Topics from “Pre-Calculus” (Explaining Calculus #1)

Before we can go into a discussion of calculus itself, it is important to set up some of the underlying concepts that calculus uses. You might view this as something analogous to learning the letters of the alphabet before you can start learning to read words and sentences. Without letters, you aren’t going to be able to grasp words. But once you start getting the words down, you don’t have to think about the letters very often any more. In the same way, there are certain mathematical ideas that are important for understanding how calculus works and what it can do, and you have to understand those first. But as you grow in understanding of calculus, you will have had enough practice with these ideas that you don’t think much about them anymore because they begin to come naturally.

My goal here is to lay out some of the key concepts that will come up at various times in the discussion of what calculus is, what it means, and what it does. You don’t have to be fluent in these ideas to understand the underlying concepts, but it certainly doesn’t hurt.


One of the most important uses of mathematics is in tracking how things are related to each other in the world around us. As an example, you might ask “if I throw a ball this hard, how far will it go?” The underlying relationship is between the force you put into your throw and how far the ball travels. In any situation like this one, you’re going through in your mind a mathematical concept called a function.

In mathematics, a function is a rule for transforming one kind of object into some other kind of object. Usually, a function turns one number into another number. The central idea of a function is that if you know the input, then there is only one way to get an output out of that. As in our example, if you know how hard you threw your ball, the the laws of physics determine exactly how far that ball will go (of course there are other factors like wind and angles, but you get the point).

Certain shorthand notations are helpful in talking about functions. Very often, we use the letter f to symbolize a function. If we need to use a few different functions at the same time, we often use the letters g and h next (just because those are next in the alphabet). You can use any symbol you want to represent your function, the choice of f, g, and h is merely a matter of convenience. We will also often write f(x) to denote the function f with the input value x. We then write f(x) = ... and in the realm of the dots, we write the rule we are supposed to use. For example, the function f(x) = 2x + 3 is a rule according to which you take your number x, multiply by that 2, then add 3. If you want to use a specific number in your function, you replace all the x‘s with the number you intend to use. For example, f(2) = 2*2 + 3 = 7 and f(-1) = 2*(-1) + 3 = 1.

There is one other slightly less common but equally important notation we should introduce. Sometimes, it is important to know the specific input values we are about. For example, you can’t throw a ball with negative force, so if you write a function describing the relationship in our example, you won’t care about whether or not you “can” plug in negative numbers. You might also care what sorts of numbers your output looks like – you can’t throw the ball a negative distance. In these situations, we use the notation f : A \to B (read “f is a function from A to B). When we write f : A \to B, what we mean is that A symbolizes all the inputs allowed for the function f, and all of the outputs of f will be somewhere in B.

When we express A and B, we usually use what is called interval notation. Interval notation is a way to express a range of values by writing down the endpoints of the ranges. When we want to write all numbers, we write (-\infty, \infty). You can basically just read this as “all numbers.” We can also use interval notation to express a limited range of numbers. For example, the interval (1,3) represents all numbers between 1 and 3, not including 1 and 3 themselves. The interval [1,3] represents all numbers between 1 and 3, including 1 and 3. In these examples, the round/open brackets ( ) translate to “not including” and the square/closed brackets [ ] translate to “including.” As another example, the interval (1,3] represents all numbers between 1 and 3, including 3 but not including 1. In intervals, \infty is used to mean that the interval never ends one some side. For example, (1,\infty) represents all numbers larger than 1. Finally, you can combine intervals using the symbol \cup (read this as “union”). So, (-3, -1) \cup (1,3) represents all numbers that are either between -3 and -1 or between 1 and 3.

Types of Functions

Functions can be just about anything you want them to be, but there are some special and helpful examples that are often used as examples in calculus and other areas of math. So, we will go through a few of those examples here.


A polynomial is any function built by adding/multiplying variables with numbers or other variables. Normally this takes the form of multiplying x by itself a certain number of times, multiplying the result by some constant number, and adding together other terms constructed in the same way. Here are a few examples of polynomials:

x^2 + 2x + 1, \ \ \ x^7 - 6x^3 + 3, \ \ \ x + 2, \ \ \ x^{101} + 53x^2 - 56x.


An exponential is formed by multiplying a constant to itself a variable number of times. One example of an exponential function is f(x) = 2^x. To calculate a few values of this exponential function, f(3) = 2^3 = 2 * 2 * 2 = 8 and f(5) = 2^5 = 32. An exponential function can use any number as its input, although the way to compute an exponential with a fraction (like 8^{1/3}) or with even weirded exponents uses some different methods… and very often you can’t really “simplify” an exponential expression.

If you use the definition of an exponential as repeated multiplication, you can deduce a few rules about how exponentials work. For example, 2^x * 2^y = 2^{x+y} and (2^x)^y = 2^{xy} will always be true. These rules form the basics of exponentials. Using these, we can also figure out a few other rules. What if, for instance, we want to know 2^0? Well, the first of our two rules would tell us that 2^2 * 2^0 = 2^{2+0} = 2^2. This quick equation leads us to conclude that 2^0 = 1. We can do something similar with negative exponents to see that a negative exponent is really just division. That is, 2^{-x} = \dfrac{1}{2^x} (see if you can see why – as a hint, you can use the first of the two rules along with the fact that 2^0 = 1). We can use the second rule to decipher the meaning of 8^{1/3}. Whatever this means, it should satisfy (8^{1/3})^3 = 8^{3 * 1/3} = 8^1 = 8. Since 2^3 = 8, this leads us to conclude that 8^{1/3} = 2. Similar ideas can be used for other fractional exponents.

The need to do calculations with exponentials is actually relatively rare, but it is important to know that there are these rules that can help us simplify expressions that have exponentials in various ways.

Graphs of Functions

Graphs are universally taught in school because they are a good way of visualizing information about mathematics. The idea is to use a two-dimensional grid to show how a function relates numbers to other numbers. This two-dimensional plane is often called the xy-plane. The reason is that the “horizontal” dimension is usually represented by an x variable, and the “vertical” dimension is usually represented by a y variable. Points on a graph are normally written as (x,y), which means “x units horizontally and y units vertically from the starting point.” This “starting point” is often called the origin (as it is the point from which the others ‘originate’ in a sense).

All equations, and in particular functions can be represented in the form of a graph. The way to do this is to plot all the points with coordinates (x,f(x)). In other words, the x-coordinate is the input, and the y-coordinate is the output. Below is an example, a graph of the function f(x) = x^2:

Circles and Distance

The concepts of distance and circles are tightly intertwined, and so we discuss them at the same time. Before doing so, it is useful to discuss a third concept – the Pythagorean Theorem – that is fundamental to understanding each. The Pythagorean Theorem tells us how to calculate the side lengths of a right-angled triangle. More specifically, if a,b,c are the side lengths of a right-angled triangle with c the longest side, then the formula a^2 + b^2 = c^2 will always be true.

How does this apply to the notion of distances? Imagine drawing two points in the plane (i.e. the xy-plane). How far are these from each other? The answer is the length of a line segment that connects them. If you’re having trouble visualizing this, draw two points on a piece of paper and connect them with a straight line. Now, draw a line in the east-west direction starting at one point and ending “beneath” or “above” the other point. Then connect this new line to the second point with a line running in the north-south direction. If the north-south line has length a, the east-west line has length b, and the distance between the two points is d, then the Pythagorean Theorem tells us that a^2 + b^2 = d^2. By taking square roots, we obtain

d = \sqrt{a^2 + b^2}.

If the two points have coordinates (x_1, y_1) and (x_2, y_2) in the xy-plane, then this distance formula turns into

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

This is because a = y_2 - y_1 or y_1 - y_2, depending on which is positive, and similarly b = x_2 - x_1 or x_1 - x_2.

But how exactly does all of this relate to a circle? Well, the definition of “circle” involves the notion of distance. A shape is a circle if it is built out of all points that are the same distance from a central point. To visualize why this works, imagine taking a piece of string. Attach one end of the string to a fixed point on a piece of paper. Put a pencil in the other string, and move the pencil around the paper, keeping the string tightly pulled the whole way. If you do this correctly, you will draw a circle. The length of that piece of string is the “distance” involved in the definition, which we call the radius of the circle. The center of the circle is point that was pinned down to the paper.

Let’s use this to make an equation for circles. We want to know whether a point (x,y) is on a circle centered as (a,b) with radius r. That is, we need to know whether (a,b) and (x,y) are distance r apart. Using the distance formula we previously came up with, this becomes

r = \sqrt{(x-a)^2 + (y-b)^2}.

If we square both sides (and swap the right and left-hand sides), we come to the easiest-to-look-at version of the circle equation:

(x-a)^2 + (y-b)^2 = r^2.


This completes the discussion of “pre-calculus” topics. You don’t have to know how to do detailed work with these ideas in order to understand what calculus is about. But I’ve found through teaching that these concepts are quite helpful and are worth thinking about.

Database: What Do History’s Great Mathematicians Think About God?

There is very often debate about whether science and religion are in conflict. I know from my own studies and will continue to affirm that these are not at all in conflict. There are quite a few ways to go about this, and I plan to take on every angle that I can. Since I am studying to be a mathematician, I thought I’d start there. For if Christianity is anti-science, then you wouldn’t expect to see very many Christians among the greatest mathematicians. Therefore, I thought it would be informative to do a sort of statistical study of the greatest mathematicians of history.

My aim here is to take a subset of the greatest who have ever lived in my professional field of mathematics – where I have the background knowledge to confirm the greatness of these mathematicians. I have used a website that has an all-time top 200 list of mathematicians (for this list of mathematicians and their accomplishments, see This list is obviously not sectioned to bias any particular religious worldview, and so I figured I’d do a skim through it and do some considerations of what these great men and women believe about God.

To be properly balanced, I ought to say plainly that nothing can be totally proved from such a list, one way or another. This is purely statistical, and so nothing can be demonstrated beyond doubt. However, this can serve as evidence that can be evaluated in a ‘scientific’ way. Because the two opinions involved – the opinion that Christianity is anti-science at its core and that Christianity is compatible with science at its core – make different predictions about how many scientists and mathematicians embrace Christianity. Thus, while not conclusive, it ought to provide at least some insight into the questions at hand.

Over time, I hope to fill in more of the listings that I have put as ‘unclear’ or as ‘likely’ something-or-other. Turns out, this kind of biographical research isn’t always easy, as many people keep their beliefs about religion/God private. There are times that I have made educated guesses, and in those situations I have listed as ‘likely’ the conclusion. If I didn’t put ‘likely,’ this means that I found something that I found convincing (although I may well still be wrong). In any case, errors and inability to find information should approximately balance out.

The Top 200 Ranking, along with Views on God

  1. Isaac Newton – Christian
  2. Archimedes of Syracuse – Likely polytheist
  3. Carl F. Gauss – Christian
  4. Leonhard Euler – Christian (Calvinist)
  5. Bernhard Riemann – Christian
  6. Henri Poincaré – Atheist
  7. Joseph-Louis Lagrange – Agnostic
  8. Euclid of Alexandria – Likely polytheist
  9. David Hilbert – Agnostic
  10. Gottfried Leibniz – Christian
  11. Alexandre Grothendieck – Theist
  12. Pierre de Fermat – Christian (Likely Catholic)
  13. Évariste Galois – Unclear
  14. John von Neumann – Likely Agnostic
  15. René Descartes – Christian (Catholic)
  16. Karl Weierstrass – Likely Theist
  17. Srinivasa Ramanujan – Hindu
  18. Hermann Weyl – Deist
  19. Peter Dirichlet – Christian (Catholic)
  20. Niels Abel – Likely Christian (Lutheran)
  21. Georg Cantor – Christian (Lutheran)
  22. Carl Jacobi – Christian (from Judaism)
  23. Brahmagupta – Hindu
  24. Augustine Cauchy – Christian (Catholic)
  25. Arthur Cayley – Unclear
  26. Emmy Noether – Likely Jewish
  27. Pythagoras of Samos – ‘Pythagoreanism’
  28. Aryabhata – Likely Hindu
  29. Leonardo ‘Fibonacci’ Pisano – Christian (Catholic)
  30. William Hamilton – Unclear
  31. Appolonius of Perga – Unclear
  32. Charles Hermite – Christian (Catholic)
  33. Pierre-Simon Laplace – Agnostic
  34. Carl Siegel – Likely Theist
  35. Johannes Kepler – Christian
  36. Diophantus of Alexandria – Unclear
  37. Muhammad ibn al-Khwārizmī – Muslim
  38. Kurt Gödel – Christian (Lutheran)
  39. Richard Dedekind – Unclear
  40. Felix Christian Klein – Unclear
  41. Bháscara (II) Áchárya – Likely Hindu
  42. Blaise Pascal – Christian (Catholic)
  43. Élie Cartan – Unclear
  44. Archytas of Tarentum – ‘Pythagoreanism’
  45. G.H. Hardy – Atheist
  46. Alhazen ibn al-Haytham – Muslim
  47. Jean le Rond d’Alembert – Atheist
  48. F.E.J Émile Borel – Unclear
  49. Julius Plücker – Unclear
  50. Hipparchus of Nicaea – Unclear
  51. Andrey Kolmogorov – Unclear
  52. Joseph Liouville – Unclear
  53. Eudoxus of Cnidus – Unclear (student of Plato)
  54. F. Gotthold Eisenstein – Christian (Protestant)
  55. Jacob Bernoulli – Christian (at least some ministry)
  56. Stefan Banach – Likely agnostic (possibly Catholic)
  57. Jacques Hadamard – Atheist
  58. Giuseppe Peano – Unclear
  59. Panini of Shalatula – Hindu
  60. André Weil – Agnostic (with heavy Hindu influence)
  61. Jean-Pierre Serre – Unclear
  62. Jakob Steiner – Unclear
  63. Marius Sophus Lie – Unclear
  64. Gottlob Frege – Christian (liberal Lutheran)
  65. François Viète – Christian (probably Protestant Huguenot)
  66. Christiaan Huygens – Christian (Protestant)
  67. M.E Camille Jordan – Unclear
  68. Joseph Fourier – Christian (Likely Catholic, considered becoming a monk)
  69. Bonaventura Cavalieri – Christian (Catholic, Jesuate monk)
  70. Hermann Grassman – Unclear (but I think Christian)
  71. Albert Einstein – Agnostic (maybe pantheist)
  72. James Clerk Maxwell – Christian (Evangelical or Presbyterian)
  73. Girolamo Cardano – Christian (Likely Catholic)
  74. Aristotle – Theist
  75. Galileo Galilei – Christian (Catholic)
  76. Michael Atiyah – Unclear
  77. Atle Selberg – Unclear
  78. Alfred Tarski – Atheist (yet also formally a member of the Catholic church)
  79. Gaspard Monge – Atheist
  80. L.E.J Brouwer – Unclear (I’d guess atheist, but am not sure)
  81. Liu Hui – Likely Taoist or Buddhist
  82. Alan Turing – Likely Theist (maybe atheist – a bit hard to tell)
  83. Jean-Victor Poncelet – Unclear
  84. John Littlewood – Unclear
  85. Shiing-Shen Chern – Unclear
  86. James Sylvester – Jewish
  87. Henri Lebesgue – Unclear
  88. Johann Bernoulli – Likely Christian (Protestant)
  89. Ernst Kummer – Unclear
  90. Johann Lambert – Christian (Likely Protestant)
  91. George Pólya – Agnostic
  92. Felix Hausdorff – Likely Atheist
  93. George Birkhoff – Unclear
  94. Pafnuti Chebyshev – Unclear
  95. Adrien Legendre – Unclear
  96. John Nash – Atheist
  97. Siméon-Denis Poisson – Unclear
  98. John Wallis – Christian (Clergyman)
  99. Omar al-Khayyam – Agnostic/Atheist
  100. Thales of Miletus – Likely Theist
  101. Hermann Minkowski – Jewish
  102. Simon Stevin – Likely Christian (Calvinist)
  103. Nicolai Lobachevsky – Atheist
  104. Andrei Markov – Atheist
  105. Daniel Bernoulli – Christian (Protestant)
  106. Mikhail Gromov – Unclear
  107. Paul Cohen – Likely Jewish
  108. John Milnor – Unclear
  109. Robert Langlands – Unclear
  110. John Conway – Likely Agnostic (Maybe Christian?)
  111. Pierre Deligne – Unclear
  112. William Thurston – Unclear
  113. Edward Witten – Likely Jewish
  114. Saharon Shelah – Likely Jewish
  115. Terrence Tao – Unclear
  116. John Thompson – Unclear
  117. Simon Donaldson – Unclear
  118. Vladimir Arnold – Unclear
  119. Stephen Smale – Unclear
  120. Timothy Gowers – Unclear
  121. Pappus of Alexandria – Unclear
  122. Sofia Kovalevskaya – Likely Atheist
  123. Leopold Kronecker – Christian (late-life convert from Judaism)
  124. Thabit ibn Qurra – Muslim
  125. Siméon Denis Poisson – Unclear
  126. Paul Erdős – Agnostic (maybe atheist)
  127. Jean Gaston Darboux – Unclear
  128. Nasir al-Din al-Tusi – Muslim (Shia)
  129. Ferdinand Georg Frobenius – Likely Christian (Protestant)
  130. George Boole – Christian (Unitarian)
  131. Hippocrates of Chios – Unclear
  132. James Gregory – Likely Christian (Protestant)
  133. John Napier – Likely Christian (Protestant)
  134. Norbert Wiener – Likely Jewish
  135. Lennart Carleson – Unclear
  136. Emil Artin – Unclear
  137. Ptolemy of Alexandria – Unclear
  138. Tullio Levi-Civita – Likely Jewish
  139. J. Müller ‘Regimontanus’ – Christian (Catholic)
  140. Abu Rayhan al-Biruni – Muslim
  141. Girald Desargues – Likely Christian (Catholic)
  142. John Tate – Unclear
  143. Alfred Clebsch – Unclear
  144. Oliver Heaviside – Christian (non-practicing Unitarian)
  145. Alexis Clairaut – Unclear
  146. Oswald Veblin – Unclear (maybe Christian, Lutheran)
  147. Colin Maclaurin – Christian
  148. Qin Jiushao – Unclear
  149. Henri Cartan – Unclear
  150. Henry J.S. Smith – Unclear
  151. Lars Valerian Ahlfors – Unclear
  152. Rafael Bombelli – Christian
  153. Michel Chasles – Likely Christian (Catholic)
  154. William Clifford – Likely Theist (?)
  155. Samuel Eilenberg – Likely Jewish
  156. Maurice René Fréchet – Unclear
  157. Lars Hörmander – Unclear
  158. Kunihiko Kodaira – Unclear
  159. Edmund Landau – Likely Jewish
  160. Peter Lax – Likely Jewish
  161. Leonardo da Vinci – Christian (Catholic)
  162. Augustus Möbius – Unclear
  163. Nicole Oresme – Christian (Catholic, Bishop of Lisieux)
  164. Roger Penrose – Agnostic
  165. Grigori Perelman – Unclear
  166. Plato of Athens – Theist (or at least close)
  167. Lev Pontryagin – Unclear (maybe atheist)
  168. Waclaw Sierpinski – Unclear
  169. Yakov Sinai – Likely Jewish
  170. Isadore Singer – Unclear
  171. Thoralf Skolem – Unclear
  172. Evangelista Torricelli – Christian (Catholic)
  173. S.G. Vito Volterra – Likely Jewish
  174. Zhu Shiejie – Likely Shamanism
  175. Jamshid Al-Kashi – Muslim
  176. Eugenio Beltrami – Unclear
  177. Bernard Bolzano – Catholic (priest)
  178. Raol Bott – Likely Catholic
  179. Luigi Cremona – Unclear
  180. Max Dehn – Likely Jewish
  181. Paul Dirac – Atheist
  182. Eratosthenes of Cyrene – Unclear
  183. Gerd Faltings – Unclear
  184. Michael Freedman – Unclear
  185. Marie-Sophie Germain – Unclear
  186. Vaughn F.R. Jones – Unclear
  187. Kazimierz Kuratowski – Unclear
  188. Jean Leray – Unclear
  189. Benoit Mandelbrot – Jewish
  190. Yuri Matiyasevich – Unclear
  191. Eliakim Moore – Likely Christian (Methodist)
  192. David Mumford – Unclear
  193. G. Personne de Roberval – Unclear
  194. Claude Shannon – Atheist
  195. Goro Shimura – Unclear
  196. Endre Szemerédi – Unclear
  197. Brook Taylor – Unclear
  198. Jacques Tits – Unclear
  199. Karen Uhlenbeck – Unclear
  200. Oscar Zariski – Atheist

Given this list, it would be interesting in light of the question of a “science/religion conflict” what the overall statistic of this group are. If, say, Christianity is anti-science at its core, we should expect to see very few Christians on this list. The totals will only be accumulated for those figures I feel confident that I have reliable information on.

Statistics on Beliefs

Total Counted: 116 (Uncounted: 84)

Theist / Atheist + Agnostic / Other: 79 / 26 / 11

Christian: 49

Atheist/Agnostic: 26

Jewish (religiously): 15

Uncommitted Theist/Deist: 9

Muslim: 6

Hindu: 5

Other: 6

Notes for the Reader

  • As in all of my database articles, this article may be updated from time to time. I will update this database both in accordance with new information that I learn as well as with any information that seems reliable provided to me by any of my readers.

Series: Databases of Resources

I am publishing this post as a preface to a sequence of posts that I will continue to update as long as I run this blog, and hope to continue updating for as long as I live – compilations of references on what I have studied during my life. I’ve decided to call these databases of resources.

Why do this? Well, whether my reader is Christian, Buddhist, Muslim, non-religious, or whatever else, I want to be intellectually responsible. If people have questions about what I believe, I want to provide responsible answers. As I see it, there are at least two things that I must do in order to do that well.

First, I should do my best to explain my reasoning in detail in important matters. Some of what I write on this blog is not quite like that. Sometimes, what I write is related to emotional issues in such a way that although I want to write clearly, I do not feel a need to address those topics in an academic writing style. However, many of the fundamentals of Christianity are not at their core emotional (though they have important emotional implications) but are based on historical and/or philosophical realities – both of which are based on objective truth rather than human perception and emotion. In objective matters, one ought to be careful to verify what you are saying. I may get things wrong from time to time, but if I make public my reasons for what I believe, then it will be easier for me to figure out where I am wrong. And if I am wrong or unclear, I can be corrected. This I welcome, that way I can correct my beliefs if I am mistaken or become more sophisticated and nuanced in my beliefs if I am merely unclear.

Secondly, I feel I should do my best to provide a way for people to double-check what I say. These databases hopefully will play that role. These are largely for people who want to learn more about a topic themselves, people who want to do their own research and come to their own conclusions. I encourage that kind of critical thinking. I believe strongly that Jesus would join me in this encouragement. I believe that the central claims of Christianity – which are sometimes called Mere Christianity in reference to CS Lewis’ foundational book by that name – are objectively true. Since objective truths that impact all of humanity must be based on some kind of evidence “out there” accessible to all of us, it is only fair that I try to provide sound reasoning based on principles that others can see – of which some examples are science, historical studies, philosophy, and ethics.

Databases may include posts like lists of useful resources, good books and authors, academic resources, or passages of Scripture that are relevant to a certain issue. I hope that as I continue to compile these kinds of database posts, they can serve as useful references for people who are interested in learning about areas that I am also interested in.

Explaining Calculus (Intro)

In our day and age, this word strikes fear into many hearts. Much like ‘rocket science,’ it is used in our culture to represent anything that is extremely difficult or even something incomprehensible. I am talking about calculus. Those who have taken a course in this topic will have a good idea of what kinds of problem calculus can answer – although in my experience those who have not taken a course in calculus don’t really know what it means. Or perhaps they do know what it is about, but are too intimidated by the word ‘calculus’ to trust themselves. Or – and I suspect most people are in this position – perhaps they actually do understand the core concepts behind calculus, but just don’t realize that they do.

Because I am convinced that most people understand at least parts of all the basic concepts that go into calculus, I am convinced that it shouldn’t be so scary. In fairness, mastering all of the computational details can be quite difficult. Just like mastering a sport, this requires a lot of practice and repetition. But, this doesn’t mean learning about what calculus is has to be difficult. Just like a person can understand the strategy and details of a sport without being a professional athlete, I also believe that we can all understand what is ‘going on’ with calculus without mastering the more difficult calculations.

I write this post at the beginning of the Fall 2020 academic semester. This semester, I am teaching a course called “Survey of Calculus 1” – which is the first calculus course a student takes. To be clear, this isn’t some sort of ‘easy’ version of calculus. The fact of the matter is that there is a natural way to split up the subject of calculus into two segments. For now, let’s just call them (A) and (B). The two segments are very closely related. In fact, if you already know (A), you can use (A) to teach people (B). And if you already know (B), you can use (B) to teach people (A). Even more, (A) and (B) both arise out of a common origin using the concept of limits. For those who already know this, I am talking about differential calculus and integral calculus. If you don’t know what these are, that doesn’t matter. When we talk about “Calculus 1” courses, we basically mean differential calculus, and “Calculus 2” is integral calculus (and a related third topic called infinite series). So, I will be teaching a semester-long course focusing on differential calculus.

Ever since taking my first course in calculus, I have found it fascinating and have been saddened by the fear others experience towards calculus. I don’t think we should have that fear. Because I have this conviction, I want to write about the course as I teach it. My goal right now is to write about this topic using the same line of thoughts and topics that I’ll be using in my actual course

I know that I have some people reading this who know calculus already, and some who know a little bit about it, and some who have no idea what it is, and I plan to write this series in such a way that I hope it will be interesting and informative for anyone. In order to do this, I will try to place emphasis on both the intuitive framework that gives calculus its intrigue as well as some of the mathematical details and how these details work to tell us not just about the world of numbers, but about the world in which we live every day.

This is my goal. I will be doing lots of writing on this subject, and hopefully it will be interesting!

Critical Thinking Toolkit: A Priori Assumptions

This is one of the most important – perhaps the most important – of the many tools in the “critical thinking toolkit.” I don’t say this because I like this topic most among the topics I want to write about – although I do enjoy this topic a lot. The main reason I think this is so important is because it is always relevant to all discussion involving two people who disagree, and I see this issue underlying almost all of “public ideological battles” today. Perhaps I am exaggerating slight here… but only slightly. The topic I want to discuss here rears its head in pretty much every discussion.

There is a need to clarify some terminology, because the terminology that is normally used here is actually Latin. The term is a priori. I think in order to most clearly define this terminology, it is helpful to introduce the terminology that is usually used as its opposite – a posteriori. For readers who don’t speak Latin (which includes myself), in order to compare the two, the phrases a priori and a posteriori can be translated basically as from the earlier and from the later. In order to understand what we mean, we must answer “Earlier or later than what”?

The answer is this – earlier or later than experience/observation. In light of this, what I mean by a priori means ‘before experience’ and a posteriori means ‘prior to experience.’ To understand what I mean, let’s give an example. Consider the statement, “Because I am 10 years old, I am more than 5 years old.” Now, numbers can be defined without reference to our personal experience of the world, so you can affirm the previous statement without referring to your own experience of the world. It is true that you can experience that 10 is more than 5, but you don’t have to experience it in order to know that it is true. This kind of situation is what is meant by a priori.

On the other hand, consider the statement “I am 10 years old.” In order to know whether this is true or not, you need some experience. You need to know, for example, who is speaking. You also need to know when they were born. Those are not aspects of reality you can understand without drawing from experiential reality. This is what is meant by a posteriori.

This is the sort of thing that is mean by a priori and a posteriori. More specifically, if you are in a debate with a person, an a priori assumption is something that you hold to be true that you hold prior to investigating the evidence from your investigation.

Common A Priori Assumptions

Below, we give some examples of commonly held a priori assumptions that can easily get in the way of having productive, intellectual conversations. I will try to point out some of the flaws involved, and later I will discuss how we can do our part to address these flaws.

Political Alignments

One example that comes to mind in particular is in the abortion debate. To be absolutely clear, I am doing my best to state what I consider to be assumptions – meaning I have rarely or never heard anyone of that position actually say these things out loud. As far as I can tell, most pro-choice supporters carry the subconscious assumption that the only reason for opposition to abortion is sexism of some form (namely, discrimination with respect to reproductive rights). In other words, “My Body, My Choice“. And as far as I can tell, many pro-lifers carry the assumption that everyone consciously believes/knows that everyone agrees that a fetus is a human being with the same rights as all other human beings. In others words, “Abortion is Murder“.

I am not here taking a side on the abortion issue. My point is entirely separate from this. If you look at mainstream dialogue, as far as I can tell, the argument rarely gets down to the important points that lie at the core. If I spent longer thinking about this, I could probably come up with of other neglected but important points of discussion within the abortion debate, and I can come up with similar underlying points within other political debates. For an example of such a position, I believe that the entire gay marriage debate basically boils down to what the word marriage means and what legitimate role the government has within that institution. As far as I can tell, it has almost nothing to do with homophobia. It just doesn’t. You could hate gay people but still support gay marriage, and you could affirm loving homosexual relationships and reject gay marriage. It all depends on what you think these words mean.

My only point here is that in politics, we often don’t every get around to discussing our real differences. We get caught up in catchy and clever-sounding sound-bite comebacks. But every time I’ve thought about a “sound-bite” from either liberals or conservatives, I have found it incredibly lacking in content. We need to do better in political discourse.

Scientific Atheism/Methodological Naturalism

There is another public debate in which I see a priori assumptions playing a significant role, and this is the debate between science and religion. Both sides tend to assume, a priori, that their own perspective is superior to others in obtaining truth about reality. That is, you have many religious people that, although both science and religion can hold truths, say that truths taken from the Bible automatically override truths from science. And many atheists do the opposite – they assume that anything science says automatically overrides anything outside of science.

But, how do we know which of these assumptions is true? And is it possible that science and religion are actually entirely compatible? I believe they are compatible, but that isn’t the point I’m trying to make here, so I will leave that for another time.

People who are very religious (let’s say Christians for the sake of discussion, although a similar idea holds for other religions) may subconsciously assume a doctrine of inerrancy and hold their atheist interlocutor to that standard. Similarly, an atheist may implicitly assume methodological naturalism (a philosophical doctrine that only non-supernatural entities can be posited as explanations) and expect a religious person to also hold to that. But these are fundamentally incompatible – so of course people who hold to these two, respectively, might not agree on something. The real question is, which is a better assumption – methodological naturalism or inerrant Scriptures? That is a question that debates about evolution and the book of Genesis cannot answer – the discussion has to be philosophical

What To Do With Assumptions When Others Disagree?

Very often, our disagreements with people boil down to our assumptions. Therefore, in order to be effective in convincing a person of your beliefs, your approach should take into account the underlying assumptions of both yourself and others.

Figure Out What the Assumptions Are

The first thing we must do is to boil down any debate to its presuppositions is to ask lots of questions. In particular, the beginning of any discussion should consist in lots of “what” questions. These help to clarify the topic of the discussion. After that, we can ask “why” questions in order to boil down the discussion to its foundational pieces. The theme of what/why questions is almost always helpful in any discussion, and is always a great thing to keep in mind.

How will we know when we are at the assumption level? One way to know would be if the “why” question no longer has a helpful answer. For instance, asking a person how they know they exist wouldn’t really have a helpful answer. Alternatively, the level of ‘assumption’ can be located by starting from the obvious claims – things like “we all exist” – and moving gradually upward in complexity/controversy until a point of disagreement arises.

We can view the abortion debate as an example. Although there are multiple ways to approach this debate, and I do not intend here to defend one side or another, I think that since people are fairly familiar with this debate, it will serve as a helpful example. As a jumping-off point, here is a “bottom-up” structure of how a pro-life advocate might arrive at his or her pro-life position.

  • God exists.
  • God is the creator of humanity.
  • God values humanity.
  • God values all human beings individually.
  • All human beings have infinite moral value.
  • An unborn child is a human being. Therefore, an unborn child has infinite moral value.
  • Unjustified killing of a human being is evil. Therefore, killing an unborn child is evil.
  • Abortion kills an unborn child. Therefore, abortion is evil.

The point here is not to agree or disagree with any of these ideas. My point here is that you could disagree with this person’s pro-life position for a variety of reasons. If you don’t think God exists, then you will have a problem with a fair amount of the points involved. If you believe God exists but you are a “deist,” then you will disagree with “God values humanity,” since on deism God is distant and removed from the physical world. You might disagree on the proper way to instill moral value on a human life, or when a human life begins, and so only disagree with this person later in their train of thought. If two people are having a conversation and misunderstand where the disagreements actually lie, then it will be an unhelpful conversation. This is one of many reasons why identifying underlying assumptions is so helpful.

Now, we can discuss some options of how to carry forward in a discussion once our assumptions are identified.

Option 1: Convince them of the assumption: One way to handle a disagreement like this is to stop debating the ‘endpoint’ but to shift the conversation to the real disagreement – the underlying assumptions.

Option 2: Convince them of your point using their assumption: This method won’t always work, but sometimes it does. Sometimes, you might be able to convince a person that they have not carried out their assumptions to their logical conclusions. This is very similar to the mathematical and philosophical method of reductio ad absurdum.

Critical Thinking Toolkit: Ockham’s Razor

When we are in debates, very often there is more than one way to explain something. When we are presented with more than one way of explaining some aspect of reality – be it scientific, historical, religious, or anything else – these situations arise. When they do, we want to be able to differentiate between the various alternatives in a reasonable way. But it is not immediately obvious how to do something like this. If Steve and Carol provide two totally different explanations for the same event, how do we analyze which one of them is more likely right? After all, we cannot evaluate the alternatives based on their conclusions, since the conclusions are identical. What then do we do?

Ockham’s Razor to the Rescue

To give some kind of intuitive explanation of the principle of reasoning that I will define here, let’s use an example. Suppose that we are tasked with explaining why there are Christmas presents under a Christmas tree. There are two possible answers available to us. For us, Explanation 1 is that Santa Claus lives in the North Pole and delivers presents to all children throughout the world on the night before Christmas morning every year. Explanation 2 is that it is actually the parents of individual children that deliver the presents, and that the story of Santa Claus is a fictional tale meant to inspire fun and imagination in children. Notice that Explanations 1 and 2 both completely describe why there are Christmas presents under the tree, so we cannot evaluate between 1 and 2 based upon which one leads to the correct conclusion (of course we could bring in other information that shows us why Santa can’t exist as described here, but for the sake of argument we will pretend we don’t know any of that). Well then, how might we choose which of the two is more likely true?

This where the principle I want to talk about here comes in. It is not meant to be an absolutely foolproof method, but it is quite reliable. The method, called Ockham’s Razor (Ockham can also be spelled Occham or Ocham) is summarized by the statement “entities should not be multiplied without necessity.” To be more specific, Ockham’s Razor tells us that if you have different ways of explaining exactly the same thing, you ought to take the explanation that makes the fewest assumptions. In the previous example, we already know that our parents exist, so the assumptions involved in explaining Christmas presents via our parents are extremely few – the only assumption we have to add is that our parents are lying to us (if we neither believe nor disbelieve in Santa, that’s really all we need). In order to explain Christmas presents by Santa visiting our house, we have to add significantly more assumptions. Therefore, unless we have new evidence that points strongly one way or the other, Ockham’s Razor tells us to accept the alternative with fewer assumptions – namely, that Santa does not exist.

How to (and not to) Use Ockham’s Razor

Ockham’s Razor is a frequently used and very important tool in critical thinking. I must make that very clear – it is a great principle and we really must take advantage of this way of thinking about complicated issues. But, it is also very necessary to emphasize both the powers and limitations of Ockham’s Razor.

Ockham’s Razor is not All-Powerful: One extremely important reality we have to acknowledge is that this principle does not always work. Ockham’s Razor is meant to ‘shave off’ additional assumptions only if those additional assumptions are not helpful in other ways. Ockham’s Razor favors simple explanations over more complicated explanations, but only when the two complicated explanation doesn’t override the simple explanation in other ways. For example, Isaac Newton’s theory of gravity is much simpler than Albert Einstein’s theory of gravity (aka general relativity). Since the two theories analyze the same concept – namely gravity – they can be compared. If that were the only information we knew, we would have to prefer Newtonian gravity to general relativity. But scientists use general relativity today in the most important applications, not Newtonian gravity. Why? Because general relativity is more accurate than the Newtonian theory. The increase in accuracy is more important than the simplicity. So, if we want to use Ockham’s Razor, we should make sure that the two ideas in question are similar in other respects.

When Occham’s Razor Applies, It Almost Always Works: When you look throughout your own life or the history of any discipline of study in human history, I am quite confident that whenever you find a proper situation in which to apply Ockham’s Razor, it will work correctly. In other words, reality tends to favor simple explanations over complicated ones whenever a simple explanation is good enough to explain whatever is going on. Even though you can’t always use Ockham’s Razor, it is very valuable.

It is Almost Always Helpful, Even if it Doesn’t Work: Ockham’s Razor is not meant to be an all-or-nothing principle. But even when you can’t make a full-blown choice of your beliefs based on Ockham’s Razor, it is still helpful. Roughly, this is because simple explanations require fewer contributing factors than complicated ones, and so in most cases simple explanations have much higher probabilities than complicated ones. Although Ockham’s Razor can be viewed as a way to choose between competing ideas that are equally good at explaining the world, it can be used in another way as well. Since simple explanations are favored over complicated ones, you can also apply Ockham’s Razor at the ‘beginning’. But if you apply it at the beginning, it isn’t conclusive. When investigating various ideas, each idea will have an “initial likelihood” in your mind – this is often called the prior probability. Ockham’s Razor has a place in evaluating these prior probabilities. For instance, the prior probability of Santa delivering presents on Christmas is much lower than the probability of everyone’s parents buying parents – but this is because, as adults, we know things that rule out Santa. But, suppose you are 3 or 4 years old. You don’t know enough about the world to decide whether Santa or your parents are better explanations of the presents underneath the tree. But, suppose your parents told you Santa brought the presents. Then since you have nothing else to go on, it is entirely reasonable to believe that Santa brings the presents, and so in this case the prior probability of Santa being the person who brings presents is quite high.

The thing to notice in this example is that the prior probability is always based upon what you already know. If you are 3 years old, the idea that Santa brings presents is the most simple, because all you have to do is believe your parents, whereas denying this requires devising why exactly your parents are lying to you – which is more complicated. But if you start off with the knowledge of an adult, then the entire situation flips on its head – because you know more information. To take this even further, if any of us were to actually live out a Christmas movie where Santa shows up, then probably the situation would change yet again to believing in Santa, because Santa’s actual existence is more straightforward than trying to explain how a bunch of reindeer were flying around. Although if you learn about special relativity theory, then the coin would probably flip again and the unlikelihood of flying reindeer might be balanced out by an outright violation of a law of physics.

Basically, Ockham’s Razor is a useful tool that states that simple ideas have an advantage over complicated ideas, so in order for a complicated idea to win it needs to gain an advantage somewhere else. There are plenty of other ways an idea can get an advantage. These will be discussed elsewhere.

Critical Thinking Toolkit: Possible v.s Plausible

In our common experience, there are a plethora of alternative explanations of the realities we see around us. Some of these are highly likely, some fairly likely, some moderately likely, some that are reasonable but not strictly ‘likely’, and some that are extremely unlikely. There are a variety of situations that we might find ourselves in when we are evaluating options of how to understand the world around us. We must be careful that we accurately assess and consider these distinctions in alternatives that are presented to us – and we should attempt as best as is possible to give alternative perspectives some benefit of the doubt and not reject them outright as impossible.

The distinction I want to deal with here is fairly easy to explain, although the implications are often missed unless very carefully thought out. Here is the idea. Suppose that you are talking to someone you disagree with, and they make (as far as you can tell) a really good point. Generally, changing your mind on a whim is ill-advised, since changing your mind on anything important enough to be worthy of a careful debate will have a lot of implications to how we live our lives. In other words, the more important something is, the more careful we ought to be in changing our minds. So, if someone presents to us a perspective that contradicts our own, we need to try our best to objectively evaluate how reasonable it is. Similarly, we have to evaluate how reasonable our responses to other people’s criticisms are.

Let me give an extreme example. Suppose that Alice and Bob are having a discussion about philosophy, and that Alice tells Bob that she believes that Bob does not actually exist, but that she is actually the only person that exists and that all other people and events are nothing more than her imagination keeping her occupied (this roughly corresponds to a philosophical view called solipsism). From Bob’s perspective, since I know that I exist, Alice must be incorrect. But from Alice’s perspective, it is at least possible (by restricting ourselves to the laws of logic) that Bob doesn’t exist. Bob will naturally respond to Alice’s statement by trying to convince Alice that he actually does exist. In this situation, Alice has two possible viewpoints being presented to her:

(1) Bob is an illusion created by my subconscious mind.

(2) Bob actually exists as an individual separate from my own mind.

Alice is faced with asking herself which of these alternatives is actually correct. As I have already laid out, both of these alternatives are possible – neither of these statements are on the same level of absurdity as 1 + 1 = 3. But I think that very nearly all of my readers would agree that option (2) is much more reasonable and likely than option (1). Option (1) seems like an unintuitive invention, and option (2) corresponds to how we normally think about the world. Our daily experience of the world seems on an intuitive level to fit into (2) better than (1). For instance, if (1) were true, then why would our brain give us illusions of things like pain and sadness? If (2) were true, it makes sense that someone who exists separately from ourselves might occasionally cause us pain or sadness.

In summary, even though both (1) and (2) are possible, only (2) is plausible. For anyone who hasn’t heard the word before, plausible means something like ‘at least reasonably likely to be true,’ whereas all that possible means is that no basic laws of logic are violated. In light of this distinction, the following statement is something that all of us have to keep in mind:

If someone who disagrees with you presents a plausible alternative, you cannot give a merely possible response. Your response to someone else’s criticism should reach the same “level of likelihood” as the criticism leveled against you.

It is true that most of the time, there is some level of subjectivity to the assessment of likelihood. It is even quite possible that someone reading this is inclined to think that solipsism is plausible. If so, then I don’t deny that you believe that solipsism is plausible – but I very strongly disagree. To me, solipsism is only a tiny bit removed from the claim that 1 + 1 = 3. But I’m also quite happy to dialogue with a person who disagrees with me, because I believe strongly that we ought to do our best to remove ourselves from our own perspectives when we have conversations like this, and I am open to being corrected if a solipsist can bring me better evidence for solipsism than I have for the contrary (although to be frank, I can’t even imagine what that evidence would look like). But if a solipsist brings to me a possible reason to reject my current beliefs, that is not good enough for me.

Similarly, I am thoroughly convinced that God supernaturally rose Jesus Christ from the dead on the third day after his crucifixion. If someone brings me some other explanation for the historical details we know about the emergence of the Christian church in the years after Jesus’ death, the explanation being merely possible is not good enough for me to doubt my faith in Christ. I would need to be given a synopsis of all relevant pieces of information that is at least as convincing as the view that Jesus actually did rise from the dead. And I fully expect that any Muslim speaking with me would require that I show them something at least as convincing as the evidence they have for their Islamic faith, and I expect that an atheist would require that I bring forth a totally of evidence that is at least as strong as the evidence they have for atheism.

All of this discussion is important, and each of us should consider our own views about the world and whether we actually have convincing evidence for what we believe that could convince people who disagree with us that we have a plausible interpretation of the world. At the end of the day, we all must remember that we cannot allow ourselves to reject what other people believe just because we can invent some alternative explanation – we must put our own explanations into a skeptical light and do our best to determine how reasonable our own explanations are and how reasonable the explanations by people who disagree with us are.

The Fallacy of Equivocation

This is one of many brief articles I am writing about how to avoid fallacious patterns of thinking. Here, we briefly discuss the fallacy of equivocation. Before I try to define it, it will be helpful to see an example of the fallacy in action. (I take this example out of the Wikipedia page for this fallacy, because I find it particularly helpful)

  • Only man is rational,
  • No woman is a man,
  • Therefore, no woman is rational.

If you use only formal logic, this argument is actually correct. It follows the following strict formula:

  • If X, then Y.
  • X.
  • Therefore, Y.

In our example, X stands for “being rational” and Y stands for “being a man.” Only man is rational – for it wouldn’t be right to say that earthworms, for instance, are rational beings. And it is true that you cannot simultaneously be fully male and fully female (this is true regardless of your political beliefs on the topic of gender). But we know that there are rational women (in fact, all human beings are rational… at least in the sense that we are capable of thinking rationally). So what did we do wrong here?

The answer is simple. We changed the definition of the word man midway through the argument. When we say only man is rational, by man we really mean mankind – both men and women. But in the second point, we are using man to refer to male human beings only – and not all human beings are male. Since we shifted our definition of the word man midway through our argument, this argument that ‘looks like’ it work doesn’t actually work.

Main Takeaways

The fallacy of equivocation is only likely to occur in a situation where one of the most important words involved in your discussion can have multiple definitions in different contexts (like man) or if the word itself is comparative and thus has a context-dependent meaning (like small). Whenever words like these are being used, be careful to ensure you are using them in the same manner throughout the discussion.

How To Overcome the Fallacy

Suppose that someone accuses you of the fallacy of equivocation. How do you overcome this objection? Actually, the solution is quite simple. All you have to do is provide a clear definition of the word in question. It might happen that some of the points you were trying to make become false when you make your definition more specific, and it may equally well be the case that the person accusing you of the fallacy misunderstood what you meant in your use of language. Things like this happen occasionally – people are imperfect and sometimes don’t realize when they use a word in a subtly different way. Nonetheless, it is important that we call each other out when we perceive equivocation going on, because like all logical fallacies, it can be used in extremely harmful ways.

Example for the Reader

For those who want to better understand the fallacy of equivocation, try to spot the equivocation in the following example. If you want to check your work, feel free to email me ( and I’ll let you know if you’ve understood the main point correctly!


  • I am Greek.
  • Greek is a language.
  • Therefore, I am a language.

Finding Patterns in the Fibonacci Sequence

This is the final post (at least for now) in a series on the Fibonacci numbers. We’ve gone through a proof of how to find an exact formula for all Fibonacci numbers, and how to find exact formulas for sequences of numbers that have a similar definition to the Fibonacci numbers. But, the fact that the Fibonacci numbers have a surprising exact formula that arises from quadratic equations is by no stretch of the imagination the only interesting thing about these numbers. In fact, there is an entire mathematical journal called the Fibonacci Quarterly dedicated to publishing new research about the Fibonacci sequence and related pieces of mathematics [1]. In fact, a few of the papers that I myself have been working on in my own research use facts about what are called Lucas sequences (of which the Fibonacci sequence is the simplest example) as a primary object (see [2] and [3]).

The goal of this article is to discuss a variety of interesting properties related to Fibonacci numbers that bear no (direct) relation to the exact formula we previously discussed.

New Recurrence Relations

In this series, we have made frequent mention of the fact that the fraction \dfrac{F_{n+1}}{F_n} is very close to the golden ratio \varphi. We can now extend this idea into a new interesting formula. Since this is the case no matter what value of n we choose, it should be true that the two fractions \dfrac{F_{n+1}}{F_n} and \dfrac{F_n}{F_{n-1}} are very nearly the same. Therefore,

\dfrac{F_{n+1}}{F_n} \approx \dfrac{F_n}{F_{n-1}} \implies F_{n+1}F_{n-1} \approx F_n^2 \implies F_{n+1} F_{n-1} - F_n^2 \approx 0.

One question we could ask, then, is what we actually mean by approximately zero. Is this ever actually equal to 0? What is the actual value? Let’s look at a few examples. Remember, the list of Fibonacci numbers starts with 1, 1, 2, 3, 5, 8, 13. Let’s look at three strings of 3 of these numbers: 2, 3, 5; 3, 5, 8; and 5, 8, 13. The expression F_{n+1} F_{n-1} - F_n^2 mandates that we multiply the largest by the smallest, multiply the middle value by itself, and then subtract the two. So, we get:

(5)(2) - 3^2 = 10 - 9 = 1,

(8)(3) - 5^2 = 24 - 25 = -1,

(13)(5) - 8^2 = 65 - 64 = 1.

Well, that certainly appears to look like some kind of pattern. It looks like we are alternating between 1 and -1. And as it turns out, this continues. The proof of this statement is actually quite short, and so I’ll prove it here.

Theorem: For every whole number n \geq 1, the equation

F_{n+1} F_{n-1} - F_n^2 = (-1)^n

is always true.

Proof: This proof uses the method of mathematical induction (see my article [4] to learn how this works). We first must prove the base case, n = 1. When n = 1, we know that F_{n-1} = F_0 = 0, F_n = F_1 = 1, and F_{n+1} = F_2 = 1. Therefore, F_{n+1} F_{n-1} - F_n^2 = (1)(0) - 1^2 = -1 = (-1)^1. Therefore, the base case is established.

Now, we assume that we have already proved that our formula is true up to a particular value of n. We want to prove that it is then true for the value n+1. That is, we need to prove using the fact that F_{n+1} F_{n-1} - F_n^2 = (-1)^n to prove that F_{n+2} F_n - F_{n+1}^2 = (-1)^{n+1}. To do this, first we must remember that by definition, F_{n+2} = F_{n+1} + F_n. Using this, we can conclude (by substitution, and then simplification) that

F_{n+2} F_n - F_{n+1}^2 = (F_{n+1} + F_n) F_n - F_{n+1}^2 = F_n F_{n+1} + F_n^2 - F_{n+1}^2 = F_n F_{n+1} + (F_n - F_{n+1})(F_n + F_{n+1}).

Now, recall that F_{n+1} = F_n + F_{n-1}, and therefore that F_n - F_{n+1} = - F_{n-1} and F_{n+1} - F_{n-1} = F_n. Therefore, extending the previous equation,

F_{n+2} F_n - F_{n+1}^2 = F_n F_{n+1} - F_{n-1}(F_n + F_{n+1}) = F_n F_{n+1} - F_{n-1}F_n - F_{n-1}F_{n+1}

= F_n(F_{n+1} - F_{n-1}) - F_{n-1}F_{n+1} = F_n^2 - F_{n-1}F_{n+1}.

Since we originally assumed that F_{n+1} F_{n-1} - F_n^2 = (-1)^n, we can multiply both sides of this by -1 and see that F_n^2 - F_{n-1} F_{n+1} = (-1)(-1)^n = (-1)^{n+1}. This is exactly what we just found to be equal to F_{n+2} F_n - F_{n+1}^2, and therefore our proof is complete.

The Pisano Period

The first four things we learn about when we learn mathematics are addition, subtraction, multiplication, and division. These are all tightly interrelated, of course, but it is often interesting to look at each individually or in pairs. Here, we will do one of these pair-comparisons with the Fibonacci numbers. The most important defining equation for the Fibonacci numbers is F_{n+1} = F_n + F_{n-1}, which is tightly addition-based. In light of the fact that we are originally taught to do multiplication by “doing addition over and over again” (like the fact that 2 \times 3 = 2 + 2 + 2 = 3 + 3), it would make sense to ask whether the addition built into the Fibonacci numbers has any implications that only show up once we start asking about multiplication. The answer here is yes.

The multiplicative pattern I will be discussing is called the Pisano period, and also relates to division. In order to explain what I mean, I have to talk some about division. When we learn about division, we often discuss the ideas of quotient and remainder. In case these words are unfamiliar, let me give an example. Imagine that you have some people that you want to split into teams of an equal size. You are, in this case, dividing the number of people by the size of each team. The number of teams you are able to make is called the quotient, and if you have people left over that can’t fit into these teams, that number is called the remainder. For example, if you have 23 people and you want to make teams of 5, then you will make 4 teams and there will be 3 people left out – which means that 23/5 has a quotient of 4 and a remainder of 3.

Remainders actually turn out to be extremely interesting for a lot of reasons, but here we primarily care about one particular reason. A remainder is going to be a zero exactly whenever everybody gets to be a part of a team and nobody gets left over. In terms of numbers, if you divide a number N by a (smaller) number M, then the remainder will be zero if N is actually a multiple of M – so N is something like 2N, 3N, 4N, 5N, etc. As it turns out, remainders turn out to be very convenient way when dealing with addition. This is because if you have any two numbers, the idea of computing remainders and adding the numbers together can be done in either order. For example, recall the following rules for even/odd numbers:

Even + Even = Even

Even + Odd = Odd

Odd + Even = Odd

Odd + Odd = Even.

Since even/odd actually has to do with remainders when you divide by 2, we can express these in terms of remainders. A number is even if it has a remainder of 0 when divided by 2, and odd if it has a remainder of 1 when divided by 2. In these terms, we can rewrite all of the above equations:

Even + Even = Remainder 0 + Remainder 0 = Remainder (0+0) = Remainder 0 = Even,

Even + Odd = Remainder 0 + Remainder 1 = Remainder (0+1) = Remainder 1 = Odd,

Odd + Even = Remainder 1 + Remainder 0 = Remainder (1+0) = Remainder 1 = Odd,

Odd + Odd = Remainder 1 + Remainder 1 = Remainder (1+1) = Remainder 2 = Even.

This interplay is not special for remainders when dividing by 2 – something similar works when calculating remainders when dividing by any number. This now enables me to phrase the interesting result that I want to communicate about Fibonacci numbers:

Theorem: Let N be a positive whole number. Then if we compute the remainders of the Fibonacci numbers upon dividing by N, the result is a repeating pattern of numbers. As a consequence, there will always be a Fibonacci number that is a whole-number multiple of N.

Proof: What we must do here is notice what happens to the defining Fibonacci equation F_{n+1} = F_n + F_{n-1} when you move into the world of remainders. With regular addition, if you have some equation like a + b = c, if you know any two out of the three numbers a,b,c, then you can find the third. The same thing works for remainders – if you know two of the remainders of a,b,c when divided by N, then there is a straightforward way you can find the third remainder (this is the sort of thing we just did with odd/even). Now, here is the important observation. If you are dividng by N, the only possible remainders of any number are 0, 1, 2, \dots, N-1. There are N possible remainders. This exact number doesn’t matter so much, what really matters is that this number is finite.

When we combine the two observations – that if you know the remainders of both F_{n-1} and F_n when divided by N, and you know the remainder of F_{n+1} when divided by N and that there are only a finite number of ways that you can assign remainders to F_{n-1} and F_n, you will eventually come upon two pairs (F_{m-1}, F_m) and $(F_{n-1}, F_n)$ that will have the same remainders. Since this pair of remainders is enough to tell us the remainder of the next term, F_{m+1} and F_{n+1} have the same remainder. This always holds, and so you arrive at a forever-repeating pattern. Because the very first term is F_0 = 0, which has a remainder of 0, and since the pattern repeats forever, you eventually must find another remainder of 0. This fully explains everything claimed.


[1] See for the Fibonacci Quarterly journal.

Finding the Fibonacci Numbers: A Similar Formula

In this series of posts about the Fibonacci sequence F_n, a very famous sequence of numbers within mathematics, we have just concluded showing how you can take the recursive formula (which uses previous values of F_n to compute the next values) and turn that formula into an exact formula that can skip right over the previous values to any F_n we happen to want to know.

The purpose of this discussion is to show how mathematicians generalize their ideas – by which I mean taking the same idea and applying that idea in as broad a context as possible. And to make things more interesting for my reader – I genuinely don’t think I’ve every derived this formula before. And I am not looking up anything as I do this. But I do go into this expecting a particular type of answer. So, this should serve for my readers as an example of how a mathematician thinks about his work.

Generalizing the Fibonacci Numbers

In order to generalize a concept, we must obtain a deeper understanding of what actually mattered for the problem we were initially dealing with. In order to understand this, it is extremely helpful to summarize our reasoning process for the actual Fibonacci sequence F_n.

Drawing from my previous articles, our basic point was to use the fact that F_n grows roughly like the exponential function \varphi^n to connect the recurrence formula F_{n+1} = F_n + F_{n-1} to the quadratic equation x^2 = x + 1. We then used the ‘largest’ of the two solutions to this quadratic equation to predict the approximate size of F_n, and we used the ‘smallest’ of the two solutions to determine how far off our initial guess was.

Now, the Fibonacci sequence is just a list of numbers generated by the two previous terms in the list. In light of this, it would make sense to ask whether a similar exact formula will work for any sequence that is generated in the same way by the previous two terms using integers. To define this more clearly, choose some random positive whole numbers a, b. Using these, define the sequence G_n by G_0 = 0, G_1 = 1, and G_{n+1} = a G_n + b G_{n-1}. To give an example, if we choose a = 2 and b = 3, then we get a list 0, 1, 2, 7, 20, 61, \dots.

Using the Same Concept

We now apply the same conceptual schema that worked with F_n to G_n to try to come up with a formula for G_n. The conceptual framework of the discussion of F_n transformed the equation F_{n+1} = F_n + F_{n-1} into the equation x^{n+1} = x^n + x^{n-1}, which is then transformed into x^2 = x + 1. Using the same concepts, we might try guessing that G_n is “basically” some exponential, transforming G_{n+1} = a G_n + b G_{n-1} into x^{n+1} = a x^n + b x^{n-1}, which then transforms into x^2 = ax + b.

Our next step for F_n transforms x^2 = x + 1 into x^2 - x - 1 = 0 and uses the quadratic formula to produce the two solutions. The larger of these was \varphi = \dfrac{1 + \sqrt{5}}{2} and the smaller was \psi = \dfrac{1 - \sqrt{5}}{2}. In our new G_n framework, we use instead the quadratic equation x^2 - ax - b = 0, and the quadratic formula gives us the roots \alpha (the larger root) and \beta (the smaller root):

\alpha = \dfrac{a + \sqrt{b^2 - 4a}}{2}


\beta = \dfrac{a - \sqrt{b^2 - 4a}}{2}.

The exact formula for F_n was expressed as F_n = \dfrac{\varphi^n - \psi^n}{\sqrt{5}}. We might notice that \sqrt{5} = \varphi - \psi, and that therefore F_n = \dfrac{\varphi^n - \psi^n}{\varphi - \psi}. We might therefore hope that

G_n = \dfrac{\alpha^n - \beta^n}{\alpha - \beta}.

This turns out to be exactly right. See if you can figure out why before moving on and reading the proof!

The Exact Formula

Now that we have developed the relevant concepts and our guess for an exact formula for G_n, we can attempt to prove our exact formula.

Theorem: For any positive integers a, b, the sequence G_n defined by G_0 = 0, G_1 = 1, and G_{n+1} = a G_n + b G_{n-1}, we have

G_n = \dfrac{\alpha^n - \beta^n}{\alpha - \beta}.

Proof: As discussed in a previous proof in this series, we only need to show that the formula \dfrac{\alpha^n - \beta^n}{\alpha - \beta} solves all three components in the statement, and as before it is not difficult to show that the n = 0 and n = 1 cases do give the values of 0 and 1, since \alpha^0 - \beta^0 = 1 - 1 = 0 and \dfrac{\alpha - \beta}{\alpha - \beta} = 1.

Now, we know that \alpha^2 - a\alpha - b = \beta^2 - a \beta - b = 0, since \alpha, \beta are the solutions to the equation x^2 - ax - b = 0. Since you can multiply 0 by anything you want and it will remain zero, we can conclude from this (by multiplying by both \alpha^{n-1} and \beta^{n-1}) that

\alpha^{n+1} - a \alpha^n - b \alpha^{n-1} = \beta^{n+1} - a \beta^n - b \beta^{n-1},

from which we can conclude that

\alpha^{n+1} - \beta^{n+1} = a(\alpha^n - \beta^n) + b(\alpha^{n-1} + \beta^{n-1}).

Dividing both sides of this by \alpha - \beta yields the equation

\dfrac{\alpha^{n+1} - \beta^{n+1}}{\alpha - \beta} = a \dfrac{\alpha^n - \beta^n}{\alpha - \beta} + b \dfrac{\alpha^{n-1} - \beta^{n-1}}{\alpha - \beta}.

This just is the identity G_{n+1} = aG_n + bG_{n-1}, which was the only piece of the puzzle that was missing. Therefore, the exact formula we have given for G_n is correct.

Here, we now have a nice example of generalizing ideas that we come up with for a specific problem to deal with a broader problem. So, if we ever come across situation with the specific kind of “generational growth pattern” that we saw in the very first post with the rabbits, we will know how to handle it.