'e'

The Mathematical Constant

Narendran SairamYale

AlgebraJanuary 4th , 2008

Table of Contents

I. 'e': The Mathematical Constant.....................................................................................................3

II. Appendix A..................................................................................................................................5

III. Appendix B: Example of continuous compounding...................................................................6

IV. Appendix C: Example of Application of Natural Log ( Measuring Age of Rocks)...................7

History:

The constant 'e' is a very important, irrational number in mathematics alongside the additive and

multiplicative inverse of 0 and 1, the square root of negative 1(i) and pi (the ratio of circumference to

diameter.) Through the years 'e' has been referred to as Euler's number after the Swiss mathematician

Leonard Euler, as Napier's constant after the Scottish mathematician John Napier who introduced

logarithms, as a logarithmic constant and a natural logarithmic base. It is approximated to be:

2.71828 18284 59045 23536...

This number was discovered accidentally by many mathematicians in. It's use is believed to

begun in the 1600s. It was introduced by John Napier who also introduced logarithms to the world.

John Napier published this constant in 1618 in an appendix table. He had the number but did not

recognize it as 'e'. The number was just a logarithmic base in one of his tables. The actual discovery of

the number is credited to Jacob Bernoulli. Bernoulli was trying to find the value of a derivative

expression(see appendix A to see the expression) when he came upon the number. It was later found

that the expression actually equaled 'e'. He used the letter c to denote the number. Later in 1727

Leonard Euler began using 'e' to denote the constant which is why e is also sometimes reffered to as

Euler's constant.

One can express 'e' using the following equation:

(1+x)(1/x)

'E' is the number that this expression approaches as x becomes smaller(See appedndix D to see

the behavior of this expression). So the smaller the x value the more accurate 'e' will be.

Uses

Following are the typical uses of the number 'e'

'e' is used in calculating continually compounding interests. If an principal amount is

compounded often then the a greater amount of interest will be earned. However there is a limit

to the amount of interest that can be earned even if no matter how many times the principal is

compounded(For example see appendix B). That particular limit is 'e'.

The resulting amount of the compounded principal is given by the equation:

A=Pert

where A is the final amount, P is the principal, r is the rate of interest and t is the number of years.

'e' is also most commonly known as a logarithmic base. Logarithms to the base 'e' are called

natural logs. Natural log is used in geology to measure the age of rocks by using atomic clocks.

They do this by measuring the amounts of potassium 40 and argon 40 in a rock. The age is

found using the following formula:

t={ ( 1.26 x 109 ) x ln[ 1+8.33 ( A/K )] } / (ln2)

where A and K are atoms of argon 40 and potassium 40 respectively.(see appendix C for example)

Natural log is also used to model Global Temperature increase. Carbon Dioxide in the

atmosphere traps heat from the sun. The additional solar radiation trapped by the carbon dioxide

molecules is called radioactive forcing. It is measured in watts per square meter. In 1896

Swedish scientist Svante Arrhenius modeled radioactive forcing R caused by additional

atmospheric carbon dioxide using th following formula:

R=k x ln (C/Co)

where Co is the preindustrial amount of carbon dioxide and C is the current level of carbon dioxide. K is a constant.

Appendix A

The expression that Bernoulli was trying to find a value was :

Later it was found that this expression indeed equaled the constant 'e'.

Appendix B

Example of continuous compounding

Problem: Suppose $5000 is deposited in an account paying 3% interest compounded continuously for 5 years, then find the total amount on deposit at the end of 5 years.

Solution:

A =Pert

=5000e0.03(5)

=5000e 0.15

≈5000(1.61834)

=5809.17

Answer: The continuous compounding of $5000 in an account that pays 3% interest, at the end of 5 years would result in $5809 and 17 cents.

Appendix C

Example of Application of Natural Log ( Measuring Age of Rocks)

Problem: If a granite specimen has 7 molecules of argon and 6 molecules of potassium, find the age of the piece of granite.

Solution:

t ={ ( 1.26 x 109 ) x ln[ 1+8.33 ( A/K )] } / (ln2)

A =7

K =6

A/K = 7/6 = 1.166666666666666

t ={ ( 1.26 x 109 ) x ln[ 1+8.33 ( 1.166666666666666 )] } / (ln2)

={ ( 1.26 x 109 ) x ln[ 1+0.19444444444444444444444444444444] } / (ln2)

={ ( 1.26 x 109 ) x ln[ 1.19444444444444444444444444444444] } / (ln2)

={ ( 1.26 x 109 ) x 0.17768117723745242184788779658444 } / (ln2)

=223878283.31919005152833862369584 / (ln2)

=223878283.31919005152833862369584 / 0.69314718055994530941723212145818

=322988089.10732982573059632824049

Answer: A piece of granite with 7 molecules of argon and 6 molecules of potassium is approximately 322988090 years old.

Appendix D

Behavior of (1+x)(1/x) as x gets smaller.

X (1+x)(1/x)

1 2

0.5 2.25

0.25 2.414

0.125 2.5678

0.0625 2.63792

...... ......

As one may observe the smaller x gets the closer the value of the expression get to 'e'