Explain what is meant by compound interest? How does the concept of continuous compounding lead to Euler’s number, e? Define the exponential function ex. Why are exponentials so useful in science? Compound interest is taken to mean that the interest earned on a principal sum (we shall call this PV which stands for present value) at the interest of r% per annum is added back to the principal at the end of the period and the total sum (FV= future value) then continues to earn the interest for a further period, and so on. Compounding of interest allows a principal amount to grow at a faster rate than simple interest, which is calculated as a percentage of only the principal amount. Thus, FV = PV × (1 + r), where r = interest expressed as a decimal (e.g.0.06 for 6%). If this annual compounding goes on for a period of n years, then FV = PV (1+r)n. This formula applies to both money invested and money borrowed. Let us consider a specific example, say RM 1000 at 6% fixed interest rate. Annually: FV = 1000 × (1 + 0.06) = 1000 (1.06) = RM 1060 (annual compounding). If annual compounding is carried on for 5 years, FV = 1000 (1.06)5 = 1000 (1.338226) = RM 1338.226. You can get this result using an exponential expression calculator (accessible freely on the internet). It is also possible to increase the frequency of compounding within the year, which is called periodic compounding. If in the above example, the interest rate of 6% is with “quarterly compounding”, it means 6% divided by 4 months, that is, 1.5% per quarter. Note that quarterly compounding does not mean 6% per quarter. Thus at the end of one year, we have FV = RM1000 (1 + 0.06/4)4 = 1000 (1+ 0.015)4 = 1000(1.061364) = RM1061.364. Similarly, for monthly compounding FV = 1000 (1+0.06/12)12 =1000 (1+0.005)12 = 1000 (1.061678) = RM1061.678. If the monthly compounding is carried out for 5 consecutive years, then the formula to apply is FV = 1000 (1+0.06/12)12x5 = RM 1348.851. The slightly larger interest gained with periodic compounding is obvious. The compound- interest problem was, in fact, first studied by Jacob Bernoulli in 1683, who discovered in the process, a mathematical constant. To show how he arrived at this, let PV in our example be RM1 and r=1, that is the interest rate is 100%. Then at the end of one year, the FV becomes RM2. With semi-annual compounding, FV becomes 1(1+1/2)2 = RM2.25. With quarterly compounding, FV becomes 1(1+1/4)4 = RM2.4414. Compounding monthly, the amount becomes 1(1+1/12)12 = RM2.613035. Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields 1(1+1/56)56 = RM2.692597..., while compounding daily yields 1(1+ 1/365)365 = RM2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100% / n in each interval, the limit for large n is the number that came to be known as Euler’s number, e. It is the limit of (1 + 1/n)n as n becomes large and has the value 2.7182818…. If the fixed annual interest rate is not 100% but r%, the compounding formula takes the form (1+r/n)n. xr By substituting x = n/r, the term (1+r/n)n becomes (1 + (1/x)) , which is just like the formula for e (as n approaches infinity), with an extra r as an exponent. r So, as x goes to infinity, then (1+(1/x))xr goes to e . Thus an account that starts at RM1, and yields (1+r) dollars at simple interest, will yield an interest of RM er with continuous r compounding in the year. e is referred to as the exponential function. If PV=RM 1000 and the annual interest rate is 20%, what is the value of FV upon continuous compounding at the end of 3 years? FV = PV (ert) = 1000 e0.2x3 = 1000 e0.6 = RM 1800 There are many ways of calculating the value of e, but none of them ever gives an exact numerical value because e is irrational (like π, √ .). But it is known to over 1 trillion digits of accuracy! The mathematical constant e constant can be defined in many ways. For example, it can be defined as the infinite series (Taylor series): (Note: The symbol "!" stands for a factorial operator; e.g. 4! = 4x3x2x1; 0!=1, by definition) The exponential function is the term used in mathematics to represent ex, first given by Leonhard Euler. x x Defining e as the infinite series, e =1+x/1!+x2/2!+x3/3!+x4/4!+….., leads us to the definition of Naperian or natural logarithm (written as loge or ln). You might recall from your knowledge of secondary school mathematics that if y = bx, then x is the logarithm of y to base b, and is written x = logb(y). That is, logb(y) is the exponent to which b must be raised to produce y. In other words, corresponding to every logarithm function with base b there is an exponential function with base b: y = bx, which is defined for every real number x. Here is its graph: Note that the x-intercept is 1 because logb1=0 and the graph passes through the point (b,1) because logbb=1. x Similarly, if y = e , then x = ln y. The natural log of e itself, written as loge (e) or ln (e), is 1 because e1 = e, while the natural logarithm of 1 [ ln(1) ] is 0, since e0 = 1. The natural logarithm of 7.389…. is 2, since e2= 7.389…. The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm which has the constant e as its base finds widespread use in pure mathematics, especially calculus. The binary logarithm uses base b = 2 and is prominent in computer science. You might also recall the following facts about logarithms: In any base, the logarithm of 1 is zero. Base changes can be accomplished: logbx = logax/ logab The three laws or identities of logarithms: logbxy = logbx + logby logb x/y = logbx − logby n logb x = n logbx Exponential functions and logarithmic functions with base b are inverses. x The functions logbx and b are inverses. For in any base b: log x i) b b = x and x ii) logbb = x The graphs of ex and ln x , showing their inverse relationship, are depicted below. The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x- axis is a horizontal asymptote. The slope of the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln x. The Taylor series for ln x is: ln x = (x-1) – (x-1)2/2 + (x-1)3/3 - … An alternative definition of the exponential function ex is that it is its own derivative: ex = ex Let's take the example when x = 2. At this point, the y-value is e2 ≈ 7.39. Since the derivative of ex is ex, then the slope of the tangent line at x = 2 is also e2 ≈ 7.39. We can see that it is true on the graph: Extending this if y = eax, then dy/dx = aeax. The derivative of the natural logarithm function is the reciprocal function: (ln x) = 1/x Integration being the reverse process of differentiation, it follows that ∫ eax dx = eax/a + C The indefinite integral of the natural logarithm function ln x is: ∫ ln x dx = x ln x - x + c Euler's formula The complex exponential function e ix has the identity: eix = 1 + ix +(ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7!... = (1– x2/2! + x4/4! – x6/6!....) + i (x– x3/3! + x5/5! – x7/7!...) = cos x + i sin x [sometimes known as cis (x)], where x is any real number and i is the imaginary number, the square root of -1 ; so i2 = -1, i3= ( i2.i )= -i; i4 = ( i3.i) = ( -i.i) = 1, etc. It follows that eibx = cos bx + i sin bx, and the exponential of a complex number such as (a + ibx) is: e(a+ibx) = eax.eibx = eax (cos bx + i sin bx) Euler is credited with finding this interesting relationship between the exponential function and the sum of two oscillating functions. The complex exponential forms are frequently used in electrical engineering and physics. For example, a periodic signal can be represented as the sum of sine and cosine functions in Fourier analysis, and the movement of a mass attached to a string is also sinusodial. The special case with x = π is Euler's identity: iπ iπ e = (cos π + i sin π ) = (-1 + 0 ) = (-1) , or e +1 =0 ix Note that ln e = ln (cos x + i sin x) = ix Hence, ln eiπ = i π = ln (-1) Euler's number is an important value for exponential functions, especially scientific applications involving decay (such as the decay of a radioactive substance). It is especially important in calculus due to its uniquely self-similar properties of integration and differentiation.