14 - What is Euler's Number 'e', Ln(x) - Natural Log & e^x Functions?

hello welcome back the title of this lesson is called the number II which is called Euler's number in math we're also going to cover the basic concepts of the natural logarithm and the natural exponential function crucial crucial crucial concepts in math so what we want to do here is explore where this special number E comes from very soon in the next couple of lessons we're going to cover the concept of the natural logarithm it's just a type of logarithm with the very special base that has this special number e Euler's number associated with it a lot of students are introduced to the natural logarithm but have no idea where this number e actually comes from and I want to take one entire lesson just to talk about where that number comes from because it's very important all right the number e pops up in exponential calculations it comes up because the number E is the natural number that pops up and when I say natural number I don't mean it's a natural number in the mathematical definition I mean I just means it that it's a number that pops up in the course of studying exponential growth and because of that it pops up in terms of logarithms and because of that it pops up all throughout calculus and advanced mathematics it's very much on equal footing with the number pi that's another one of these very special numbers that you already know about the number e has I would say equal statue with the number pi it's just that we don't learn about the number e until a little bit later in math so let's explore where this number actually comes from so this number is called the number e and the person it's named after is a mathematician last name is it looks like the last name as Euler but actually it's pronounced Euler so his name is oil this is Euler's number okay it's extremely important it crosses all disciplines of science and math and engineering it's used everywhere like I said you don't typically study it until you get a little more advanced math but once you get there you can't get rid of the number e it pops up everywhere okay where does it come from first of all let's talk about what this number actually is so the number E is approximately defined to be the following number two point seven one eight two eight dot dot now the reason I have the dot dot is because the number E is called an irrational number it's an irrational number just like pi is an irrational number so the decimals here they go on and on forever in a non repeating pattern I could grab a computer and calculate the number e oilers number two a hundred million decimal places if I wanted to and I would never see any repeating pattern of decimals of course some of the numbers would be repeating but the pattern would not repeat because it's an irrational number all right so in order to understand where the number e comes from let's take and why it's useful let's take just a second to talk about ultimately once we know where that number comes from what are we gonna use it for it's just a number first of all when you see the number E or the letter E written in a mathematical equation or formula in your mind just replace it with a number it's just a number that happens to be a very very special number we're going to discuss exactly why in just a second but ultimately once we know what this number E is and once you understand where it comes from we're going to later on define a few very special functions the first function is remember we talked about exponential functions we could have 2 to the power of X we could have 10 to the power of X we could have 35 to the power of X those are all exponential functions so a very special exponential function is e to the power of X where e is just this very special irrational number I don't expect you to know why this exponential function in particular with this base this base being 2.71828 is just a number here ok I don't expect you to know why it's important we're gonna study that later and in fact we study why it's so important a lot more even in calculus but just to know that this exponential function with this particular base is extremely important in math has very special properties and because this is the exponential function and because we now know that logarithms are the inverse function of the exponential function then if you take this as your exponential function and you create the inverse of this guy then you might come up with something called the logarithm with a base e of some number X so these are inverses of each other but we never write log base E what we really do is we have a very special we have its own name we call it the natural logarithm so these two things are exactly the same thing so if you were to see in an equation somewhere if you were to see Ln of s Ln of X Ln of some number it means natural log of that number it just means logarithm with a base e of that number so the number E as a base is so special that we have a its own logarithm for it and the name of that logarithm is called the natural logarithm so this is called the natural log now you might guess by the name natural log that it's something natural about it there's something very special about that number and that's because the number E as I mentioned before is that natural number that just pops up when you study exponential growth it just falls out of the equations just like the number pi falls out of the equations when we talk about circles okay so what we want to do for the rest of this lesson is start to go down the road of trying to understand where this number e comes from why it's important and then in the next couple of lessons we will dive a lot more detail into the natural logarithm and the exponential function will do a lot more detail work in that area now I want to talk about where this number comes from before we go there what are some other numbers in math that are special that have these special status well I could argue the number zero has of this very special number I can argue the number one is a very special number it's number one identifies is the first number in a set that denotes the amount of something the number zero denotes a nothing at all which is kind of it's only in math it's it's only kind of a special class of number the number 0 and 1 are both very special above and beyond that there's some other numbers that you all know about that are also very special for instance we talked about the number pi you all know that pi is 3.14159 and it goes on and on and on forever where does the number pi come from the number pi comes from looking at the geometry of circles if I have some circumference of a circle and I have in some known diameter of this circle then I know that the circumference is pi times the diameter so pi serves to be a conversion factor it converts diameters into circles if I know the diameter of a straight line and I multiply by this very special number it gives me the length of a circle that would have that diameter going around that's what we call the circumference so the reason PI pops up is because it's a feature of our geometry of our universe we live in it's what relates linear distances to curved circular distances is what we call the circumference of a circle so it comes from geometry what are some other special numbers in math all right the number square root of 2 is obvious is also a number that pops up in math quite a bit we have other radicals of course but this one pops up a lot because it comes from the Pythagorean theorem if I have a right triangle with a length of side length of 1 on one side and the length of side on the other side a length of 1 on the other side and then I would denote the hypotenuse to be C this is different than the circumference of the circle here and then I can write C squared is a squared plus B squared but this is just 1 squared plus 1 squared and so what's going to happen here is C squared is going to be this is 1 plus 1 is 2 and then C is going to be equal to square root of 2 so the square root of 2 is the very special length that comes when I arrange a triangle like this and measure the diagonal along one side so again it comes from geometry the circumference of a circle comes from geometry the definition of the square root of 2 is another irrational number that comes from geometry ok what are some other very special numbers we're not going to go through all of them but probably the other one that is very very important is the number I which is the imaginary number and this comes from the definition being the square root of negative 1 so these guys here these first two definitions of these two irrational numbers they come from geometry one comes from looking at how circles behave one comes from looking at the Pythagorean theorem both of these are geometric definitions this one is not a geometric definition it's just the it's just the definition of the square root of negative 1 we call this thing the imaginary number I so these come from geometry this one comes from a definition the number e comes from the study of exponential growth so it's not a geometric definition it comes from studying how things grow exponentially and it turns out when you study how things grow exponentially a number falls out of those equations when you study it and that number is two point seven one eight two and eight and then infinite decimals Beyond and that number has very special applications in math all right so in order to figure out where this thing comes from ultimately it comes from one way to talk about how where it comes from I should say there are many ways to approach where Euler's number e comes from the way we're going to talk about let's talk about what we've discussed in the past which is the compound interest formula which talks about how much money grows over time in terms of an exponential function we discussed this extensively in the past so recall compound interest right and I would say if you haven't watched my lessons on compound interest I would highly recommend you do it first because I extensively talk about this formula in the past but never one write it down again the value of money at some time in the future is equal to whatever you invest we call that the principal multiplied by one plus whatever the annual interest rate is divided by the number of compounding periods and this is all raised to the exponent to the number of compounding periods times the time in years okay if you don't know where this formula comes from if you've never seen it before it just means you haven't watched my lesson on my lessons on compound interest so I highly recommend that you go watch that because I explain exactly where every little part of this equation comes from but just to make sure we're all in the same page this is the principle you invest this is how much money is worth time T years down in the future R is what we call the annual interest rate and n is really important it's the number of compounding per year so it's how many times I'm compounding per year if I'm only compounding annually I'm only getting free money once per year so then n is one if I'm compounding two times a year than Ennis - if I'm compounding quarterly four times a year then n is equal to four so this is equation that governs how much money you're gonna have in the future when I give you growth of a certain interest rate compounded so many times per year that's what this is and this number is ultimately going to allow us to calculate this very special exponential base that we call E which is Euler's number so here's how we're gonna tackle this all right let's say I'm a bank and I'm gonna give you an excellent deal I'm gonna give you an amazing deal the deal of the century what I'm gonna do is I'm going to give you 100% interest rate which isn't really not gonna ever happen normally interest rates in real life or 5% 6% interest per year or something like this but that's not the point we're deriving Euler's number E the way you go about it is you say okay I'm gonna give you 100 percent interest and I'm gonna compound that annually and I want to start off with just $1 and I want to see how much money I'm going to have at the end of a year now you might already know the answer if I give you 100 percent interest per year then after one year I'm gonna gain an extra dollar so in my mind I know that I'm gonna have $2.00 at the end of the year because I gave you 100 percent interest over that year but I still need to go through the equation because we're gonna make some changes along the way and you'll see how we end up driving with Euler's number e right here so let's say that I'm gonna give you 100 percent interest and that's compounded that's what comp means compounded annually that means only one time per year all right what do we get and it's a one dollar investment basically I want to see how much of my $1.00 grows whenever I do this so the amount of money I'm gonna have in the future is equal to the principal times one plus the interest rate over the number of compounding periods raised to the power then T this is the same equation we wrote above here's how you go about calculating this we know we're gonna have two dollars after one year but let's just do it anyway the number of dollars I have to start with one then it's one plus what is my interest rate I said it's a hundred percent interest now you didn't you never want to put 100 and you never want to put the percentage in you have to convert the percentage to a decimal so if you take a hundred percent convert to a desk when you move it two spots to the back so you end up with one point oh oh that's what you get when you convert 100 percent to a decimal shift the decimal spot two times I'm only doing it annually that means one time per year I'm only doing it annually one time per year and I'm looking down the road one year in the future again if you don't know how this works it's because you have it watched my compound interest formula example like my lesson on that so here we go we have one on the inside this is going to be one plus one so I'm gonna have to win here raise to the power of one now you can see what's happening is I'm gonna have two dollars right because two times one that's what we get and that's exactly what we expect if I'm giving 100 percent interest compounded annually I only give you the free money once per year I only do the calculation once per year than you expect after one year I should earn 100 percent of what I had and so I add them together I get two dollars that's what I get so I'm gonna go ahead and I don't want to circle that actually that's just a intermediate answer now let's change it a little bit now what happens if I do the exact same calculation I still give you 100 percent annual interest rate but now I'm not going to compound it annually let's compound it every six months so every six months means you compound it two times per year so for the next thing what we're gonna do it's is gonna be one hundred percent compounded every six months but again I'm only looking one year in the future all right so let's go ahead and calculate that the amount of money I'm gonna have is the principal times 1 plus R over N I'm not going to write this every time but I went ahead and did this time I'm investing $1 my interest rate is still 100 percent I have to move the decimal so I'll put 1.00 but I'm compounding two times per year right I put a two there and then it's going to be per year here I'm looking one year in the future I just want to find out how much money I get if I can pound it twice a year instead of once a year again I'm looking over a one-year period so what I'm gonna have is one this is going to be one half 1 plus 1/2 is 1 point 5 raised to the power of 2 so if you do 1.5 square and you multiply it by one what you're gonna get is 2.25 $2.25 so then look at this is very important for you to understand before we go on what's happening is if I tell you I'm gonna double your money and I'm gonna compound it only one time per year you'll get exactly what you expect I get double my money right but if I give you the exact same interest rate but I compounded every 6 months what this equation is doing is I'm taking that interest rate I cut it in half because I'm gonna give you half of the interest in each half of the year I'm gonna give you point five percent or fifty percent interest really in two halves of the year and so I'm gonna do that smaller interest rate one point five two times with two compounding periods in a year because I'm doing it every six months so when you get this you actually get a little bit more money than in the case whenever I just do it on a yearly basis and that's a general thing that's true if you keep the interest rate the same but I can pound it more frequently even though the annual rate is the same I actually make a little bit more money doing that that's a general thing that's true for all calculations but let's go on further let's say I have an incredible bank that's gonna give me a hundred percent interest let's do it even more often let's calculate how much I'm gonna get if I compounded monthly right so what if I give you 100 percent interest rate then that's what I R means interest rate compounded monthly compounded monthly let's see what we have here so I'm gonna have an amount that's going to be equal to the $1.00 that I spend one plus the interest rate still the one hundred percent interest rate but now I'm compounding 12 times per year and up here I have the 12 times per year I'm looking one year in the future and so I need to calculate what I get here so what am I gonna have I don't have one you do the division here and add one you'll get 1.08 three with a repeating decimal on the three raised to the power of 12 so you see what's happening I took my one hundred percent interest but I chopped it up into 12 pieces because I'm gonna actually compound it twelve times per year so I give that a take that one hundred percent but I split it in two into twelve equal parts and I give that to you twelve times that's what compounding means I take the annual rate I cut it into however many compounding periods I have I cut it into a smaller number and I give you that amount of interest throughout the year that's what compounding is doing this smaller rate is smaller than this one which is smaller than this one because I had to write here right so I'm basically chopping two rate into a smaller piece but I'm giving it to you more often so what happens whenever I go through this is I get two point six one dollars two dollars and 61 cents so notice when I compounded annually just one time of year I got two dollars when I did it every six months I got 225 and whenever I do it every month that's two dollars and 61 cents so people like Euler wondered what would happen if I can pound it even more often what if instead of compounding you know monthly or something like that or even weekly let's go and compound it daily so I take that 100 percent annual interest rate and I chop it into three hundred and sixty-five little tiny interest rates and I give that to you every day when I'm when I make more money than this or not and you can see the general trend is that I should make more money so let's say that I take 100 percent interest rate and I compound it daily and we do have real loans in real life that do compound daily so what if we have that we have one dollar I'm investing one plus the interest rate which is still one but I'm dividing it into 365 pieces and then I'm gonna have 365 looking one year in the future so you can see what's going to happen is it's gonna be one on the inside here when you do this division and you add one you're gonna get 1.0 to seven four this is rounded when I divide that this is a rounded number raised to the power of 365 so you see what happens is I take that interest rate that I was doing annually I get a very small rate that's 365 pieces right but I'm giving that to you and I'm compounding that every day that's why there's so many compounding periods I'm giving you a little bit of interest every day rather than the whole thing over the whole year what does this number come out to and what are you gonna get you're gonna get two point seven one four six notice what's happening as I increase the number of compounding periods I go from two dollars to 225 to two sixty one to two point seven one four six I'm getting closer and closer and closer to a limit right because notice that it was going it was changing rapidly here but then as I increase increase increase the point to the point where I'm giving you daily interest daily compounded interest the difference between this and this is not as great as it was over there and you might have guessed by now that this number that I'm approaching is going to become the number E but let's prove it to ourselves by taking this one step further what happens if I don't compound monthly or even daily what happens is it even possible to compound continuously what if I compound every second I mean think about it then I would take one year's worth of seconds there's got to be thousands of seconds in a year I could go calculate what the number is there's thousands of seconds in a year right but I could chop that interest rate into that many pieces and I could do that many compounding period so this exponent would be really really big and this interest rate will be chopped up into even more pieces but I would be giving you those more and more and more often what would happen if I did it to the point where I did it down to the second what about down to the millisecond what if I took a year's worth of nanoseconds and then tried to figure out what this number is what if I could do it continuously what if I could shrink that compounding window down so close to zero that effectively was almost quite zero but it wasn't quite there and I compounded what we call continuously what would this number be in that point at that in that limit what would it be and that's what we want to basically calculate now so let's go off to the next board and do that the equation that we would use to do that 100 percent interest rate compounded continuously that's what we're trying to do the amount of money that I would have in the future would be the $1 that I would invest times 1 plus the interest rate that never changed but now instead of dividing by 365 and then figuring out how many seconds in a year and how many milliseconds I'm gonna actually instead of putting a number here I'm gonna put the number in down here because I don't I'm just gonna make it really really small and then in the exponent it's gonna be in times 1 this is exactly the same thing as I was doing here notice I had 365 and then 365 times 1 I had 12 here and then 12 times 1 now all I'm gonna do is I'm going to change it to a variable and then make it a very that same variable times 1 so what I'm going to end up getting is 1 times 1 plus 1 over N to the nth power that's what this is okay now we said we want to compound it continuously I can't I don't really know exactly how to calculate it continuously but let's try to get close as we can let's let in equal to one thousand equal one thousand so instead of compounding daily where we have in is equal to 365 compounding periods in a year let's increase it so that n is equal to a thousand a thousand slices of a year that's how many compounding periods I'm gonna have any year if I put a thousand here then it's 1 over a thousand plus 1 raised to the power of a thousand times 1 then what ends up happening is the amount of money you get is two point seven one six nine two all right what if you let in equal to ten thousand I take a year and I chop it into ten thousand pieces so I put ten thousand here and I put ten thousand here I add this together raise it to the power of ten thousand then the number I'm gonna get for the money is gonna be two point seven one eight one five notice how it's getting close to some critical number here let's go to our last one one hundred thousand one hundred thousand here I chopped the year into one hundred thousand pieces I raise it to a hundred thousand and then the number that I get here is two point seven one eight two seven again very little difference between as you can see as I increase the number in I'm rapidly approaching some critical value I'm not gonna get I'm never gonna get past two point eight right so here's what you have if you let in approach infinity infinity is not something you can ever get to I can never get to infinity if I think the biggest number I can think of one billion trillion kajillion bazillion that's not even a number but if I make it up then I can always think of a bigger number I just add one to it so the number infinity is not a number it's an idea the number infinity is not a number I keep saying the number infinity it's a concept it's the largest possible thing that you never ever get to but we can still approach infinity by letting in get larger and larger and larger but if we let in approach infinity then what happens is the number e ends up becoming by definition what we call in advanced math the limit as n approaches infinity of 1 plus 1 over N to the power of n and this is approximately equal to 2.71828 dot dot dot and I'm gonna go ahead and put in big letters that this is irrational in other words these decimals do not repeat there's not a repeating pattern I cannot write this number as a fraction that's what that's what that means and because the mathematician euler figured this out this is called Euler's number okay is what you will see in a textbook you'll probably see it in a advanced algebra book or an algebra 2 book you'll definitely see it in a calculus book in an advanced math book what basically means is the typically the definition that you will see is if you take the limit as n approaches infinity of this expression you're doing exactly what we did up here start with a thousand or something like this calculate the number go to 5,000 go to 10,000 go to a hundred thousand go to a million go to two million go to five million and you're going to get numbers they get closer and closer and closer to the limit is what we call now how do we calculate this exact number is what this limit is these methods to arrive at this exact number of course we could get closer and closer by picking numbers but to know that this is the exact answer those methods come from calculus that's beyond the scope of this discussion how do we actually calculate what the limit is but you can see that these numbers are approaching this and to the fact we get to 2.718 to seven that's very very close up to six decimal places or something like that so you can certainly get enough accuracy for a calculation just by doing it manually but we know that this number goes on and on and on forever we can calculate with computers and in the methods of calculus we can find out what this limit actually is this is what you will see in a textbook unfortunately this doesn't tell you anything about why the number is important you'll see it in a book that the number E is by definition equal to this but and it's approximately equal to this but it doesn't convey why it's important that the special number e comes about because exponential growth is is when you take a number and you grow it exponentially over time right so if you pick the special case of trying to if you pick the special case of compounding annually where I'm gonna give you I'm going to double your money in a year then that's exactly what you would get if I take that same rule trying to double your money but I chop it up into two times a year doing it every six months I get closer to Euler's number if I instead compounded monthly again giving you an annual rate of a hundred percent trying to double your money then I get a little bit more than double a double them one dollar that I started with and then of course as you go to three sixty-five and as you give the 1,000 compounding periods and 10,000 compounding periods 100,000 compounding periods you end up with more and more and more money but only up to a limit so Euler's number is that limit it's how much money you would gain if I offered you a hundred percent annual interest rate compounded every single nanosecond of the year that's what it's basically telling you you're gonna get a little bit more than double your money but it's only gonna go up to a certain point and that very special number is called Euler's number 2.71828 I cannot express to you enough how important Oilers number is it's basically impossible for me to do it because you haven't learned enough math to know how much this number pops up but I can tell you with certainty that the number e pops up as much or more than the number pi probably more than the number pi in all advanced math classes all of a wave interactions you know sine and cosine waves that travel through space that means all of our theories of light all of our theories of oscillation they all boil down to this number E because all of those equations sines and cosines which we'll learn about a little bit later they can all be written in terms of the number E it's a very fundamental number that comes about as a result of exponential growth just like the number pi is a very special natural number that pops out in the study of geometry of circles and straight lines okay now this is not a class on calculus but I do want to do one more thing before we go and finish up everything here and what I want to do next let's see what board space we'll have what I want to do next is I want to talk about a little bit at least why this number is so important okay so E is special remember in the beginning I told you I said that it's a very special number that comes about because of exponential growth and so when we make an exponential function with E as the base it's a special function because the inverse of this is the logarithm the natural logarithm which is the inverse of this function okay is also a very very special number now I want to take just a second to tell you a little bit why it's a special this is not a class in calculus it's not a class in advanced math but I think everybody watching this can definitely understand why it's so special and I want to take it will be criminal for me not to tell you why it's special so let's take a second to do that let's say we have the function f of X is equal to the number E to the power of X we take that very special number that comes about when studying exponential growth and we make it the base of an exponential function okay what does this equation look like what does this function look like if I were to graph it so I'm gonna go and draw a graph this is gonna be freehand so it's not gonna be exact alright so it's not gonna be exact in any way shape or form so don't yell don't hate me right but anyway if we have one and we have two and we have three let's say this is 1 this is 2 and this is 3 we know that this exponential function is going to go let me use red we know it's going to go through the point 1 0 comma 1 because if I put 0 here e to the 0 it's just a number number to the 0 power is 1 so that's going to go through that point and we also know that the number E is actually 2 point 7 1 what did you say it was 2.71828 so it's about two point seven so this number is about somewhere right around here this right here this is where the number E is is about two let's see two point seven one is roughly equal to e it's somewhere right around there and so whenever I go and take a look at the number this is one this is one so if I put the number one in here e to the first power is just e so this point right here if I can draw it correctly is going to be on the graph to this number right there right because if I put one and I'm gonna get e to the first power which is e and that's what he actually is so this exponential function is gonna go through both of these points something like this I'm gonna do terrible job drawing it but I'm gonna try I'm gonna go up like this up like this up like this something like that that's actually not too bad so this is the famous function e to the power of X right e to the power of X what is so special about this thing you're not going to learn this until you get study calculus later but you can take a look at functions and you can figure out you know how we talked about the slope of the line in algae a long time ago we said the slope of the line is rise over run so it's easy to calculate the slope of the line but what is the slope of this thing I mean forget about the actual number you know the slope of this function is very high right here it's very steep and you know that the slope of this function is very low right here it's very steep a very shallow and as I travel along this curve it's a very low slope this is like a medium slope and this is a very high slope so the slope of this thing whereas the slope of a line is the same thing the same slope all the time the slope of this exponential function is always changing the slope is very low close to zero here and the slope is very high maybe the slope is like 50 or 100 right here and it's every number in between as I go now the way that you figure out what the slope is of this curve if you want to know what is the slope right here at this red dot what you do is you draw a line tangent to this guy you draw a line tangent tangent just means that it touches the red curve only in one spot only at this dot right here okay so what this very special function e to the X does is the slope of the line tangent the slope of the tangent line okay at this point right here is equal to e so that's something that kind of flies over your head if you don't study calculus but I want to make sure you understand how incredible that is what it's basically saying is this function e to the X is the only function so that wherever I am on the function the slope of the function is equal to the value of the function itself so right here at this point the value of this function is e right but look at this you look at the dotted line you go up e to the power of 1 is e the value of this function right here is e and because the exponential function e to the X is very special the slope of a line tangent to the curve at this point is e it's the value of the curve itself that's not true for any other function when you look at functions in general you can calculate the slope later on down the road in calculus you can calculate the slope of the line tangent but the slope is going to be whatever it is you're going to calculate it using using the methods of calculus but the very special exponential e to the X is really the only function so that the the slope of the line tangent is just equal to whatever the value of the function is itself so let's say right here the value of this function right here is two so if you were to go through and say what is the slope of the curve right here I'm not gonna touch it because I'll mess up the ink but if I say what is the slope here if you were to draw a line tangent there the slope of the curve right here is equal to two it's the equal to the same value of the function what is the slope right here the slope right here is one what is the slope right here well it's whatever the value of the curve is down here you can figure out the slope anywhere you want on this exponential function just by looking at the value of the function itself it's the only function it really behaves that way okay now the other thing I want to talk about let's plot the thing one more time f of X is e to the power of X and I'm gonna do the same exact curve it probably didn't give myself quite enough room all right but let's just do one more thing let's say one two three this is one two three and I'm gonna do not if you do a little faster so I know that I have a dot right there I know this is basically the number E and then I can go right here and I can say this number is gonna cross basically right here because I'm gonna put the number one into this equation e to the power of one it's II which is right here so this curve has to go up something like this hopefully I won't mess it up I got to go through here through here something like this this is the curve so I've tried to draw that thing freehand okay another thing that's interesting about it so the slope of the line that you get is equal to the value of the curve at that point itself and also if you look at this e to the X curve the area under the curve which is if literally if I were to draw this thing and actually somehow physically measure the surface area when I say the area under the curve I mean literally if I could like pour a bucket of paint and measure how many square centimeters were underneath the area of this thing up to this point then what I would find is that the area under curve up to this point right here is equal to the value of the point e so the slope of any point on this curve of a tangent line is equal to the value of the curve itself and the area under this curve up to this point all the way from negative infinity all the way up to and including this point is also equal to the value of the curve so if I go over here so that I have I'm looking at the number two and then the value over here is e to the power of of two or something like that then the surface area underneath would be equal to the whatever the value of the curve is at that point so the interesting thing about this and this really the only function that behaves this way is that the slope of the line or the slope of the curve at any point is equal to the value of the curve itself and also the area under the curve from negative infinity all the way up to any point you want to pick on that curve the area under it is also equal to the value of the curve itself so it's a really special function and the reason it pops up that way is because the base of this exponential function is a very special number it's the number that pops out by looking at the natural course of exponential growth and so that's that's Euler's number so these properties here the fact that the the slope of this curve is equal to the value of the curve itself and the area under this curve is equal to the value of the curve itself at that point those are crucial crucial crucial properties that we study in calculus in fact you study the slope of a line tangent to occur for about the first half of calculus one and then the last half of calculus one you study areas under a curve in fact if I could boil calculus one down to one sentence it would be you would study slopes and areas of functions and because this function is so special in that the slope is always equal to the function itself and the area under it is also always equal to the function itself all of that will be proved later on in calculus but because that is true then if this number this exponential function pops up everywhere and because this exponential function pops up everywhere also the inverse which is the natural logarithm also pops up everywhere okay last thing I'm gonna leave you with is something that's just interesting that you might have heard or some people you know read books or watch videos or whatever there is the most voted equation that's the most beautiful equation in all of mathematics and it's not something we're going to study here but I'm gonna just present it to you and show you why it's so beautiful and it involves the number e that we have been studying here it's called Euler's formula we're not going to study it but we're going to look at it okay Oilers formula is the following e that beautiful number either we just learned ^ I which is the imaginary number I times pi plus the number 1 is equal to 0 this is called Euler's number it is very frequently discussed and talked about as being the most beautiful equation in all of mathematics and this is the reason why Oilers formula is an equation because it has an equal sign that includes the 5 most you know beautiful or core numbers in all of mathematics remember I told you the number zero was one of those beautiful perfect numbers the number one it was also one of those beautiful perfect numbers the number pi pops up in geometry the number I the imaginary number is extremely important and the number E and this equation relates all five of those core fundamental numbers together if you grab a calculator and take the number E and raise it to the imaginary number I times pi and you add 1 to it you're going to get zero so it's not something that we're gonna do any problems on it's not something I'm gonna test you on or anything like that it's mostly just trying to tell you why the number E or at least one example of why the number E is so incredibly important it even pops up in one of the most famous equations and all of math so in this lesson we really wanted I really wanted to give you a background to where this number E comes from this Euler's number because I do get questions on it we're going to study natural logarithms and exponential functions extensively and I want to give you an idea of where it comes from and the number itself is 2.71828 repeating non repeating decimals like this we have some numbers that come from geometry some numbers that are defined but are not geometric but ultimately the number e comes from the exponential growth of money the compound interest formula if you compound annually a 100 percent interest rate and then keep compounding more often - 6 months - every month - every single day and then on to finally you go a thousand times per year 10,000 times per year a hundred thousand times for you eventually approach this very special number e which falls out of the exponential growth equations and then lastly when you take that number e and you raise it to the power of X you get an exponential function with a base of e this is stuff that you learn in calculus but I'm just telling you what ends up happening is the slope of this curve at any point is just equal to the value of the curve itself the area under this curve from minus infinity up to and including whatever point you're looking at is just equal to the value of the curve itself and then finally this number e is so important that it's involved in Euler's formula which is one of the most beautiful and fundamental mathematical formulas ever derived so it's really one of those fundamental numbers that pops up again and again and when you go on to more advanced math to advanced calculus to differential equations which is even more advanced calculus you are gonna see e pop up so often that you're your heads gonna spend it pops up more often than pie does so I wanted to give you a background as to where it comes from now follow me onto the next lesson we're gonna study the natural logarithm in a lot more detail and do a lot more problems with the natural logarithm in math

Channel: Math and Science

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Keywords: euler's number, ln(x), ln x, e^x, e x, natural log, natural logarithm, what is eulers number, what is eulers formula, what is euler equation, what is a natural logarithm, what is a natural log, what is an exponential function, natural logarithms, natural logarithms equations, natural log rules, eulers number, properties of logarithms, natural log equations, natural log explained, natural logarithms and e, the number e and the natural logarithm, calculus, algebra, math, tutorial

Id: 6MQGrGvRDlE

Length: 43min 15sec (2595 seconds)

Published: Tue Mar 24 2020