Monday, December 14, 2015

Math What?

Nature of Math

The saga continues! Is math a science? Is math discovered or invented? I thought I had a solid answer on the former question, but Alas! Mathematics Strikes Back!

The debate on these two questions were brought front and center in my History of Math class. The class was divided on this issue with a large portion of the class believing math was a science. A small portion believed math is completely separate from science. A small few were still agnostic on the issue. The class was less divided over whether math is discovered vs invented. So, I'll start there.

Math Discovered or Invented?
Discovered. Definitely. As our class has discussed and from what Pickover has discussed in The Math Book (blog post), there are math milestones that were worked out by different people simultaneously, but independently from each other (1). Credit for Calculus has been given to both Isaac Newton and Gottfried Wilhelm Leibniz (1). Another example was when hyperbolic geometry was being worked out by three different mathematicians at the same time: Janos Bolyai, Nikolai Lobachevskii, and Carl Friedrich Gauss (4). If math was invented then it would just be a random collection of concepts that were pulled out of thin air by the imaginations of great mathematicians, and it would be hard to make the case that these simultaneous creations by different mathematicians occurred by coincidence. How could they just happen to have the (almost) exact same creation at the same time? That would be like Leonardo DaVinci and Michelangelo painting the exact same Mona Lisa at the same time. Don't Even.  

Furthermore, mathematics is far too ordered, structured, and interconnected to be randomly invented. In BBC Horizon's documentary, The Last Theorem, a connection was established between three seemingly unrelated math topics: elliptic curves, modular functions, and Fermat's Last Theorem (5). To prove Fermat's Last Theorem, Andrew Wiles had to prove that modular forms and elliptic curves are the same, because when someone believed that Fermat's Last Theorem was false, those equations looked like an elliptic curve that wasn't modular (5). In the end, Andrew Wiles proved Fermat's Last Theorem was true by showing elliptic curves and modular forms were the same (5).

The reason behind the order, structure and interconnectedness in mathematics is logic. Jordan Ellenberg in his book, How Not to Be Wrong: The Power of Mathematical, describes math as using common sense, and applying rational thinking or logic to problems (the problems don't have to be mathematical) (4). Eugenia Cheng, in a presentation at Grand Valley State University, explained that "mathematics is the logical study of logical things" (6). Logic is the underlying pinnings that unites all of mathematics together.

A part of mathematics logical structure are axioms. Ellenberg describes axioms as self-evident rules that form a system of mathematics (4). They are used to manipulate elements in the system to prove logical statements about the system, and these logical statements are based on truth tables, the language of logic. When Janos Balyai decided to remove the Parallel Postulate from Euclid's Axioms, he worked out a new geometry - hyperbolic geometry (4). He found that the parallel postulate was neither contradictory, which would make Euclid's geometry false, nor mandatory, which would make it the only possible geometry (4). Hyperbolic geometry is just another geometric world in addition to Euclid's geometry or, more broadly speaking, mathematical space in mathematics (3). When another world was discovered with a different set of axioms, the nature of axioms changed (3). Axioms are more like hypotheses that lay the framework for the world one is in (3).  But even though there is an important difference between hyperbolic and Euclidean geometry, there is overlap because there is overlap of axioms between the two worlds and the logic that is used to prove statements in all of mathematics is the same. Another example is with integers and rings. Integers are a specific type of ring, and depending on which axioms are used, proven statements (theorems) in the integers can be used or slightly modified to prove similar results in rings.

Furthermore, when I think of something that is invented, I think of art and games. When children come up with a new game, they create their own rules. Most of the time, these rules aren't logical. Take Monopoly for example. Why do players get free money whenever they pass Go? That doesn't make sense or even apply to the real world. It's a whimsical rule, and mathematics are not governed by whimsical rules.

Mathematics are governed by logical rules, specifically truth tables and axioms. Axioms, which are the foundation of proof and mathematical theorems, can be compared to rules of a game. With games, rules can be changed and the game can still be played. To continue with the Monopoly example, if the rule of getting money when passing Go disappears, all the other elements of the game remain. But in math, the rules, or axioms can't just change or disappear and leave the theorem or concept fundamentally different. If they do, all the mathematical concepts that relied on those axioms and theorems crumbles (4 and 5). This is the key difference between games (rules without logic) and math (rules with logic). Axioms and truth tables, the logical rules, makes math a discovered endeavor, and rules without logic are of human invention.

Lastly, mathematics is not a solely human invention. If math is truly a human invention, then only humans could do it, understand it, and come up with it. But primates and other animals can be trained to count, while a type of Saharan Ant knows the number of steps and the lengths of their stride to find their way back to their nest (1). But as someone in the class mentioned, if math is discovered, then it has to be found in nature, which would make it a science. And that brings us to the next question.

Is Math a Science?
I was a firm believer that math is not a science. Now, there is room for doubt. I have written a detailed blog post on this topic, so I will only give a brief description of what I already discussed. The pursuit of science is to discover how natural processes work, while the pursuit of math is discovering relationships with numbers. Math is found and can be applied to nature. In fact discoveries in math started with the ancient Greeks describing our 3-D world with geometry (3). But even though math can be found or be used to describe nature, there are abstract ideas that transcends the physical world. The example I gave before was a one-dimensional line expanding to infinity. Furthermore, math's capability to describe our natural world with accuracy does not make it the reason why it works (2). Just because math is accurate in modeling natural phenomenon, it does not mean it's the explanation behind the process (2). There could be a completely different reason that makes the process act that way (2). Accuracy does not necessarily make something true (2).

But then I came across the Mathematical Universe Hypothesis. Max Tegmark is the man who came up with the hypothesis (1). According to Clifford Pickover's explanation of the hypothesis, "...our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics" (1). If this hypothesis is proven to be true, then mathematics is not just something that can be applied and describe our natural world with accuracy - it is the foundation, the reason, and the process behind how everything in our natural world works. And that would make math a science....and it would make science provable with mathematical proof. But at this point, the Mathematical Universe Hypothesis is just a hypothesis, and the divide between science and math, nature and number, clear. And I can continue believing, a little shakily, that math is not a science.

(1) Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(2) Wigner, Eugene. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." College Course 9. 

(3) Space (mathematics). Wikipedia. 

(4) Ellenberg, Jordan. How Not to Be Wrong: The Power of Mathematical Thinking. The Penguin Press. New York, NY. 2014. 

(5) Simon Singh and John Lynch. Fermat's Last Theorem Documentary. BBC Horizon. 1996. 

(6) Eugenia Cheng's Presentation at Grand Valley State University. Quotation from MathHombre's Tumblr page:

Monday, December 7, 2015

The Spider and the Fly: An Exploration in 3-D Geometry and Graphing

Doing Math

The NRICH asks an interesting math question at this Link. Let's say you're a spider, and you see a fly across a room. What would be the shortest, and therefore fastest, route you would have to crawl to catch the fly?

The spider is in a room, so it's 3-D. The usual 2-D graph of x and y, rise over run, doesn't setup the problem accurately. There is width, height, and length: an x, y, and z coordinates respectively. But even though the spider and the fly lives in a 3-D world, we can break down the problem into a 2-D one, since the spider can only crawl on the wall in a 2-D way - side-to-side (x), up-and-down (y).

We are given that the room is 4 meters wide, 2.5 meters high, and 5 meters long. The spider is in the middle of the smallest wall on one side of the room. The fly is on the opposite, smallest wall and it's position is 1.5 meters high and 0.5 meters away from the adjacent wall next to the window. Below is how I visualized the room, and how I labeled the coordinates. The room is not drawn to scale.

Then, I thought how to move the spider from wall to wall to get to the fly in the shortest distance. If the spider goes towards the back wall, then the spider would have to cross the window to get to the fly, which is 3.5 m away from the fly (4-0.5 = 3.5). So, the better way for the spider to travel is to move towards the front wall. Next, I imagined the fly on the spider's wall in it's (0.5, 1.5) position. Eventually, the spider has to get to the height of 1.5 meters, so I figured it should do so while traveling to the edge of the front wall. That way, the spider will only need to crawl in a straight line to get to the fly. Below is the route indicated by the arrows.
Since the spider is crawling a straight line on the front wall and the opposite wall, then the spider will be crawling a distance of 0.5 + 5 = 5.5 meters for those two walls. I needed to use the distance formula to calculate the diagonal distance the spider crawls to get to the edge of the front wall at the height of 1.5 meters.
Distance Formula

The coordinate of the spot I want the spider to crawl to is (0, 1.5, 5), since x=0 is the edge of the front wall, 1.5 is the height of the fly, and 5 is the z coordinate for end wall. I don't need to use the z-coordinate, 5, because the spider is moving in 2-D, from side-to-side and up-and-down. My calculation using the distance formula is below.
This distance is approximately 2.02 meters. Therefore, the total distance traveled for the spider to reach the fly is 0.5+5+2.02= 7.52 meters. Then the problem asks another question: If the fly starts moving towards the floor (I'm assuming the fly  moving straight down), then at what point should the spider change it's route to catch the fly?

Not knowing when the fly starts moving downwards, and where the spider is on it's route, I decided to redo the problem with the fly already at the bottom of the wall. Then I thought I would do the problem in the same way as above. But then I had an Eureka! moment. I was already breaking down spider's movements or the walls of the room into 2-D. The two end walls had (x, y) coordinates and the front and back walls had (z, y) coordinates. So, I decided to remove the back wall, and flatten the 3 remaining walls into one large wall. Then, the spider just needs to travel diagonally to get to the fly, instead of changing direction to go straight once the spider reaches an edge. Above is how I visualized the new problem (the fly is moving straight downward).

Now, we have just have (x,y) coordinates to deal with. The fly's new coordinate is (9.5, 0) and the spider's new coordinate is (2, 1.25). I used the distance formula again to calculate the distance the spider will travel on the diagonal line. Below is the calculation.

The answer is approximately 10.68 meters. Since, I thought of a new way to look at the problem I decided to re-calculate the distance of the first problem using this method. So, the spider will, again, move diagonally only, and not change direction to go straight. Below is the new visual for the first problem.
So, the spider has the coordinates of (2, 1.25) and the fly has the coordinates of (9.5, 1.5). The calculation for distance using the distance formula is below.
The answer is approximately 10.61 meters. So, it would seem that the best route was my first one. Going diagonally all the way across adds distance instead of shortening it. So, going diagonally over a shorter distance and then going in a straight line to the final destination point is the best route. But the fastest route, instead of going around the room, is to go through the room. So, what if, instead of a spider that has to crawl over a 2-D space (side-to-side/up-and-down), there is a dragonfly that can fly through a 3-D one (side-to-side/up-and-down/forward-and-backward)?

3-D Extension
I was very interested in exploring 3-D geometry, and the first thing I wanted to learn was how to obtain a line in 3-D space. So, I watched David Butler's "Example: Finding the equation of the line in 3D through two points" YouTube Video from Maths Learning Centre UofA channel, and applied it to this problem. My work is below.

Then, I wanted to find the distance the dragonfly would have to travel to reach the fly. My professor explained I could use the distance formula with the z-coordinates added. My work is below:

The square root is about 5.226. So, the dragonfly would have to fly about 5.226 meters to get to the fly.

A great problem, like this one, inspires the imagination. The original problem, even though the setup is in 3-D, asks a 2-D question. The concept that needs to be understood is that a spider can only move in two ways on the wall or plane. The problem can be thought of and solved in different ways, as shown above. So, the original problem has lots to explore, but it also offers more. As soon as I read the problem, my mind went straight to "But what if the spider could fly?", turning the 2-D question into a 3-D one, and then I sought out ways to answer my question. Curiosity is an indication of interest and readiness to learn something new. Offering opportunities for students to imagine "what if" is the best way to engage and encourage their learning. It will motivate them to learn on their own, fostering a sense of responsibility for their education. And as a teacher, that is the most important skill I can encourage in my students.

Figures created using GeoGebra and Microsoft Word. I do not have any affiliation with NRICH, GeoGebra, or David Butler, and none have endorsed my work. I am a college student. 

Sunday, November 1, 2015

Arctan and Euler

 Communicating Math

Leonard Euler has made many contributions to math, and one of them is a formula for arctan. To see and investigate Euler's formula, I used GeoGebra's An Equation of Euler by timteachesmath (Link). There are 5 sets of examples of the formula in each of the 3 options in the GeoGebra exploration. Below is my explanation of the equation in relation to the GeoGebra activity.

Option 1

First, at T=0, there is a square with a diagonal line (Figure 1). As I move T along the slider, the green square becomes longer to create a rectangle with the original square. I also get a blue square and blue rectangle that has been rotated. The reason the blue square/rectangle gets rotated is so the two corners of the blue rectangle remain touching the corners of the green rectangle.

Figure 1: Option 1 - T=0

When T=1, the formula appears (Figure 2). Essentially, it's saying the angles b and c will add up to equal angle d. Since tangent of an angle = opposite/adjacent, the first arctan corresponds to angle b (green line to the red diagonal) in the green rectangle, since tan (b) = 1/3. The second arctan corresponds to angle c (from blue line to red line) in the blue rectangle, since tan (c) = 1/2. The final arctan corresponds to the angle d (from green line to blue line) in the green square, since tan (d) =1/1.
Figure 2: Option 1 - T=1

Option 2

Again, Option 2 starts with the same square as Option 1 when T=0. When T=1 though, the first square gets divided into fourths, there are 3 large green squares and 1 large rotated blue square. Again, the two corners of the blue square remain on the two corners of the green rectangle. Now, the equation asserts that angle b of the green rectangle and angle c of the blue square add up to angle d of the green rectangle comprised of the 2 smaller squares (Figure 3). This is because tan(b)=1/3 , tan (c ) = 1/1, and tan (d) = 2/1.
Figure 3: Option 2 - T=1

Option 3

Here, Option 3 starts with 2 green squares stacked vertically at T=0 (Figure 4).
Figure 4: Option 3 - T=0

 At T=1, the 2 green squares expand to 6 (3 columns and 2 rows). There are 5 blue rotated squares, and again the 2 corners of the bigger blue rectangle remain on the two corners of the bigger green rectangle (Figure 5). Because we have two rows of squares the numerator of the first arctan is 2. In other words, tan (b) = 2/3 (the big green rectangle), tan (c ) = 1/5 (the blue rectangle), and tan (d) = 1/1 (the green square).
Figure 5: Option 3 - T=1

The key for this equation is keeping the two opposite corners of the blue square/rectangle inline with the expanding green square/rectangles. This makes the blue squares rotate and expand with the addition of the green squares. This creates a blue diagonal that goes through the green squares and rectangles. Thus, the red and blue lines divide the green rectangle in different ways to make the equation possible.

There were two other interesting things I noticed. The first arctan term was the answer for the next formula in Option 1, and all the numerators and denominators are numbers in the Fibonacci Sequence in all the options. Also, the Fibonacci Sequence was in reverse order in Option 1. As the blue rectangle rotates and expands with the green rectangle, a relationship between the green squares and rectangles and the blue squares and rectangles is maintained. So, the equation only works when the original square (T=0) can divide the green rectangle equally and the blue rectangle can be divided equally by a different size square.

Update (11/9/2015): Eureka!  The rotated blue rectangle creates a parallelogram with the green rectangle. The red diagonal splits the parallelogram in half, creating two congruent triangles. We can also see the blue lines that create the sides of the parallelogram as two congruent transversals that cut the green rectangle. So, the properties of parallel lines and parallelograms is why Euler's arctan formula works.

Screen shots are of timteachesmath's An Equation of Euler on GeoGebra. I added the angles b, c, and d. I do not have any affiliation with GeoGebra or timteachesmath, and neither has endorsed my work. I am a college student. 

Sunday, October 18, 2015

From Primitive to Modern: The Evolution of Mathematics

Book Review: Clifford A. Pickover's The Math Book

From the very beginning in the Introduction, Pickover makes it clear that this book is not the complete, definitive list or the final word on what the milestones of math should be. As he mentions, these are his selections that cover a wide range of mathematical topics, from games to important theorems, and are written in such a way that an average person could understand, in a general sense, what the topic is about. And to his credit, Pickover does just that, and does it well.

The list is in chronological order and the topics have one-page summaries with an accompanying picture that can help the reader visualize the math concept or milestone featured. The breadth of time Pickover covers is from BC to the early 2000s. The list starts, surprisingly, not with a human mathematical milestone, but with an animal one. In fact, the first two milestones describe how animals such as ants and monkeys have mathematical capabilities. And the next milestone describes a mathematical pattern in Cicada (insect) behavior. The first human mathematical milestone, which is the fourth milestone in the book, is the presence of knots. Then the milestones become more historical with the mathematical advancements from ancient civilizations, like the ancient greeks. Famous games, equations, numbers, and texts are featured in this book, as well as technological advancements, like computers that can help prove conjectures.

The reason Pickover showcased the milestones chronologically, as he states in his introduction, is to have the reader see the evolution of mathematical thought, and how one milestone is connected and/or spurred another milestone. Pickover delivers on this goal, since I saw how mathematics is ultimately a compounding network of many topics, each connected - influencing, advancing and creating others. He not only points out these connections, he even lists the related topics at the bottom of the page. Furthermore, Pickover also points out how a milestone is important or useful outside the field of mathematics like in economics or the sciences. The best example of this are knots. Pickover mentions knots as the first primitive human milestone, and the topic continues to progress in complexity with advances in knot theory like Jones Polynomial, which has applications regarding the DNA molecule and proteins (pg. 24, 478, 490).

But why start the book, which is about mathematical milestones, with a mathematical capability of an animal, instead of a human? Why would it even be considered a milestone? Pickover discusses how mathematics is not solely human, and it implies how human mathematical capability has evolved from our animal ancestors.

All in all, this book is a great read for those who want a condensed history of mathematical milestones. Since this is my first major exposure to the history of mathematics, it's hard for me to compare this list to other milestones and say "Why isn't such-and-such in this book?!?" However, 3 out of my 5 milestones that I could barely come up with in this blogpost were featured in Pickover's book (Zero, Calculus, and Alan Turing). The first one that didn't show up was Albert Einstein's E=mc2, even though Pickover mentions Einstein in the book, and specifically that equation in the Introduction. The second was a milestone that happened after the publication of the book, which was the first woman to win the Fields Medal (1).

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.

Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(1) Ball, Phillip. "Maryam Mirzakhani Becomes First Woman To Win Prestigious Fields Medal." The Huffington Post.

Sunday, October 11, 2015

Math: Making Science Possible

Nature of Mathematics

Math has always been taught as a separate subject from science throughout my education. Furthermore, math was taught first. Addition and subtraction were 1st through 3rd grade concepts. Multiplication and division were in 4th and 5th grade. And that's when science was introduced as a subject - not until the 4th and 5th grade. Based on the structure of my education alone, it seems that the concepts of math (numbers, operations etc.) had to be taught first before the concepts of science could be learned, since science uses a lot of mathematics. But could math be considered a science field because of this connection? 

In my class the other day, we conducted an experiment where we dropped objects from a bridge that were attached to a certain number of rubber bands. We were supposed to determine the right number of bands that would make our object come closest to the ground without touching it. For me, I felt like I was doing both science AND math while doing this experiment. The scientific component: forming a hypothesis, factoring gravity, weight, and elasticity to make that hypothesis, and testing that hypothesis with an experiment. The mathematical component: the acceleration of gravity is a mathematical constant, the mass of the object, and the force of the weight, which is described in a mathematical equation. Even though they overlap, I still see them as two separate fields.

To help illustrate this point further, I've made a Venn Diagram showing how I see math and science as two separate fields that do overlap. 

Math/Science Venn Diagram

So why did I organize the Venn Diagram the way I did? I won't be able to explain my decision-making process for each subtopic (my blogposts are long enough), but I will try with a few examples. I put geometry squarely in Math, and Astronomy in Science, even though I can think of ways they overlap. For example, orbits around the sun in astronomy can be described as ellipses and math was applied to the data to discover this. But shapes, like ellipses, exist and have mathematical properties in geometry independent of astronomy. Furthermore, there are concepts in astronomy that are independent of math, like which planets have or don't have atmospheres and how that impacts that planet's environment. So, that's why geometry and astronomy, and other subtopics, are not in the overlap part of the diagram.  

So, what connects the two? What is in the overlap? Data most importantly. Science collects data during experiments. Math and Statistics - a field I consider separate from Math, but is included as a subfield of math for simplicity - are used to make sense of the data. Data and math were used to discover that planets orbit the sun in an ellipses shape. Physics, even though it's usually referred to as a science, is so dependent on math, I put it in the overlap. I can't think of physics (gravity, force, acceleration etc) without thinking of math, like trigonometry and quadratics, which are all in the overlap.

There are two concepts in Math and Science that are similar but have different processes and functions that help separate Math and Science from each other, and they are Proofs and Experiments, respectively (look at the bottom of the circles). Scientists conduct experiments to test hypotheses, which are predictions based on observations of the natural world. Mathematicians, when they observe a pattern with numbers or shapes, make a conjecture and then they try to prove it. When they do, the conjecture becomes a theorem and it is considered fact in the mathematical community. This is where the line between math and science is most visible for me. In science, hypotheses are never considered fact. When the data is inline with the hypothesis, scientist say the hypothesis is supported, never proven. It takes multiple experiments done repeatedly before a hypothesis or hypotheses are considered to have enough supporting evidence to be a theory, which is still not considered a fact.

And there are other fundamental differences that separate math and science. Science is about understanding how things in the physical world/universe work. As my class discussed, math is about quantifying the world/universe and understanding numerical patterns, which are often found in nature, like the Fibonacci Spiral in pine cones (1). But finding patterns in the natural world that can be described using numbers doesn't make mathematics a science. Math exists without science, as illustrated with subtopics like geometry. In geometry, there are abstract concepts that don't exist in nature, like a one-dimensional line that expands to infinity. Also, discoveries or advancements in math are concerned with patterns in numbers, not physical processes in the world/universe, like in science. Furthermore, it seems that math makes science possible since math is used to interpret scientific data. And even though, math describes gravity with quadratics, science would not perish if the connection between science and math was less strong. For example, what if gravity was random, and couldn't be represented by a quadratic? (Let's not think about the fate of humanity and all life for the moment). Scientists would still be trying to find out how things work, but it would have less of a mathematical connection.

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.


(1) Vi Hart. "Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]." YouTube Video. Date of Access: October 12, 2015.

Figure created using GeoGebra.

Monday, September 28, 2015

The Value of Nothing: How Did We Get From One to Zero?

History of Mathematics

At the very beginning of my K-12, and maybe even my preschool education, zero has always been a number. The concept of zero - as a number that stands for something that isn't there - is as acceptable, and natural to me as the numbers that represent things that are there (1 thing, 2 things etc.). Mathematics wouldn't make sense without it. For counting, I could either start at nothing, 0, or 1. For arithmetic, 2-2=0. Two things exist. Take those two things away and those two things aren't there anymore. Nothing is left.

But the concept of zero as a number isn't natural at all, especially in Western Civilization (Europe) (0). The concept of zero started in the East, most notably in India, and it took until the Renaissance for Western Civilization to accept a representative number for nothing (0 + 1). So how did Western Civilization finally place a value for nothing? How did we get from one to zero? The time period and from what civilization zero was born depends on how one defines a number.  

According to the MacTutor History of Mathematics Archive, the expression of zero throughout history can be separated into two categories (1). The zeroeth category (see what I did there) is zero as an indicator for empty space in a number system or as a positional notation in context with other numbers (1). The first category is zero as an abstract, independent number with value and properties, which is the way we think of zero today (1+ 2).

We will first discuss the first category: zero as an abstract number. Brahmagupta, an Indian mathematician living from 598 to 665 AD, is considered to be the first to think of zero this way (0 + 3). In his work titled Brahmasphutasiddhanta, he developed properties for negative and positive numbers, and for the number zero (1 + 4). The idea of zero as an actual number spread to China and the Middle East (1). Fibonacci in the 1200s tried to get Europe to replace Roman numerals with the 0-9 number system that we use today, but met opposition (0 + 1 + 7). In Christianity,  God was associated with infinity and the idea of nothingness, the opposite of infinity, was therefore associated with the devil (0). Furthermore, Fibonacci was campaigning for the new number system during the Crusades, and the new number system was associated with Islam, since they had already taken the same number system from the Hindus (6 + 7). Finally, in the 15th century , Europe converted to the 0-9 number system (1).

But before zero was seen as an abstract number, the concept of zero started as a positional value, which is the zeroeth category (1). In an interview on BBC's In Our Time Radio program, Robert Kaplan, Ian Stewart, and Lisa Jardine describe the importance of the number zero in history, and the significance of it's use in positional notation. The concept of zero started as a symbol to signify no-number about 5,000 years ago in the Sumerian number system, which used positional notation (0). Eventually, zero as a positional value became common in bookkeeping in different civilizations over time because of trade, except in Western Civilization (0). The ancient Greek mathematicians considered basic arithmetic that was used in trade not intellectually stimulating enough to think about (0). Furthermore, the ancient Greeks decided that the concept of nothing couldn't exist in the real world (5). So, the zero that meant no-number in trade didn't cross over to become an abstract mathematical concept for the ancient Greeks, nor the Romans, who based their number system off of the ancient Greeks (0).

So when in mathematical history should the advent of the number zero be marked? Was Sumerian Civilization the first when they used zero has a positional value? Or was it Indian Civilization the first when they treated zero as an abstract, independent number?

As I discussed in a previous post, I believe counting to be a mathematical concept, and therefore consider assigning numbers and values to the physical, tangible world as one of the first mathematical activities humans did. Historically speaking, humans did not start counting at zero, as indicated by the number systems of many ancient civilizations, since nothing isn't tangible and therefore has no value (1).

Yet, the concept of zero was used as a positional value in number systems, and arithmetic was used in trade, even though people of these civilizations didn't treat zero as a number that had value, and therefore didn't compute it in their calculations like we do today (0). They probably skipped right over the symbol, considering that the symbol for zero in position notation was often represented as an empty space (1). But they were inadvertently placing a mathematical property on that empty space when they did arithmetic, a property that Brahmagupta would attribute to the number zero: any number plus or minus zero is the same number (4). With that said, this action of skipping over the symbols for zero as a positional value is equivalent to us using the number 0 in 101+190 = 291.

Therefore, I believe the number zero was born when it was just a positional value at the time of Sumerian Civilization. Very humble beginnings indeed, especially when we look ahead in time and see all that was achieved during the Renaissance when zero was finally accepted as a number in Western Civilization, like Newton discovering calculus (0 + 2). Who knew nothing could amount to so much?

Sources: *No person or organization has endorsed my work, nor am I affiliated with anyone or organization. I am a college student.*

0. Podcast. Interviewees: Kaplan, Robert. Stewart, Ian. Jardine, Lisa. Interviewer: Bragg, Melvyn. "In Our Time: Zero." BBC Radio 4. Date of Access: September 28, 2015.

1. O'Connor, John J. and Robertson, Edmund F. "A History of Zero." MacTutor History of Mathematics Archive. Date of Access: September 28, 2015. 

2. "Who Invented the Zero?" Ask History. Date of Access: September 28, 2015.

3. Hayashi, Takao. "Brahmagupta: Indian Astronomer." Encyclopedia Britannica. Date of Access: September 28, 2015.

4. Mastin, Luke. "Indian Mathematics - Brahmagupta." The Story of Mathematics. Date of Access: September 28, 2015.

5. "A Brief and Early History of Zero (ca. 2nd C BC - Onward)." The Ancient Standard. Date of Access: September 28, 2015.

6. Mastin, Luke. "Islamic Mathematics - Al-Khwarizmi."  The Story of Mathematics. Date of Access: September 28, 2015.

7. Mastin, Luke. "Medieval Mathematics - Fibonacci."  The Story of Mathematics. Date of Access: September 28, 2015.

Sunday, September 13, 2015

Riddle Me This, Diophantus

Blog post for Doing Math

Diophantus was a great, ancient greek mathematician, who played a very important role in the subject of algebra ( To honor this man, my MTH 495 professor gave us a riddle - a riddle of algebra - to solve that would give Diophantus' age when he died. Here it is, as given on the class handout, below:

" 'Here lies Diophantus,' the wonder behold. Through art algebraic the stone tells how old: 'God gave him his boyhood one-sixth of his life, one-twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; in five years, there came a bouncing new son. Alas, the dear child of master and sage after attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' "

The language seems clear but riddles are tricks of the mind. What seems straightforward, isn't, and the challenge is finding where the riddle deliberately leads the mind astray, so it would make an erroneous assumption. In other words, how has the riddle tricked us into thinking one interpretation, when it really meant something else?

To demonstrate how challenging the riddle was for me, I went through 7 attempts or 7 different interpretations of the riddle before I got the correct answer. For the first attempt, I tried solving the riddle without using algebra, and that meant picking an age for Diophantus, and doing the series of computations laid out in the riddle to see if it was correct. Basically, guess-n-check. I picked 50 because people rarely lived for very long back then.

On this first attempt, I thought the challenge, or the trick in the riddle, was in the line "And then one-seventh ere marriage begun." The word ere means before, according to ( So, I had to subtract 1/7 from his boyhood and youth before continuing on with addition. So, the sequence of events laid out in the riddle is not in chronological order. Everything else seemed straightforward. I interpreted "God gave him one-sixth of his life," as 1/6 of his age. "One-twelfth more as youth," I interpreted as his youth was 1/12 more than his boyhood, or 1/6 of his age. "One-seventh ere marriage begun," I took as 1/7 of youth subtracted from his boyhood and youth. My interpretation for the rest: then 5 years after his marriage, a son was born, who grew to half the age that Diophantus eventually reached, 4 years after his son's death. As mentioned before, I picked 50 for Diophantus' age. So, I did the following computations:

Since the ages don't match, my guess was wrong. But the closeness between the two numbers gives a false impression that 50 was close. However, the fractions would divide any number in such a way that would make the two numbers seem close. At this point, I decided that I could keep on blindly guessing or try to solve the riddle in the most efficient way possible, with algebra.

When I continued with my other attempts, I realized the challenge was no longer with the chronological wording of the riddle. The trick lay with the wording of the fractions. In my middle school teaching course, the class went over fractions, and whenever someone said "Take 1/2," our professor would always say "1/2 of what?" The fraction needs to be applied to another number, but sometimes the riddle doesn't tell explicitly what the fraction is being applied to. For example, nowhere in the line "One-twelfth more as youth" does it say what 1/12 is being taken from. Is it 1/12 more of his entire life that is being added to his boyhood, or is it  1/12 of his boyhood, or is it just simply 1/12 of his life was his youth, and not in addition to his boyhood?   

I interpreted the riddle differently, with different algebraic equations, 6 more times. On the last attempt, I got it right. I will show the different combinations or ways I interpreted the riddle with the variations for each section of Diophantus' lifespan below. The straightforward interpretations are on top and the multiple interpretations are on the bottom:

Finally on the last, overall 7th attempt, I got the answer right, at least according to Wikipedia ( The interpretation necessary was youth equaling his boyhood plus 1/12 of his entire life, and his marriage was 1/7 of his boyhood before the end of his youth. My solution is below:

I enjoyed this riddle very much. It reminded me how the process of mathematics should be like. Doing math is about looking at problems or situations with different interpretations to see how it plays out and what's discovered in the process. After all, learning requires making mistakes. Figuring out what went wrong is necessary to find out what will make it right.

Figures were created using Microsoft Word and Geogebra.