11.3 The Number e

The Number e John Napier is famous

for computing tables on logarithms.

In a 1618 book of logarithm tables, William Oughtred (an English mathematician famous for his invention of the slide rule) wrote an appendix that eludes to a “magical” number that should exist.

The Number e

In 1683 Jakob Bernoulli looked at compounding interest and tried to find the value that the interest would approach as the number of compounds approaches infinity.

Jake knew that it was a number between 2 and 3, but closer to 3.

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A = P 1 +r

n ⎛ ⎝

⎞ ⎠

nt

The Number e

The number ‘e’ was first defined in 1690 in a letter written by Leibniz (he called it ‘b’).

In 1748, Leonard Euler proved ‘e’ and defined it as an infinite series that we’ll look at later.

The Number e

Let’s use the calculator to examine Bernouli’s approximation further.

Type the numbers 0 through 12 into L1.

The Number e

In L2, type the function:

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L2 = 1+1

10L1

⎛ ⎝

⎞ ⎠

10L1

The Number e

As the number ‘n’ gets really large, the value in L2 approaches e.

The Number e

It’s important to remember that e is a mathematical constant.

e is also irrational – meaning the decimal approximation cannot be expressed as a quotient of integers.

It’s just like Pi, only it’s approximately 2.72ish.

Be careful! Even though it might look like the decimal repeats (indicating it’s rational)… the decimal approximation does not repeat!

Decimal Approximation for e

Truly Exponential

The equation to the right is what we call a truly exponential function.

The number e is called the natural number because it is one of the most naturally occurring numbers on Earth.

€

f x( ) = ex

Truly Exponential

Exponential growth vs. exponential decay.

€

f x( ) = e−x

€

f x( ) = ex

Compounding Interest

Let A represent the amount of money in an account. Let P represent the principal investment. Let r represent the interest rate (APR) as a decimal. Let t represent the time in year. Then the formula for continuously compounding

interest is:

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A = Pert

Compounding Interest

Assume you have $5000 to invest. You have 2 accounts to choose from, one that pays 2.5% APR compounded continuously, or one that pays 2.75% compounded monthly. Which would you choose?

Homework

Pg. 617 # 12-17, 25-27