Explain what is meant by compound ? How does the concept of continuous compounding lead to Euler’s number, e? Define the 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 ) at the interest of r% per annum is added back to the principal at the end of the period and the total sum (FV= ) 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 . 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 . 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 compounding in the year. er 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 .

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 (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 log x/y = log x − log y b b b 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 :

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: eiπ = (cos π + i sin π ) = (-1 + 0 ) = (-1) , or eiπ+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.

Exponential functions are so useful in many real world situations. They are used to model populations, determine effective medicine dosages, carbon-date artefacts, help coroners determine time of death, compute investments, as well as in many other applications.

The size of computer files, processors and programs, for example, are expressed as an exponent. Computer file size is measured using “megabytes”. But megabytes are actually an expression of an exponent; a megabyte is equivalent to 106 bytes. Computer programmers use their understanding of exponents to determine the most efficient way to write a computer program and the minimum specifications a computer requires for a program to work.

The most common example of exponential growth of population is that of bacterial kt colonies. The formula to apply here is: N =No e , where N is the population at time t, No is the initial population and k is the growth rate (k>0). It is to be noticed that the equation is identical to the expression for radioactive decay discussed in my earlier article in the Science Corner, except that the exponent was -kt rather than kt. An exponential decay model also -kt applies for determining the concentration of a drug in a patient's body: C(t) =Co e , where

C(t) represents the concentration at time t, and Co represent the concentration just after the dose is administered. Suppose that Co of a drug administered intravenously is 2.5 mg/ml and after 3 hours the concentration of the drug in the bloodstream drops to 0.5 mg/ml. Using the formula we compute the value of k as follows: -3k -3k 0.5 = 2.5 e or e = 0.5/2.5 -3k Taking logs on both sides, ln e = -3k = ln 0.5 –ln 2.5

Hence, k = (ln 2.5 – ln 0.5 ) /3 = (1.6094379)/3 = 0.5364793 .

A problem facing physicians is the fact that for most drugs, there is a concentration, m, below which the drug is ineffective and a concentration, M, above which the drug is unsafe. Thus the physician would like to have the concentration C(t) between these values. This requirement helps determine the initial dose of a drug and when the next dose should be administered. For example, if for the drug in the experiment, the maximum safe concentration M =5 mg/ml and the minimum effective concentration m =0.4 mg/ml, and the initial dose administered is 4.5 mg/ml. How many hours later will the drug concentration reach the minimum effective level? Inserting the value of k, the equation for the evaluation of t now becomes - 0.5364793t 0.4 = 4.5 e , whence t= (ln 4.5 – ln 0.4)/ 0.5364793 = 2.420368128/0.5364793 =

4.5 hours

Logarithmic functions are used in sounds. We perceive sound intensity as loudness. Decibels are a logarithmic scale to quantify loudness: β = 10 log (I/I0). The reference value, I0 is the - threshold of human hearing and is roughly equal to 1x10 12 watts/m2 (pin drop!). Threshold of pain occurs at an intensity of about 1.0 watts/m2 (rock concert!), which is 12 orders of magnitude of hearing! A factor of 2 in intensity corresponds to 3 decibels. A factor of 10 in intensity corresponds to 10 decibels. An increase to 20 decibels is equivalent to a sound intensity that is 100 times greater.

The Richter scale used to measure the magnitude or intensity of an earthquake is also logarithmic. A simplified equation for the scale is R = log I, where I is the intensity of the earthquake measured relative to a reference value, I0, which is the smallest seismic activity that can be recorded on a seismograph (Io =1). Every increase of 1 in the Richter scale means the magnitude of the earthquake is 10 times greater. The increase is in wave amplitude. That is, the wave amplitude in a level 6 earthquake is 10 times greater than in a level 5 earthquake. However, in terms of energy release, a magnitude 6 earthquake is about 31 times greater than a magnitude 5. The amplitude increases 100 times between a level 7 earthquake and a level 9 earthquake; earthquakes of such magnitudes cause major destruction and loss of life over wide areas.

Another well-known logarithmic scale is the pH scale (pH = -log [H+]) which is widely used in chemistry to measure the levels of acidity or alkalinity, including measurements of acidity in swimming pools and of acid rain.

Yet another example is the star magnitude scale which measures the intensity of brightness of stars. This scale is logarithmic because the brightness of any star is 2.5 times brighter than the star of 1 magnitude greater, so a star of fourth magnitude is 2.5 times brighter than a star of the fifth magnitude, and a star of the second magnitude is 2.5 times brighter than a star of the third magnitude. To find out how much brighter a 3rd magnitude star is to a 5th magnitude star, take 2.52, and you get 6.25, so the 3rd magnitude is 6.25 times brighter.

vg kumar das (07 September 2012) [email protected]