Mercator projection when in actuality Africa is nearly fourteen times larger!

The Mercator Projection with red dots showing size distortion. Image: Stefan Kühn, via Wikimedia Commons.

The Mercator can be created by projecting the Earth onto a vertical cylinder with circumference equal to the circumference of the Earth. The next projection, called the Transverse Mercator, is obtained using a horizontal cylinder instead. This projection does not maintain straight rhumb lines like its counterpart and distorts scale, distance, and direction away from the central meridian used.

Transverse Mercator projection. Image: Public domain, via Wikimedia Commons.

Next among the more famous projections is the Robinson. This map features a flat top and bulging sides, with meridians starting and ending equidistant to each other but spreading out as they approach the equator. This projection can be seen as a compromising projection: it loosely preserves size, shape, and distance by not being exact in any one of them in particular. The Robinson, like the Mercator, is frequently used in classroom maps due to providing good guesses for relative shapes and positions and being very easy to understand.

Robinson projection. Image: Strebe, via Wikimedia Commons.

Next up are the Stereographic and Orthographic projections. These projections have existed for thousands of years. They were even used by the ancient Greeks! These two methods are projections of the sphere onto a plane, resembling what it would look like if you were to view the earth from space. The Orthographic projection maps along straight lines perpendicular to the tangent plane of the sphere (think looking through a window from space) while the Stereographic projection maps each point by constructing a line through a predefined point (like the north pole) and drawing where it intersects the tangent plane (think the image in a mirror that the earth is placed on). Thus, the main difference is that Orthographic takes the projection from infinity while the Stereographic takes the projection from a point on the sphere. This means that the Orthographic projection only shows one hemisphere, where the Stereographic can show the entire sphere (except the pole) but in a more distorted way. For both of these projections, directions are true from the center point. With the Orthographic projection, any line going through the center is a great circle.

Orthographic projection. Image: Strebe, via Wikimedia Commons.

Stereographic projection. Image: Strebe, via Wikimedia Commons.

There are many more projections to be found online and in books. Indeed, Wikipedia has a stellar list of some of the many different types. These projections all have different uses and are able to convey information in their own clever way, but they share at least one thing in common: they take quite a bit of ingenuity and creativity to come up with and they reflect a deep love and understanding of math in their creators.

The Path of Chinese Astronomy

“Second star to the right and straight on ‘til morning.” I’m pretty sure that if most people now took Peter Pan’s directions to Neverland, they’d never get there. In these modern times, we rely significantly less on the sky to navigate and it’s only because of the knowledge accumulated and developed over time. Finding new information usually isn’t an easy task without using prior knowledge of another subject or idea as a starting point. For example, astronomical observations used known mathematics to become more pertinent to modern-day science. Such was the case for China, as time was measured and kept constant with the usage of the cycles of the sun and moon as well as intercalation– the insertion of days and months to make the lunisolar calendar follow the moon phases. The study of the night sky flourished for the Chinese during the Han dynasty (206 BCE – 220 CE), continuing through to the modern day.

Mathematical proof for the Pythagorean Theorem from the Zhou Bi Suan Jing. Image: Chinese Pythagorean theorem, from page 22 of Joseph Needham’s Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth, published in 1986 by Cave Books Ltd., based in Taipei, via Wikimedia Commons

The lunisolar calendar was used to mark the passing of the seasons and special occasions. The Chinese used advanced algebra for this purpose. It was mainly equatorial based, which focused on circumpolar stars and ecliptic frameworks that stemmed from Western science. The Zhou Bi Suan Jing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), one of the oldest and most famous Chinese mathematical texts dating back to the Zhou dynasty (1046 BCE – 256 BCE), used the Pythagorean Theorem on astronomical calculations as well as for multiple equatorial based problems. To go into detail, one of the 246 problems in the compilation was to find the height of the sun from the earth, as well as the diameter of the sun. One person was to wait until the shadow of a 264 cm gnomon (the part of a sundial that casts the shadow) was 198 cm so that a large 3-4-5 right-angle triangle could be formed. This larger triangle would be from the sun straight to the ground, along the ground to the gnomon (forming the right-angle), and from the gnomon to the sun (angle of elevation). The smaller triangle, consisting of just the gnomon and its shadow, was used to find the equivalent measurements of the larger triangle so that the Pythagorean Theorem could be applied. As a result to this problem, the base of the triangle would be 24,900,000 m, the height of the sun 33,200,000 m, and the hypotenuse going toward the sun 41,500,000 m.

Su Sung’s diagram for the Cosmic Engine. Image: Page 451 of Joseph Needham’s book Science and Civilization in China: Volume 4, Part 2, Mechanical Engineering, via Wikimedia Commons

Moving through to the Tang dynasty (618 CE – 907 CE), Yi Xing was a well-known monk, engineer, and astronomer who had used the knowledge of the previous dynasty to work on an astronomical clock, which displayed the relative positions of the sun, moon, zodiacal constellations and major planets occasionally. The improvements on the function of an astronomical clock would later be succeeded by Su Sung during the Song dynasty (960 CE – 1279 CE), when he created a water-driven astronomical clock for his clocktower, and designed  and constructed a Cosmic Engine that operated as an astronomical hydromechanical clock tower. Su Sung had worked off of the achievements of Zhang Heng, an astronomer, inventor, and guru of mechanical gears who lived from 78 CE – 139 CE. Along with that, Su Sung was among the first during the dynasty to work on empirical science and technology.

Caption: A Ming dynasty (1368 – 1644) mariner’s compass diagram developed from Shen Kuo’s studies.. Image: Unknown, via Wikimedia Commons

Another genius during this time was Shen Kuo, who was most known for finding the concept of the geographic north pole (true north) and the magnetic declination (angle on the horizontal plane between magnetic north and true north) towards the north pole by using a more precise measurement of what’s known as the astronomical meridian (a large circle that passes through the celestial poles, the nadir (vertical direction pointing in the direction of the force of gravity), and the zenith (vertical direction opposite to the force of gravity, opposite of the nadir) for a given location). He also used advanced math to calculate the position of the pole star that had moved over many centuries, which made sea navigation more accurate using a magnetic needle compass. Shen Kuo also theorized that the sun and moon were both spherical and used cosmological hypotheses to predict planetary motion. He worked with his colleague, Wei Pu, to record and plot the moon’s orbital path for a duration of five years. However, much of their efforts were wasted thanks to political rivals who only used part of the corrected plots calculated by Shen Kuo and Wei Pu for planetary orbital paths and speeds.

The Song dynasty was followed closely by the People’s Republic of China from 1912 to modern-day, during the time of rapid development in science and technology. They moved away from the study of celestial objects and focused more on the application of past astronomical studies on mechanical technology and modern science. So where the use of calculation, measurement, and logic was previously aimed toward the shapes and motions of celestial objects, it was now applied to military technology, arsenals, shipyards, steamships, and artillery. In short, the Chinese did not reduce observations of nature to mathematical laws until much later, since for a short period after the Song dynasty the focus was mainly on literature, arts, and public administration. Chinese mathematics had shifted more towards Western mathematics in terms of being used for technology and modern science rather than for astronomical studies. Despite this, China and many other Asian cultures still use the lunisolar calendar, remade each year using the same mathematical calculations from the Han dynasty.

Sources

Applications of Imaginary Numbers

The concept of imaginary numbers has always been a fascinating one. The Greek mathematician Heron of Alexandria, born around 10 AD, is noted as being the first person to have come up with the idea of imaginary numbers. It wasn’t until the 1500’s, though, that rules for arithmetic and notation for complex numbers really came to fruition. Of course, at the time most people thought imaginary numbers were just stupid and pointless. Heck, today I’m pretty sure most people still think imaginary and complex numbers are stupid and pointless. Surely they can be used for more than just generating pretty looking fractals (like the Mandelbrot set), right?

Yes, because of imaginary numbers there is a solution to any type of polynomial equation… but there has to be more use to them than that, right? The topic I wish to present in this article is about some of the other applications of imaginary numbers. Imaginary numbers are really useful and they can be used to do all sorts of awesome things!

While presenting this information, I do not claim to list every single practical use of imaginary numbers. There are many useful applications that involve some crazy complicated mathematics and are admittedly beyond the scope of my understanding at the present time. Rather, I wish to share a few of my favorite applications of imaginary numbers. It is my hope that the reader will learn more about why mathematicians have studied so much about imaginary and complex numbers.

First, complex numbers have a remarkable application in triangular geometry. There is a fascinating theorem called “Marden’s theorem”. I read about this theorem in an article written by Dan Kalman, a doctor of mathematics who works in the Department of Mathematics and Statistics at American University. He claims that this theorem is “the Most Marvelous Theorem in Mathematics.”

caption

A visualization of a Steiner inellipse with its foci. The ellipse is based on the polynomial p(z)=z3-(9+9i)z2+(3+52i)z+(33-39i). The black dots are the zeros of p(z), and the red dots are the zeroes of p'(z) and the foci of the inellipse. Uploaded by User Kmhkmh for Wikipedia on 2/6/2010. Creative Commons license. Reuse permitted.

Basically, this theorem can help one find the foci of a Steiner inelipse. A Steiner inellipse is simply an ellipse that is inside of a triangle and is tangent to the midpoints of the three sides of the triangle. Such an ellipse is shown in the following diagram.

The foci of a Steiner inellipse can be found by using complex numbers! The triangle’s vertices can be written as points in the complex plane as follows: a = xA + yAi, b = xB + yBi, and c = xC + yCi. Marden’s theorem states that if you take the derivative of the cubic equation (x-a)(x-b)(x-c) = 0 and set it to zero, then the solutions of this equation will be the two foci of the Steiner inellipse in complex numbers. Isn’t that a really bizarre theorem? If you think about it, though, it makes some intuitive sense. When you take the derivative of an equation and set it equal to zero, the solutions of that equation give you the maximum and minimum values found on the arcs in the equation. A regular cubic equation could have up to two arcs, so it’s natural that there would be two max/min values. The fact that these two values are the two foci of the inellipse is really interesting.

As it turns out, using complex numbers here gives us a very amazing and useful geometric tool to use. There are also a few generalizations of this theorem that apply to different types of polynomials and other geometric shapes!

So it seems that first we have geometric applications for complex numbers. Now I would like to present a second category of applications. These are related to phasor calculus. A phasor is a complex number that represents a sinusoidal function. Thanks to the amazing Euler’s formula (e= cosx + sinx), sinusoidal functions can be rewritten as complex numbers. This allows for easier problem solving and analysis for many types of problems.

For instance, in electrical engineering alternating currents can be a pain to analyze sometimes. After all, they have voltages that exhibit sinusoidal behavior. With the use of phasors, one can analyze aspects of AC circuits more easily. Analysis of resistors, capacitors, and inductors can be combined into a single complex number, which is called the impedance. Phasors are comparatively easy to interpret, so it’s a lot easier to study AC circuits when studying them in the complex plane! In addition to AC circuits, complex numbers are similarly useful when studying electromagnetic fields, where the quantities of electric and magnetic field strength are combined into a single complex number.

The last application I wish to bring up involves the usage of imaginary numbers to solve integration problems. As it turns out, we can use the aforementioned Euler’s formula to simplify real integration problems and help us find real answers. This is done by using a base integral that has a complex solution. An example of a base integral would be∫ e(1+i)xdx. Using simple u substitution, we can find the answer to this integral, which is ((1-i)/2)e(1+i)x + c1 + ic2. With this known imaginary answer, we can compute the answer to a real integral.

Consider, for example,∫ excosxdx. First, we rewrite the previously mentioned base integral as: ∫ exeixdx. Then we can use Euler’s formula to alter this integral further:∫ exeixdx = ∫ ex(cosx + isinx)dx. This will further simplify to∫ excosxdx + i∫ exsinxdx. We can set the known solution of the base integral equal to this complex integral and solve for ∫ excosxdx , which is the real integral we are trying to compute. We will see that the imaginary parts must be equal and the real parts must also be equal. Solving in this manner will show us that ∫ excosxdx = .5ex(cosx + sinx) + c. Hopefully I don’t have to explain how useful integrals are! The fact that complex numbers can help us solve integrals alone means they are really useful.

I think in general it seems that whenever there’s an oscillatory phenomenon of any kind then complex numbers are naturally helpful in describing said phenomenon. Complex numbers have multiple substantial applications in a multitude of scientific problems. In addition to the few I’ve mentioned, complex numbers are also used in: quantum mechanics, control theory, signal processing, vibration studies, cartography, and fluid dynamics. Dang. Since a long time ago complex numbers have been thought of as trivial and inconsequential. Descartes himself (who coined the term “imaginary”) called these types of numbers imaginary because he meant for this to be derogatory. However, as we have learned more about math throughout the ages we have found many a useful application for imaginary numbers.

The aforementioned Mandelbrot set. This is a fractal involving a set of complex numbers. Uploaded by User Localhost00 on 10/13/2013. Creative Commons license. Reuse permitted.

Sources:

-Imaginary Number. (n.d.). Retrieved February 24, 2015 from Wikipedia: http://en.wikipedia.org/wiki/Imaginary_number#cite_ref-1

-Complex Number. (n.d.). Retrieved February 24, 2015 from Wikipedia:

http://en.wikipedia.org/wiki/Complex_number#Applications

-Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Loci (March 2008)

– P. Ceperley. 8/28/2007. Phasors. Retrieved from:

http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html

-Integration Involving Complex Numbers. Retrieved February 24, 2015 from:

http://www.mathnotes.org/?pid=109#?pid=109

Googols

During class, we spoke briefly about Graham’s number.  It’s a number so vast that it can’t be written in our conventional decimal number system.  There are other very large numbers as well, such as the googolplex.  We run into a problem when talking about numbers on this scale, because we rarely encounter really large numbers in day to day life.  In fact, very few people have a frame of reference for how large numbers like a trillion really are.  One of my favorite numbers is the googol.  It’s the number that nerdy kids throw around when they want to sound smart when describing innumerable quantities.  The definition of a googol is pretty simple, 10100.  The one question that remains:  What would it take to have a googol?

An easy way to relate large numbers is to compare them with money.  Consider a $100 bill.  It’s a rather light, thin piece of paper, but it holds real value to a lot of people.  Its dimensions are 6.14 in. x 2.61 in. [1], and it is .0043 in. thick. [2]

Assembling a million dollars in one location using $100 bills is easy (if you have the money, of course).  If you were to stack them neatly into a cube, the cube would be about a quarter of a foot long, a quarter of a foot wide, and a quarter of a foot high.  This is about the size of a basketball.

This may be all well and good, but what about a billion?  Each side of the cube now is 7.36 feet long. This cube would also weigh 11 tons. This is about the weight of 3.5 Subaru Foresters.

How about the glorious trillion?  A trillion dollars would form a cube 73.6 feet long.  If stacked on a football field, the pile of money would be 7 feet deep.  The current national debt of the United States is around $17 trillion.  This money on a football field would be 117 feet deep.  It’s about the money required to half-fill Rice-Eccles stadium with $100 bills.

As the numbers grow, the money pile gets more and more ridiculous. Let’s consider how the short scale number system works.  A million is a one with six zeroes.  A billion is a one with nine zeroes.  Every ‘–illion’ thereafter has an additional three zeroes.  How do we determine what should prefix ‘-illion’?  We count in Latin.  Therefore, we get prefixes like ‘quad-‘, ‘quint-‘, and ‘sept-‘, just to name a few.  A googol in the short scale system is 10 duotrigintillion, or ’32-illion’.

A quintillion dollars (that’s $1,000,000,000,000,000,000) would form a cube 7,360 feet long, or about 1.4 miles on each side.  This is more money than the purchasing power of the Earth by a factor of 11,300. [3] In addition, if we divided up this money equally among the world’s population, each person gets $140,350,877.  This would also likely cause the world’s economy to completely fall apart.

A septillion dollars would form a cube measuring 139x139x139 miles.  This is rapidly getting harder to comprehend.  This cube of money would weigh 11 quadrillion tons.  It’s also enough to completely cover the Earth in money 83 feet deep.

A nonillion dollars would form a cube 13,940 miles across.  This is an enormous amount of money, and would in fact form a cube much larger than the Earth.

An undecillion dollars would create a cube 1.4 million miles across, which is larger than the sun.  Oddly enough, the bakery chain, Au Bon Pain, was once sued for $2 undecillion dollars. [4] While the lawsuit may have been filed, it was under spurious claims, and was subsequently dropped.

We’re starting to run out of comparable objects.  The sun is the largest object that people are most people are familiar with.  However, there are still some valid comparisons to be made.  1 AU (Astronomical Unit) is 92,955,806 miles across.[5]  It’s defined as the median distance from the Earth to the Sun.  One tredecillion dollars (for reference, that’s $1,000,000,000,000,000,000,000,000,000,000,000,000,000,000) would form a cube of $100 bills that is 1.5 AU across.  You could line up one corner of the cube on the center of the sun and have the other corner reach Mars.

By now, we’ve reached Mars from the Sun.  If we increase the value of our cube to $1 quindecillion, our cube has grown to the point that we could set a corner of it on the Sun, and have the other corner of the cube reach the Voyager 1 probe, the farthest man-made object from Earth.  We’re now at 150 AU.  How much further do we need to go to reach a googol?

Well, to get a cube that reaches from the Sun to the center of the Milky Way galaxy, we need $1.2 duovigintillion (That’s 22-illion).  This is around 1.7 billion AU.  Having exhausted our units again, we’ll need to switch to light-years.  There are 63,241 AU in one light-year.  One trevigintillion dollars would form a cube 237,000 ly across.

It takes roughly $1.1 quattorvigintillion to form a cube of money that stretches from our sun to the Andromeda Galaxy.  We’ve also long passed the point at which the cube of money would collapse to form a black hole.  The cube would now be 2,400,000 ly across.

We're going to need a lot of these. Image by Bureau of Engraving and Printing, via Wikimedia Commons.

We’re going to need a lot of these. Image by Bureau of Engraving and Printing, via Wikimedia Commons.

Once we reach $4 octovigintillion, we have a cube of money that is the size of the observable universe. Unfortunately, we’ve really run out of reference points.  The only thing that we can do now to reach a googol is to increase the value of our bills!

Finally, we’d need to increase the value of our bills to quintillion dollar notes, but at last, we are successful.  We filled the observable universe with bills that are worth thousands of times more than the world economy, but we’ve finally gotten a googol!  Congratulations!

Sources:

[1] http://en.wikipedia.org/wiki/United_States_one_hundred-dollar_bill

[2] http://www.ehd.org/science_technology_largenumbers.php

[3] CIA Factbook: https://www.cia.gov/library/publications/the-world-factbook/geos/xx.html

[4] http://www.loweringthebar.net/2014/05/2-undecillion-dollar-demand.html

[5] http://en.wikipedia.org/wiki/Astronomical_unit

Six Times Seven is What Again?

Mathematics is complicated. I’m pretty sure most normal people would agree with me on that opinion. There are always rules to be understood and those special circumstances in which they may not apply, things to be memorized and foundations to gain before you can move onto the next complicated thing. At the foundation of all math, though, is the need to multiply.

We all remember growing up memorizing times tables. Gross. Taking those silly timed tests over and over again until we could finally (and usually lastly) get the 6’s, 7’s and 8’s. And once that foundation is built, we start doing bigger multiplication. Hundreds times thousands and the like. Even more gross. But there are so many ways, outside of our standard multiplication algorithm, that we can visualize multiplication! We learned of one such method in class, the Egyptian Doubling method. But there are many others as well; the Russian Peasant, Sieve Multiplication, Finger Multiplication and a new fun one, Line Multiplication.

The Russian Peasant method of multiplication is very similar to the Egyptian method. The reason these 2 methods are so similar is because they are both based off of a base 2 algorithm, or a binary system. In the Egyptian method, we can see that if we break a number into its binary components and then multiply those by our other number, we can get our product. In the Russian Peasant method, the same idea is applied in a different way.

85×18 done using peasant multiplication. Image: A. Bogomolny, Peasant Multiplication from Interactive Mathematics Miscellany and Puzzles.

Let us begin by trying 85 times 18. In this method, you start as the Egyptians did with 2 columns. 85 at the top of the left column and 18 at the top of the right. In the left column you begin dividing your number in half disregarding any remainders. So 85/2 would be 42. 42/2 would be 21. 21/2 would be 10 and so on until you are left with 1. On the right side, you then double your number for as many rows that were created in the left column. Finally, you will add up the right columned numbers associated with the odd left column numbers (disregard any evens). So in this case, you will disregard 42, 10 and 2 and add 18, 72, 288 and 1152. This gives you your answer of 1530 [1].

Now notice that in this method, the odd numbers play an important role as they are the ones that you keep while you disregard the evens. The odd numbers (or the numbers which you can break up as 2k+1) show you the binary decomposition. Wherever you have a one left over (aka an odd number), that is where you will place a 1 in your binary. So 85 would be 1010101. This is how you discover which numbers to add to get to your answer.

The Sieve method is commonly known as the lattice method and originated in the Middle East in the late 1300’s [2]. This method is most similar to our multiplication method of today because it uses a base 10 algorithm much like our own system.

58×231 done using the Lattice Method. Image: Javier Rosa, via Wikipedia

Let us multiply 58 by 213 by the Sieve method. First, create a grid where your first number is listed across the top and the second down the right side with each number getting a row or column. Next, place a diagonal through each square. Then multiply your row by your column. So the first square will by 5×2, the second, 8×2 and so on. Place the tens digit of your answer in the top half of your square and the ones digit in the bottom half. Now, add down your diagonals beginning with the bottom right corner, so in this example, 4. So the next number would be 8+2+5 or 15. Make sure you only write 5 and carry your 1 to the next diagonal. Finally, you read your answer by reading the numbers down the left side and across the bottom. So our answer is 12,354 [2].

8×7 done by Finger Multiplication. Image: G. Patrick Vennebush, via Math Jokes for Mathy Folks.

Remember how I said earlier that we all hate 6’s, 7’s and 8’s? There is a method of finger multiplication that is said to have come out of Italy and was widely used during the Medieval Period that was specifically designed for all times tables 5×5 to 10×10 [3]. To begin this method, understand that every raised finger is one more than 5. Let us do 8×7, one of the most brutal of the multiplication tables. On the left hand, you will raise 3 fingers (because 8 is 3 more than 5) and 2 fingers on the right (7 is 2 more than 5). You should have a total of 5 raised fingers and 5 closed fingers, 2 closed on the left and 3 on the right [6]. You will multiply all raised fingers by 10 and then add that to the number of closed fingers on the left multiplied to the number of closed fingers on the right. In this case we have (5×10)+(2×3) or our answer, 56. Pretty nifty huh?

The final method of multiplication I wanted to talk about is one that is commonly attributed, on the internet, to the Chinese or Japanese called Line Multiplication. But as I did my research a little bit more, there are no real references to this being a Chinese or Japanese method. One person found that the earliest reference to this method was a YouTube video in 2006 but no sources have been found to accurately date it [5].

22×13 using the Line Method. Image: Francis Su, via Mudd Math Fun Facts.

Let us do 22×13 for this method. Begin by drawing diagonal lines slanting up for the tens and ones digit of your first number (2 lines, a space and then another 2 lines). Then create diagonal lines slanting down (that intersect with your first lines) for the tens and ones digit of your second number (1 line, a space and then 3 lines). You will then circle where each set of lines intersect and count the number of intersecting points and place this number just under each group. In this case we have 2 on the left side, 6 and 2 down the middle column and 6 on the right side. Add these numbers moving down your column from right to left (carrying tens when needed). Here we get 286 as our answer [4].

The one thing that I love most about each of these methods is that each can apply to different types of learner. Some are more hands on, some are more visual and others are more mental. As I am going into teaching, I can implement each of these methods into my classroom so that each of my students is able to learn this foundational mathematical concept in their own learning style.

References

[1] A. Bogomolny, Peasant Multiplication from Interactive Mathematics Miscellany and Puzzles
http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml, Accessed 30 January 2015

[2] Lattice multiplication. (n.d.). Retrieved January 30, 2015, from http://en.wikipedia.org/wiki/Lattice_multiplication

[3] West, L. (2011). An Introduction to Various Multiplication Strategies. 2-3. Retrieved January 30, 2015, from http://scimath.unl.edu/MIM/files/MATExamFiles/WestLynn_Final_070411_LA.pdf

[4] Su, Francis E., et al. “Visual Multiplication with Lines.” Math Fun Facts. <http://www.math.hmc.edu/funfacts&gt;.

[5] What is the origin of “how the Japanese multiply” / line multiplication? (2014, July 24). Retrieved January 31, 2015, from http://math.stackexchange.com/questions/877120/what-is-the-origin-of-how-the-japanese-multiply-line-multiplication

[6] Vennebush, G. (2011, May 28). Finger Multiplication. Retrieved January 31, 2015, from https://mathjokes4mathyfolks.wordpress.com/2011/05/28/finger-multiplication/

Galley Division

Up until about the 18th century, one of the most common methods of division taught was the galley method. The algorithm is very similar to modern long division (after all, how many different ways can you really see how much of one thing fits into another?) but is seemingly more space intensive and, in my opinion, less easy to follow. The method is essentially to remove multiples of the divisor from the dividend, counting the amount removed as we go, until there is no longer enough left over to fit a full multiple.

As a quick introduction, let’s do an example with some numbers. We begin by writing the dividend with a solid line next to it and writing the divisor beneath the dividend, lining up the leftmost digits. For example, if we were calculating 4892385 / 362 we would line up the problem as follows:

4892385 |
362          |

We can see that 362 divides into 489 exactly once, so we write a one on the other side of the bar and then calculate the difference between 489 and 362 * 1, writing the answer above the original dividend, whose first three numbers we now cross out. We also cross out our original divisor and rewrite it one space to the right. Our work would now look like this (periods are for formatting purposes, which wouldn’t be an issue if we were writing this on a paper):

127               |
4892385      | 1
3622             |
..36               |

Now we are dividing 362 into 1272, which goes in three times. We do 1272 – 3 * 362 = 186 and repeat the process stated above. We continue repeating the steps until the divisor does not fit in to any of the remaining dividend. Once all of that is done, our work will look like this:

…..13        |
..18571     |
1276367   |
4892385   | 13514
3622222   |
..36666     |
….333       |

We can therefore see that 4892385 / 362 = 13514 with a remainder of 317. That wasn’t so bad at all! Comparatively, we can write out the same problem using long division as follows:

………__    13514
362 | 4892385
…….-362
……..1272
…….-1086
……….1863
………-1810
………….538
…………-362
………….1765
…………-1448
……………317

Notice that many of the numbers (indeed, all of the remainders after subtraction is finished) appear in both methods. Galley division is much more compact, but long division looks much less cluttered. While galley division writes the divisor many times and does not right the subtrahend at each step, long division writes the divisor once but writes each subtrahend we come across. The underlying algorithm for both is the same: go from highest place value to lowest subtracting as many multiples of the divisor from the dividend as possible, keeping track of the amount as you go. Since the algorithm is the same, it should go without saying that the answer they arrive at is the same.

Now, if you’re anything like me, you’re probably thinking: what does that have to do with a galley? Well, in the words of Dr. Lamb, “If you draw a boat around it when you’re finished, it looks like a boat!” The image below shows what the problem can look like if you choose to make your work look pretty at the end instead of recognizing that math is pretty in its own right and doesn’t need flowery decorations.

Image by Francis Rolt-Wheeler via Wikimedia commons.

Perhaps the biggest flaw I see in galley division is that it is very cluttered and it can be hard to follow each step since one number can end up being written across multiple lines. By contrast, in long division each step can obviously be seen down and to the right of the preceding one. One strategy that has been used to combat this lack of clarity in the galley method is performing each step of the division and erasing any unnecessary numbers before moving to the next step. I think this is a good compromise. After all, if your work isn’t going to be followable you may as well not show it at all.

Like most really old things, we’re not too sure on the particulars of the galley method’s origin. The method is thought to have originated somewhere in China or the Middle East around 400 CE and was probably designed for a sand abacus or abax, which was a table covered in sand where stones and other objects would be used as counters and columns of objects were used as place value markers. The galley method is no longer taught in Western schools, but is still taught in northern Africa and the Middle East and is probably equally as misunderstood and hated by the general public there as long division is here due to a focus on teaching algorithms rather than actual underlying mathematics. Galley division and long division lack an intuitiveness, in my opinion, that seems to be inherent in other methods of division, such as the bubble method (which consists of pulling out “bubbles,” aka multiples, of numbers and adding them together to get the quotient) or the Ancient Egyptian method (which relies on building up the quotient from powers of two). Even though the underlying mathematics are more or less the same, additive methods seem much more intuitive since adding is simpler than subtracting. However, galley division was used for centuries upon centuries before being replaced, so perhaps it’s not as hard as I’m making it out to be.

The Wikipedia page on galley division is woefully sparse, but in truth only so much can be said about one really old algorithm and Wikipedia covers most of it. For more information on performing the method, check out http://www.math.wichita.edu/history/activities/arith-act.html#worse or http://en.wikipedia.org/wiki/Galley_division