Boris Rarden 0 Report post Posted September 28, 2011 (edited) I was asked a question on an interview, what is my favorite number. My Answer: my favorite number is 2.718 approximately. I will try to describe what it is and why it is my favorite, in the not-so-short explanation that follows. a) The exact number can't be written out explicitly, since it is irrational. I am going to call it B, and will describe how to calculate it exactly. b ) For this number B, the curve y = B^x has a rate of change curve (derivative) that is the same curve again: y = B^x. So if you start to work out a derivative of y=2.718^x, you will get again the curve y=2.718^x approximately. Here's this curve plotted: http://www.wolframal...i=y%3D2.718%5Ex You can see in a lower section on that page that the "Derivative" formula is about the same. It is y = 0.999*2.718^x. (When you multiply something by 0.999 it doesn't change much.) What is a derivative anyway, and how can I get a feel of this curve in a practical sense? Well, suppose you are driving a car and keep track of your speed by plotting a curve of how fast you were going. As well, you also plot a curve of how much gas you were using up. Gas usage has to do with how much you press on the pedal. Let's think about those two curves. Ordinarily these curves will look different. Why? Doesn't more gas mean more speed ? Yes and no. In the city when you go slow you use a lot of gas because you constantly stop and start. However, on a highway when you keep a steady high speed, you don't use much of it. So the curves for speed and gas usage will look different because you need gas to change the speed, but you don't need gas to keep the speed steady. Is it ever possible that these two curves would look the same? Yes, they would look the same if you accelerate according to the special curve with the shape y=2.718^x. Then, the speed will also change according to the same formula. This is because acceleration is a derivative of speed curve and B, approximately equal to 2.718, is a magic number for which derivative of y=B^x stays y=B^x. c) How did I find and calculate B in the first place? I assumed that such number exists for which the derivative of y=B^x still stays y=B^x. Given this assumption, I worked out a very simple Taylor series for a function f(x) = B^x. Recall that Taylor series for function "f" at center 0, is an alternative expression for this function through its derivatives. It is a sum of terms [c_k*x^k] where c_k is equal to the k-th derivative of f, evaluated at 0 and divided by k factorial. The k runs through all positive natural numbers. In our case, since the derivatives are equal to the same thing, which is B^x, and zero to any positive power is 1, it works out that the Taylor series is just a sum of terms [ x^k / k! ] for all natural numbers k. This means that at x=1, f(1) = B and B is sum of inverse of all factorials. To approximate B, I can take k up to very large numbers and then stop. That is how I figured out that B is approximately equal to 2.718. d) I also found it very convenient to calculate angles using B. First, I'm going to play a game that whenever I see (x^2 + 1) I can remove it. In other words, in this game x^2 + 1 = 0. It turns out that if I take all polynomials in x where coefficients are real numbers, then I can, sort of, work modulo (x^2+1). A random example of a kind of polynomials I am talking about is [ 7 * x^21 + 5 * x^10 + 0.333 * x^14 ]. I can add and subtract and divide these polynomials, and I just keep in mind that x^2 + 1 = 0. Working mod (x^2+1) I get a system that works in a consistent manner and I can't easily produce some kind of contradiction like 5 = 3, for example. So whenever I work, I just have to remember that I can add to any equation x^2 + 1 without changing it, cause it is equal to 0. Alternatively, if I see x^2 somewhere I can replace it with -1. This is because x^2 = x^2 + 1 - 1 = 0 - 1 = -1. Continuing with the same idea, I can replace x^3 with -x, because x^3 = x^2 * x = -x. So I can never have any integer power of x in my system since it just gets reduced to either x or -x. For convenience I will use letter "i" instead of "x", so that I can use "x" for other things. So now i^2 + 1 = 0, and "x" is unused. I can only ever see -i or i in my formulas. Of course, we must remember that this is restricted to the game of working with polynomials modulo (i^2+1). I am now going to use "i" to look at expression B^(i*a), where "a" stands for angle. It is a bit surprising to use "i" inside exponent, since our setup involved "i" only as the polynomial variable. However, since we know that B^x can be expanded in terms of power series, we will in a moment see that we get back to the polynomial domain. More precisely, in part © I got an expression for B^x in terms of power series (x^k / k!). Well, if I stick in that series formula "i*a" instead of "x" then I will get power series for B^(i*a). I am going to show in a moment that that power series is equal to a sum of two other power series, the one for cosine and the one for sine multiplied by "i". Lets work out Taylor series for sin(a) and cos(a). It is not hard to do that because the derivative of sin(a) is cos(a) and the derivative of cos(a) is -sin(a). The derivatives just alternate among themselves. Also, when we evaluate them at 0 they are going to give zeros and ones because cos(0) is 1 and sin(0) is 0. So you can imagine that we will get some kind of simple regularity in the power series. Consequently, sin(a) works out to be a simple pattern: (a - a^3/3! + a^5/5! - a^7/7! ...). You get only the odd terms since in the every other term we get sin(0) = 0. For cos(a) we get a similar expression cos(a) = 1 - a^2/2! + a^4/4! - a^6/6! ... etc. Now it is time to connect what I have been discussing. The curious thing is that B^(i*a) = cos(a) + i*sin(a). Please read this formula several times because it is a very famous one. We can verify this equality if we match up the series expressions for both sides of the equation and work out the terms, taking into account that i^2 = -1. If I replace "i" with "-i", I get B^(-i*a) = cos(a) - i*sin(a). This gives me a second formula, which together with the first can help me find a shortcuts for doing trig calculations. Lets solve for cos(a) using the two equations derived above. If the two equations are combined, the sine part cancels out and I get cos(a) = 0.5 * B^(- i*a) + 0.5 * B(i * a). This is the first shortcut formula --- it gives me a way to avoid working with cosine and instead work with exponents of B. I can get a similar formula for sine, by subtracting the two equations and getting: sin(a) = 0.5 * B^(- i*a) - 0.5*B^(i*a). In conclusion, I have two formulas that enable me to make shortcut calculations with sine and cosine. I just turn them into exponents using B with above two formulas, and use basic algebra to simplify expressions before I turn them back into real cosine and sine. It turns out to be pretty useful trick, especially for integration of trigonometric expressions. Too bad I didn't know about it when I was in grade 11 high-school, and had to make trig calculations. e) For a special angle a = Pi, since sin(Pi) = 0, and cos(Pi) = -1, and using formula B^(i*a) = cos(a) + i*sin(a), I get equation B^(i*Pi) = -1. If I move the -1 to the left side, I get a pretty formula B^(i*Pi) + 1 = 0. Why do I find the formula B^(i*Pi) + 1 = 0 pretty ? Well, because it has cool numbers in it. B is a cool number that we have been discussing all along, and which I claim to be my favorite. The number "i" is the one that we invented and is not really a number, since in our game we have i^2 = -1. That means that "i" can't be any real number, since any real negative number, when squared must be positive. Remember the rule that negative times negative equals positive. So the thing "i" is imaginary, but it helps in calculation of real functions like sine and cosine. Some people believe that "i" is not an abstract number but actually exists in nature, inside atoms. In any case, we can interpret the non-real nature of "i" in the same way as we think of the non-real nature of negative numbers -- negative numbers don't exist in nature, everything you could measure is a positive thing (time, distance, weight, etc). However, we have learned to understand that negative numbers represent an intermediate step in calculation, and once we use them or interpret them properly, we get back positive numbers. For example, a negative temperature of -3 degrees Celsius is really a positive quantity because we assumed that 0 Celsius is 273 degrees of true temperature in Kelvin. If we interpret the negative sign of -3 degrees Celsius according to its meaning and assumptions, we would get +270 kelvin. In a similar fashion, we try to think of "i" as intermediate step in calculation or representation. When final answers are needed we expect that "i" will magically cancel out. Unfortunately, this does not happen for quantum physicists. The "i" doesn't cancel out and they are forced to conclude that "i" really exists. However, since quantum physics and relativity theory are still in contradiction, we may assume there's a mistake somewhere so "i" may not be a real thing after all. The number "1" is also a great number, since it is the main building block to make all other natural numbers. Just add many ones enough times to get any natural number. Actually, 1 is a representative of a "block", because we like to group things into blocks or units. When moving apartments, we put all our stuff into boxes, so that we can just count the boxes, and not the individual items. Also, all measurements and prices are given in terms of 1: price per 1 pound, meters per 1 second. Another example is price per square meter for ceramic tile. If 1 ceramic tile is 50 centimeter wide, then 4 of them arranged in a square make 1 square meter. From this point, we can work with a unit of 1 square meter and forget that 4 ceramic tiles go into it. For example, we can convert between square meters and square feet directly, without considering that there are 4 ceramic tiles in there, or that an integer amount of tiles doesn't fit into 1 square foot. This is important, since in Canada price is given per 1 square foot, however, for me it is much easier to measure out my apartment in meters and centimeters. So, we can conclude that 1 is a very basic, natural, and important number. Now we get to the number "0" that shows up in that formula. The number "0" was the last number invented or discovered. It is different from all other numbers since it has a property of uniqueness: 1 apple is not equal to 1 plum, but 0 apples is equal to 0 plums. It was discovered when people tried to calculate the number of years that has passed between 1 BC and 1 AD. It is actually two years, and not 1 year. Clearly, something must be in the middle, and that is 0. The invention of 0 also gave the ability to solve equations by moving stuff from one side to the other. As well, it gave us a positional number system that we use today: ... 100, 10, 0.1, 0.01 ... etc. This positional system is the foundation of the binary, octal, and hexadecimal system that allows us to program computers by working with two voltage levels, represented by "off" and "on". The zeros in 1000 represent empty buckets into which blocks of different amounts can be placed. Conversion between different position systems, is arranged by relabeling those boxes and making more or less of them, adjusting whats in them accordingly. So to conclude, 0 is an important number. The last number in the formula that we haven't talked about is Pi. Well, Pi is a famous number. It was first observed that in any circle the ratio between its circumference and diameter is the same. If you make the circle bigger, both the diameter and the circumference grow by just the right amount so that if one is divided by the other, you get the same number, about 3.14. During the last 10 thousand years every civilization tried to calculate this number as precisely as possible. Today we can calculate it to thousands of digits, but as this number is irrational it can't be written out precisely in full. It can also be approximated as power series, just as well as B, of course, with a few math tricks. Unfortunately, two thousand years ago those tricks were still unknown, and those people had a hard time figuring Pi out. That is why we can link historic scientific development with how many digits of Pi were known at that time. Pi is now used to measure angles in radians. Imagine, cutting an extra-large pizza with a diameter of two feet into four slices, so that one slice is a quarter of a circle. The round edge of the pizza slice is a quarter of the circumference, which is Pi * D / 4, because Pi * D is the circumference of the whole pizza. In a circle with diameter equal to 2 this works out to be Pi / 2. Thus, it is taken that 90 degrees (a quarter of a circle) corresponds to the angle of Pi /2 radians. The reason that using Pi is better than using degrees, is because taking that a circle has 360 degrees, is a rather arbitrary thing -- we could have taken the circle to have 400 degrees for example, so that a quarter would be 100 degrees. Alternatively, working with radians represents a true property of a circle that the diameter and the circumference are related in their sizes in a fixed ratio. This turned out to make all the math formulas that use Pi to work better, rather than if they used degrees. You can see how Pi get's into all the math formulas, since cos(Pi) = 0, and sin(Pi) = 1, and cos(a) = sin(Pi - a) for any angle, and we can expand sin(Pi - a) in terms of individual angles of Pi and a, through working with shortcut formula worked out in part (d). As well, many shapes have areas and volumes that can be expressed with Pi. For example, the volume of a sphere (our Earth) is 4/3 * Pi * r^3, where r is the radius, which for Earth is about 6370 km. The radius of Earth was calculated by measuring the shade length from two posts located far apart, at the same time. This calculation was done by Eratosthenes, an ancient greek librarian of the Alexandria, who knew that Earth is round and that sun's rays arrive parallel to each other, because the sun is so far. Knowing the radius of Earth also gives a way to calculate the length of the equator, which turns about to be about 40 thousand kilometers. Too bad that with the rise of Christian religion the knowledge that Earth was round was lost, and was replaced by the belief that the Earth is flat.So you can agree with me that Pi is a pretty important number, and probably most widely known special number in math. I'd like to conclude that I really like the number B = 2.718.... As I have shown, it leads to very useful and unusual formulas, one of which brings together many important numbers in mathematics: i, Pi, 0, 1. Please note that most people don't use the letter B for the number 2.718. I leave it to you to find out what is the standard letter. It plays an important role in many other formulas and applications an mathematics, physics and engineering. Edited September 28, 2011 by Boris Rarden Share this post Link to post Share on other sites

Weston 3 Report post Posted September 28, 2011 We all know it's e dude, tenth grade math is not beyond most of the posters on this board. That being said, I'm still unsatisfied with the commonly accepted explanation for why e^(pi*i)=-1, care to comment? Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 28, 2011 Irrational ah? Well you know what Ayn Rand would have to say about those irrational numbers, right? Bad puns aside.. "e" ( 2.718 etc) is pretty good, and one of the ones I find the most interesting / useful a lot of the time. Unless I am mistaken you forgot to mention one thing that makes it pretty useful : That it makes solving/working with a lot of very useful Differential Equations A LOT easier than using some other method. Your explanation of differentiation in that context was a little circular . If you were trying to reduce it fully : then you did not really manage it. Though I think anyone here has a good grasp of the concept of differentiation should not really have needed the explanation anyway and should get the idea. e^(i *x) = cos(x + i*sinx, which you allude to is known as Eulers Formula ( or Eulers Identity sometimes ) and is in fact very simple to prove in only a few steps ( three or four steps using the Power Series Expansion definitions for sine and cosine). e^(i*pi) + 1 = 0 is a "special case" of the formula derived when you set x = pi. As for the actual nature and usefulness of "i" : It is simply a mathematical *tool* which is required to solve certain mathematical problems ( quite a large number of them really, but that is besides the point) properly. Obviously it does not correspond to an "actual quantity". It is merely one of many mathematical concepts which have been created in order to allow us to use real-world input values in order to derive equations which give us answers which allow us to find values for properties ( and so forth) of things in reality. Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 28, 2011 (edited) Weston, lets use Wikipedia to help prove it shall we? sine and cosine can be defined as follows : For complex z : And ( of course ) : So : cos(pi) = -1 sin(pi) = 0 so i(sin(pi) ) = 0 so... e*(i*pi) = cos(pi) + i(sin(pi)) = -1 Edited September 28, 2011 by Prometheus98876 Share this post Link to post Share on other sites

Boris Rarden 0 Report post Posted September 28, 2011 Unfortunately, this is the way it is taught in university, and many people don't feel 2.718 as clearly as they do 3.14. Many know of "e" but don't know or forgot how much it is, and don't really know what it is for. They remember the formulas of integration and differentiation, D(ln(x)) = 1/x, D(exp(x)) = exp(x) without appreciating that number 2.718.. that stands behind them. Share this post Link to post Share on other sites

ObjectivistMathematician 3 Report post Posted September 28, 2011 Personally, I like this proof for Euler's formula better: So, both of those functions solve the differential equation y' - iy = 0, and they both equal 1 at x = 0. Therefore, they are equal. Then you just plug in pi for x, and you have e^(pi*i) = -1. Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 29, 2011 I dont know, at least where I went ( technical college ) most decent students had no problem with the fact that e was in fact just easy notation for approximately 2.718. Maybe I just had better lecturers / classmates... Yeah, that is a pretty cool and eloquent proof. Though I still prefer the way I mentioned. For one it does not require that you either assume the first step or figure it out yourself. But heh, that is not unreasonable. For my one you just have to take for granted the definitions of e ( for complex zs) and the power series definitions of sine and cosine . Or better yet understand a little Then again, very few people are aware of the power series definition of sine and cosine. Heck, even some "mathematicians" are not aware of it or have to be reminded sometimes. I guess I just like my one better because it makes the "deep" connection between e, i and trig functions a little more explicit.... Share this post Link to post Share on other sites

RationalBiker 39 Report post Posted September 29, 2011 My favorite number is 42. Share this post Link to post Share on other sites

Steve D'Ippolito 30 Report post Posted September 29, 2011 Actually e is easy to remember to 9 decimal places if you can just get to five places. 2.718281828 (the 1828 repeats). Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 29, 2011 It would have been much more fitting if you had put it like this : "i easily pick 42!" Share this post Link to post Share on other sites

Black Wolf 2 Report post Posted September 29, 2011 (edited) Weston, lets use Wikipedia to help prove it shall we? sine and cosine can be defined as follows : For complex z : And ( of course ) : So : cos(pi) = -1 sin(pi) = 0 so i(sin(pi) ) = 0 so... e*(i*pi) = cos(pi) + i(sin(pi)) = -1 Now you just need to prove WHY the infinite series for e^x, cos(x), and sin(x) are what they are ;-) I actually wrote a seven-page book about it, so I couldn't really do it here. Edited September 29, 2011 by Black Wolf Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 29, 2011 (edited) I suppose I could do that But I think that would take a little more time than I am willing to spend on it *here* I am just going to assert that those are axioms of my argument and leave the derivation of those as an exercise for the reader Here you can find a partial proof , not quite seven pages ( would be somewhat closer if it was more complete perhaps), but it gives the general idea for anyone that is curious : http://mathforum.org/library/drmath/view/53746.html Edited September 29, 2011 by Prometheus98876 Share this post Link to post Share on other sites

Weston 3 Report post Posted September 30, 2011 Promestheus, I'm aware of the Wikipedia article on the subject (it's the first place I looked for an explanation, being the lazy bastard I am), but my issue is accepting that (ix) works the same as (x) or any other real value when it comes to the Taylor series for e^__ Why should (ix) be treated as just another value when it's not? Warning: I have only taken math up through Calc II (multidimensional), so if the answer to this is obvious, I apologize in advance. Share this post Link to post Share on other sites

Prometheus98876 3 Report post Posted September 30, 2011 (edited) Well, supposing that you accept the "definitions" of e, cos, sin and "i" presumed by my proof ( I think I already gave a link if one wants to see the reasoning for why it is valid, at least for sin and cos) : Then the rest must hold, regardless of the value of "x". See the definitions for sin and cos which I gave at the start? Good. See where it says e^z? In this case we set z = ix, i times x. In this context, z can take ANY complex value, that is the point. i * x is complex number ( not a real value anyway). e^(anything) by definition takes the following form : The only difference is that we were using z not x before. However the same identity holds if x is actually a complex value. There is nothing special that says it only holds for *real* values. So just replace x with z and the same holds. The same holds if you then replace z with i*x. I do not seem what is so hard to accept. If you accept this, the proof clearly follows from there. Edited September 30, 2011 by Prometheus98876 Share this post Link to post Share on other sites

ITWT 0 Report post Posted September 4, 2012 Further to the topic, here is a website about the favorites numbers. Everyone can participate. http://www.thefavoritenumber.com Share this post Link to post Share on other sites

Devil's Advocate 103 Report post Posted September 10, 2012 3 Share this post Link to post Share on other sites

dream_weaver 421 Report post Posted October 2, 2012 ITWT stated in #15: Further to the topic, here is a website about the favorites numbers. Everyone can participate. http://www.thefavoritenumber.com The number 1 is ranked n° 11 in the ranking with 5.3 % It is the : - 3rd favorite number in : United States - 2nd favorite number for the : 51 -> 60 years old Share this post Link to post Share on other sites

ITWT 0 Report post Posted November 9, 2012 Thank dream_weaver for participating. For the moment there is not much to vote and the results are not significant. Should make the pub but I don't reach attract more people. Share this post Link to post Share on other sites

tadmjones 40 Report post Posted November 10, 2012 (edited) crikey I was gonna say Sheldon's favorite number but I dont remember what it was Edited November 10, 2012 by tadmjones Share this post Link to post Share on other sites