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.

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Presentation transcript:

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.

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:

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.

Truly Exponential  Exponential growth vs. exponential decay.

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:

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