Euler’s Number

1. Euler’s Number RajeshRathod42@gmail.com

2. A Problem in Compound Interest Clearly at the end of the 12 months, we get 112 Rs. Let us say we have 100 Rs. invested @ 12% p.a. interest for 12 months. Now instead if we invest it @ 12% p.a. interest, but compound the interest at the end of first 6 months and then at the end of the 12 months: At the end of the first 6 months, the interest is 6 Rs and the principal is 106 Rs. At the end of the next 6 months the interest is 106*(6/100) = 6.36 Rs. The total at the end of 12 months = 112.36 Rs. Now instead if we invest it @ 12% p.a. interest, but compound the interest at the end of 3 months, then at the end of 6 months, then at the end of 9 months and then at the end of 12 months. At the end of the 12 months the interest can be computed as = 112.550881 Rs. We can continue this exercise by compounding at the end of every month, every week, every day, every hour and so on The question is what would happen if we kept on compounding at shorter and shorter intervals. How much would be the principal?

3. Compound Interest for N Intervals Divide the year into N equal intervals; the interest at the end of each interval being (12/N) % Rate of Interest p.a. = 12%; Amount invested: 100 Rs. We can setup a calculation table as follows: At the end of the 1st Interval Interest= Amount = At the end of the 2nd Interval Interest= Amount = At the end of the 3rd Interval Interest= Amount = At the end of the nth Interval Amount = In general, if Amount=P, R=(% Annual Interest)/100 Amount =

4. What if N is very large? The question is, what happens if we make N very large in the formula What is ? One would expect to make much more money if it was compounded more frequently – what if we compounded it daily, for example? And here is the surprise: its not that much more money! Its just about 2.718 times, if R, the annual rate of interest was 100% - that is, you doubled the money annually at simple interest. We will show that: e being defined as: This e is called Euler’s Number

5. Expression for e Expanding using the Binomial Theorem Letting n→∞ each of the terms 1/n, 2/n, … (r-1)/n vanish to 0. We then get: Note that: which is our definition of e! e= From our definition of f(x):

6. Evaluating e Now that we have defined an expression for e, does this expression yield any result? What if the value of e keeps on increasing as we add together more and more terms? Note that: As we add more and more terms of series for e together, the terms themselves become “very small” since they go as 1/r! For example, for r=10, the term (1/10!)=2.8x10-7. So we are adding together “incredibly small” positive numbers and intuitively one we may believe that the total sum of these positive numbers of the series would still be a finite number (even if we were to “somehow” add all the infinite terms)! But, we cannot trust such an argument! For example, the sum of the terms of this series actually keeps on increasing as we add more and more terms, even though the terms become smaller and small as r increases. It so happens that the series sum for e, as we have defined, does converge to a finite value.

7. 2<e<3 We compare the respective terms of the two series: The first 3 terms are the same. The 4th term of S>4th term of e The 5th term of S>5th term of e and so on for all subsequent terms Thus we get e<S RecognisingS as a geometric series from the second term onwards, we get S=3 and hence e<3 Thus : 2<e<3 and yet it continuously increases since each of its term is positive This is the criteria for e to converge to a definite value between 2 and 3 2.718281828459 The value of e up to 12 decimal places is:

8. Calculating e Just to illustrate, let us calculate e using the formula: The tables give the value of the series for e when evaluated for n terms: Notice that the additions due to the 17th term onwards do not make any difference to the 9th digit after the decimal. The graph depicts the same pictorially.

9. Each term of the series addition for e is a rational number; the addition of two rational numbers is another rational number, but the addition of infinite rational numbers leads to an irrational number. Very counter intuitive!! e is Irrational! Assume e is rational. Then: Since 2<e<3, p, q can be assumed >1 (Multiply by q!) (Re-arranging) Now, the left hand side of the equation is a positive integer. On the right hand side, the terms in the first parentheses are all positive integers. Hence it must be that: Hence by contradiction we show that e cannot be rational. is an integer. Hence: is an integer. Comparing S and S’ term by term we see that S<S’ Now consider: However, Hence, S’ is a geometric series with ratio But S should be a positive integer! Hence: q>1 ⟹ S’<1; and S<S’ ⟹ S<1

10. Extending the Definition of e We defined that: This definition can be extended as: For any given x, there will be some n such that n<x<(n+1) and hence Note that: We thus get that is between 2 numbers which both converge to e as x, n→∞ Thus by the famous “squeeze theorem” of limits: Note that we can also rewrite as:

11. Calculating We have defined Hence: Applying the Binomial Theorem for real numbers as exponents: Leading to: Note that for series evaluates to 1 as it should since And for we get back the familiar:

12. Differentiating • Given: • Thus: is the only number to possess this property!

13. Calculating • For , we know that, • This is based on the very definition of the log function • Thus: • Using the series expansion of • This is a very useful way to calculate in general • Of course we need to know the

14. Calculating • We now have a means to define what happens when a real number has a complex number as an exponent • Firstly consider: where • We DEFINE using the infinite series representation of for real exponents • Since • It is however not enough to simply define in this manner – it has to be ensured that: • The infinite series CONVERGES to a value, and, • The basic property of exponents: remains valid • Then we can extend all the methods of manipulation of real exponents to complex exponents

15. Calculating • Consider . where . Then: • We can write this product using the distributive law as: • … • Collecting all the terms with : • Collecting all the terms with : • In fact, collecting all the terms with will yield: • Thus, by our DEFINITION of for being a multiple of • Hence: . remains valid for complex exponents Verify this!!

16. Calculating • If we notice carefully, the proof previous proof of . is equally applicable for any quantities as long as: • Operations of addition and multiplication are defined • These operations are Commutative and Associative • The operation of multiplication distributes over addition • Thus could very well be any real of complex number • So, even if we were to choose A, where is a square matrix, we could actually compute and get a square matrix as the result!! • Clearly the set of complex numbers satisfies all the above properties and hence we can say: • where is any complex number • The set of complex numbers of course includes all real numbers

17. Does Exist? • We need final step: to ensure that the infinite series for CONVERGES • Earlier We mentioned that: • The two series within the braces are respectively DEFINED as the Cosine and Sine functions and we will show that they converge for any value of so that: • Since Sine and Cosine series converge, too converges • And since we can see that converges for all • Of course , always converges for NOTE: Cosine and Sine are DEFINED as infinite series, without recourse to Trigonometry or Analytic Geometry

18. Convergence of Sine and Cosine • Ratio Test for Convergence: • Let where is the term of the series. • If the series converges absolutely • Convergence of : • Calculating the Ratio • Hence, the Sine function converges for all • Similarly, the Cosine function, converges for all

19. Properties of Sine and Cosine - 1 • Both Sine and Cosine are continuous functions since each of the terms of their series are continuous. Also: • For , each term in the brackets is for as also • For , all the terms in the bracket are positive and some hand calculation will show that • Thus is a continuous function such that: • Therefore there must exist ,DEFINED as the smallest positive number such that and NOTE: is defined purely in algebraic terms, without reference to Geometry/Trigonometry

20. Properties of Sine and Cosine-2 • We have DEFINED: • and • C and • The Cosine sequence consists of even powers of the variable and the Sine sequence consists of odd powers, hence: • and • Differentiating each term of Sine sequence successively yield terms of the Cosine sequence; terms of Cosine sequence similarly yield the negative of the Sine functions: • and NOTE: The properties of Sine and Cosine are derived without recourse to Trigonometry or Analytic Geometry

21. Properties of Sine and Cosine - 3 • Also: • Equating the real and imaginary parts of and : • and

22. Properties of Sine and Cosine - 4 • Applying the properties derived so far: • (by DEFINITION of ). • , and • If , but by definition, is the smallest positive root of the Sine function so this cannot be • So it must be that

23. Properties of Sine and Cosine - 5 • We prove that: , by induction: • Case: [Defn of ] • Let • + • Thus, by Induction, for any integer • The “any integer ” is true because: • and • Also: • Combining with: and

24. Properties of Sine and Cosine - 6 • Now, • Thus, is periodic with period • Thus, is periodic with period • We have utilised the earlier results that and

25. Properties of Sine and Cosine - 7 • The famous limit: can be shown as: • Also: for small • Hence: for small • For example, if , the … evaluates to around which is around 4% error against the more accurate value for

26. thus giving us the expression: This is a beautiful equation, relating 5 fundamental mathematical quantities

27. Representing Complex Numbers with e We usually represent Complex numbers as: c=a+ib Since: eiθ=(cos θ)+i(sin θ) We can express: Where: We can always find a pair of values r,θ such that this is true. Thus: Also: Hence: Where: This is particularly useful when we want to multiply 2 complex numbers:

28. Logarithmic Function We assume that we can find a function, named log(x), such that for a,b ∈ R, a,b>0: We can prove that log(x) is the inverse of the function y=rx for some real number r>1 Let a=rpthen p=log(a) and b=rq then q=log(b) That is: if p=rqthen we say log(p)=q Now, a.b=rp.rq = rp+q⟹ log(ab)=p+q=log(a)+log(b) Also: a/b=rp/rq = rp-q ⟹ log(a/b)=p-q=log(a)–log(b) Also consider: r(b.log(a))=(rlog(a))b=ab Since ab=r(b.log(a)) by definition of the log function: log(ab)=b.log(a) Also, since 1=r0, log(1)=0 A very usual choice of r is the number 10. Also, another convenient choice or r is the number e! The number r is called the base of the function log(x) So, when we say log(x) or log10(x), it implies r=10; when we mean to say r=e, we write ln(x) or loge(x) Finally, it is evident that: logr(r)=1 for any choice of r>0

29. Derivative of the Logarithmic Function The function loge(x) [base r=e] as the following properties, for any a,b>0 Since logr(r)=1 for any r We find the derivate of loge(x): Furthermore: For any given value of x, h/x→0 Hence, the derivative of loge(x) is: By the chain rule of differentiation for any u(x):

30. What is the big deal about e and log? is considered a beautiful since it ties up the 4 mathematical entities 0, 1, π, and e in a simple equation. is true only for e and for no other number. e crops up in the mathematical description of several physical phenomena: Cooling of a hot body by radiation Liquid draining our from a tank under gravity Discharge of a capacitor in an electric circuit Radioactive decay of an element Calculation of compound interest Population growth models

31. A Question! • We have defined as the small positive number such that and of course and as infinite series • Geometry and Trigonometry defines using circles and triangles • Which one is the “real” definition? • What is real about a definition?!! • Why is it that should have same value as when defined using a circle, as when defined using an infinite series?