Growth, Reduction, Simple and Compound Interest and the Value of "e"

Numerical entities can either grow or reduce with time. For e.g., age of living and non living things increase with time, radioactivity of radioactive substances decrease with time, bank balance increases (or decreases) with time etc. There are two aspects associated with the process of growth or reduction.

1. Growth or reduction happens on some initial value of entity. This value is known as Principal.
2. Growth or reduction happens by some value. For process of growth this value is known as Interest. For process of reduction this value is known as Decay.
3. Growth or reduction happens to some value. For process of growth this value is known as Value After Interest. For process of reduction this value is known as Value After Decay.

1. Initial quantity of entity present: The principal quantity P.
2. Current quantity of entity present: This may or may not influence the extent of growth or reduction depending on the type of growth or reduction.
3. Time period of growth or reduction: Time taken for a single unit of growth or reduction. It is denoted by letter T. This may or may not influence the extent of growth or reduction depending on the type of growth or reduction.
4. Rate of growth or reduction: Amount by which the quantity grows or reduces during time T. It is denoted by letter R.
5. No. of Intervals of growth or reduction: Total no. of intervals of time T for which the growth or reduction has taken place. It is denoted by letter t.

1. Arithmetic growth and reduction: In arithmetic growth or reduction, both the values by which and to which the entities grow or reduce are independent of the current existing quantity of the entity, the actual interval for growth or reduction T and the no. of intervals of growth or reduction t.
2. Exponential growth and reduction: In exponential growth or reduction, both the values by which and to which the entities grow or reduce are in terms of some factor of the current existing quantity of the entity and also depend on the actual interval for growth or reduction T and the no. of intervals of growth or reduction t.

1. An increase in time period T by a factor increases the rate R by same factor and decreases the number of intervals t by same factor. This means if the value of T becomes T x 3, then value of R will become R x 3 and value of t shall become t/3. That is, when we multiply T by a factor, we must multiply R by the same factor and divide t by the same factor
2. A decrease in time period T by a factor decreases the rate R by same factor and increases the number of intervals t by same factor. This means if the value of T becomes T/5, then value of R will become R/5 and value of t shall become t x 5. That is, when we divide T by a factor, we must divide R by the same factor and multiply t by the same factor.

Arithmetic Growth and Reduction

In arithmetic growth or reduction, both the values by which and to which the entities grow or reduce are independent of the current existing value of the entity, the interval of growth or reduction T and number of intervals t. The only factors that influence the growth and reduction are the initial value of the entity and the Rate R of growth or reduction.

For example let's assume a numerical entity P which continuously changes by K units after time T.

Initial value=P
Value after time T = P + K
Value after time 2 x T = (P + K) + K
Value after time 3 x T = (P + K + K) + K
Value after time t x T = P + t x K

The value of K can be any real number. If K is greater than 0 the quantity increases with time. If K is lesser than 0 the quantity decreases with time. If the value of K is 0, quantity does not change with time. Generally the value of K is given in terms of some factor of initial value P (especially for reduction, otherwize the quantity can get less than 0 which should not be possible). This factor is known as rate of growth or reduction R. So,

Value after time t x T = P + t x (R x P)

For the process of arithmetic growth R must be any positive real number greater than 0.

For the process of arithmetic reduction R must be any negative real number value between 0 and 1.

Properties of Arithmetic growth and Reduction

1. The value of the entity can become zero after certain intervals of time in case of reduction
2. The value of amount generated after any no. of intervals of time is independent of the time period T. (This is because any change in T shall bring about mutually canceling changes in both R ant t). This property is illustrated below.
   
   For e.g. suppose P=10, T=4, t=2, R=5
   Initial value= P (=10)
   Value after time t (=2) x T (=4) = 10 + 2 x (5x10) = 110
   
   Now, suppose we increase the time period T by a factor of 2, so T becomes Tx2= 4 x 2= 8
   The corresponding new value of R will be = R x 2 = 5 x 2 =10
   The corresponding new value of t will be = t / 2 = 2 / 2 =1
   So, value after time t (=1) x T (=8)=10 + 1 x (10x10) = 110
   
   Now, instead of increasing, if we decrease the time period T by a factor of 2, T becomes T/2= 4 / 2= 2
   The corresponding new value of R will be = R / 2 = 5 / 2 =2.5
   The corresponding new value of t will be = t x 2 = 2 x 2 =4
   So, value after time t (=4) x T (=2)=10 + 4 x (2.5x10) = 110

Simple Interest

The formula P + t x (R x P) is also known as the formula for amount generated after simple interest. In this formula the meaning of symbols is the following.

P= Initial amount deposited in the bank
R= Rate of interest generally provided on a yearly basis and in percentage (therefore it needs to be divided by hundred whenever such is the case).
T=Time period for interest calculation. Generally this is a single year.
t=Total time in years for which interest is calculated. It is some multiple of T. It can be in fraction.

Whenever the value of R is not given for the time period T of interest calculation, or the value of t is not given as a multiple of period T of interest calculation, they need to be multiplied by some factor so as to bring them all in same time units. In such scenarios the formula for amount after simple interest becomes

P + m x t x (n x R x P)

Here m and n are the factors by which R and t need to be multiplied so as to bring them in terms of T. The factors m and n are reciprocals of each other.

Exponential Growth and Reduction

In exponential growth or reduction, both the values by which and to which the entities grow or reduce are in terms of some factor of the existing value of the entity.

For example let's assume a numerical entity P which repeatedly doubles after a certain time T, then P changes as given below.

Initial value=P
Value after time T = P x 2
Value after time 2 x T= (P x 2) x 2
Value after time 3 x T= (P x 2 x 2) x 2
Value after time t x T= P x 2^t

Instead of doubling, if the value of P would have changed by any constant factor F of existing value then after time T, then after a time of t x T the value shall be P x F^t. This is illustrated below.

Initial value=P
Value after time T value = P x F
Value after time 2 x T= (P x F) x F
Value after time 3 x T= (P x F x F) x F
Value after time t x T= P x F^t

The value of F can be any real number greater than 0. If the value of F is > 1, the quantity grows with time. If the value of F > 0 and <1 quantity reduces with time. If the value of F is 1, quantity does not change with time.

The rate at which the quantity grows or reduces by per unit value of P per unit time period T is denoted by R and can be expressed in terms of the factor F.

Rate for the Process of Growth

Initial value=P
Value after time T = P x F
Growth in value after time T = (P x F) - P = P x (F-1)
So,value after time T = P + P x (F-1) = P x (1 + (F-1)) = P x F = P'

Similarly value after time 2 x T = P' x F
Growth in value after time 2 x T = P' x F - P' = P' x (F-1)
So, value after time 2 x T =P'+ P' x (F-1) = P' x (1 + (F-1)) = P x (1 + (F-1)) x (1 + (F-1)) = P x (1 + (F-1))² = P x F²

Similarly, value after time t x T = P x (1 + (F-1))^t = P x F^t

The quantity (F-1) in the above formula is known as the rate R at which the quantity grows by in a unit Time Period T per unit value of P. The relation between Growth Factor F and Rate R for growth process is:

R=F-1
\(\Rightarrow\)F=1+R

The rate R for the process of growth can be any real value greater than 0

Also, in terms of R, value after time t x T = P x (1 + R)^t

Rate for the Process of Reduction

Initial value=P
Value after time T = P x F
Reduction in value after time T = P - (P x F) = P x (1-F)
So, value after time T = P - P x (1-F) = P x (1 - (1-F)) = P x F = P'

Similarly value after time 2 x T = P' x F
Reduction in value after time 2 x T = P' - P' x F = P' x (1-F)
So, value after time 2 x T =P' - P' x (1-F) = P' x (1 - (1-F)) = P x (1 - (1-F)) x (1 - (1-F)) = P x (1 - (1-F))² = P x F²

Similarly, value after time t x T = P x (1 - (1-F))^t = P x F^t

The quantity (1-F) in the above formula is known as the rate R at which the quantity reduces by in a unit Time Period T per unit value of P. The relation between Reduction Factor F and Rate R for reduction process is:

R=1-F
\(\Rightarrow\)F=1-R

The rate for the process of reduction can be any real value must be a real value > 0 and < 1.

Also, in terms of R, value after time t x T = P x (1 - R)^t

Properties of Exponential Growth and Reduction

1. The value of the entity can never become zero after any intervals of time in case of reduction.
2. The value of quantity after any no. of intervals of time is dependent on the time period T. A decrease in time period of growth (or reduction) always increases the growth factor. This property is illustrated below.
   
   For e.g. suppose P=10, T=4, t=2, R=5
   
   Value after time t (=2) x T (=4) = P x (1+R)^t = 10 x (1+5)² = 360
   
   Now, suppose we increase the time period T by a factor of 2, so T becomes T x 2= 4 x 2= 8
   The corresponding new value of R will be = R x 2 = 5 x 2 =10
   The corresponding new value of t will be = t/2 = 2 / 2 = 1
   So, value after time t (=1) x T (=8) = P x (1+R)^t=10 x (1+5x2)¹= 110
   
   Now, suppose instead of increasing, we decrease the time period T by a factor of 2, so T becomes T/2= 4 / 2= 2 The corresponding new value of R will be = R/2 = 5 / 2 =2.5
   The corresponding new value oft will be = tx2 = 2 x 2 =4
   So, value after time t (=4) x T (=2)= P x (1+R)^t = 10 x (1+2.5)⁴= 15006.25
   
   The above two examples clearly show that the value of the factor of exponential growth or reduction is actually dependent upon the value of Time Period T. The value of the Growth / Reduction Factor increases with a decrease in Time Period, and decreases with an increase in Time Period.

Compound Interest

The formula P x (1+ R)^t is also known as the formula for calculation of amount generated after compound interest. In this formula the meaning of symbols is the following

P= Initial amount deposited in the bank
R= Rate of interest generally provided on a yearly basis and in percentage (therefore it needs to be divided by hundred whenever such is the case).
T=Time period for interest calculation. Generally this is a single year.
t=Total time in years for which interest is calculated. It is some multiple of T. It can be in fraction.

Whenever the value of R is not given for the time period T of interest calculation, or the value of t is not given as a multiple of period T of interest calculation, they need to be multiplied by some factor so as to bring them all in same time units. In such scenarios the formula for amount after compound interest becomes

P x (1+ mR)^nt

Here m and n are the factors by which R and t need to be multiplied so as to bring them in terms of T.

Continuous Growth (or Reduction) and the value of 'e'

If we keep on decreasing the time period T for a given rate R and time interval t such that the process of growth or reduction takes place every moment.then such kind of growth or reduction is known as continuous growth or reduction. In such cases, actual factor by which exponential growth takes place is mathematically represented as

Continuous Growth Factor = (1 + R/n)^nt (such that n tends to infinity)

This growth factor is generally represented by powers of an irrational constant number 'e'. This number is also known as the Euler's Number or Napier's Constant. Its approximate value is 2.71828,18284,59045 (or 2.7183 for short). So,

e^x = e^Rt = (1 + R/n)^nt (such that n tends to infinity)

where x is the product of rate R and number of time intervals t

Since, 'x' is given as a product of rate R and number of time intervals t, 'e^x' (or e^Rt) can be defined as the maximum factor by which any quantity can grow (or reduce) in 't' number of time intervals when growth rate is 'R'.

If the value of R is 1 (i.e. 100% growth rate), 'e^x' can be defined as the maximum factor by which any quantity can grow in 'x' number of time intervals when growth rate is 100%. For this definition, 'e^x' is mathematically given as,

e^x= (1 + 1/n)^xn (such that n tends to infinity)

On binomial expansion of the RHS of the above formula we get

e^x = 1 + x + x²/2! + x³/3! +x⁴/4! +x⁵/5! ...

which is the exponential series

If the value of t is 1 (i.e. unit time interval), 'e^x' can be defined as the maximum factor by which any quantity can grow or reduce in a unit interval of time when the growth (or reduction) rate is 'x'. For this definition, 'e^x' is mathematically given as,

e^x= (1 + x/n)ⁿ (such that n tends to infinity)

On binomial expansion of the RHS of the above formula we get

e^x = 1 + x + x²/2! + x³/3! +x⁴/4! +x⁵/5! ...

which is, once again the exponential series

Based on above definitions of 'e^x' we can define the value of e as the maximum factor by which any quantity can grow in a unit interval of time when the growth rate is 100%. Mathematically it is defined as,

e= (1 + 1/n)ⁿ (such that n tends to infinity)

On binomial expansion of the RHS of the above formula we get

e = 1 + 1 + 1/2! + 1/3! +1/4! +1/5! ...

which is the exponential series for 'e'