The important thing about doing most math is that we really want more than the answer, we want to generalize the answer, to find the answer for any problem like we're workin on.

Ian, at age 11, came to me with the problem- what would his Dad pay in monthly installments on a house worth $10,000 at a 10% annual interest, with a 30 year mortage- he really wanted a number for an answer. This got us working on the simpler problem of investing and finding the interest.

I must say that I don't know everything, and make a lot
of mistakes, in spite of what my students might believe. So I wasn't able
to give Ian a quick answer, but he and I worked very hard to solve his problem.
We started this way.

**1. Simple interest. **Find the amount you have in the bank after 2
years if you put in $1, at a 6% annual rate of interest. (It will always be an
annual rate of interest with all the problems below.)

The Interest = Principal * rate * time

I(2 yr) = $1 * .06 * 2 = $.12 and the
amount (A) you would have after 2 years

A = P + I = $1 + $.12 = $1.12 Notice that the interest is **not added each year
to get a new principal.
2. Compound interest and 3, and 4. leading to a very important infinite
sequence.** Here the interest is added after each compounding period.

Find the amount you have in the bank after 1 year, with a principal of $1, at @6%, compounded

The interest earned during the first (1/2) year = I(first 1/2 year) = $1*.06*(1/2) = .06/2 ($.03)

The Amount you have in the bank after the first (1/2) year

A = P + I =

I(2nd 1/2 year) =

A(after 2 (1/2)years)= P + I =

Factoring out the term

A(after 2 (1/2)years) =

What would the amount you have in the bank after 1 year, putting in $1, at 6%, compounded quarterly (

What would the amount you have in the bank after 1 year, putting in $1, at 6%, compounded monthly (

What would the amount you have in the bank after 1 year, putting in $1, at 6%, compounded daily (

What would the amount you have in the bank after 1 year, putting in $1, at 6%, compounded

What would the amount you have in the bank after 1 year, putting in $1, at 6%, compounded

New things, not in my books! See the answer page
to find out what Kirsten and I did with graphing and exploring **(1
+ .07/x) ^{x}**

**5. Naming e^{x}
as an infinite series.**

Using the binomial expansion from chapter 9, the first 5 terms of (A + B)

**6. Patterns** with **i**

**7. Leading to the most exciting single mathematical statement, that contains
the 5 most important numbers in mathematics!**

This is not
in my books, but happens regularly.When a student works on logs, as in Ch
13., I'll have them find ln(-1). Looking at the graph at the right, you can see
that ln x is the inverse function of **e**^{x}
(they are mirror images of each other in the line y = x), but also there is no
ln of a negative number! Now I'll have the student find ln(-1) in *Derive*
and lo and behold we get ln x = **i***Pi, which is not
a real number but an imaginary number... and that shouldn't show up on our graph
at the right. That means that **e**^{i*Pi}
= -1 by definition of logs. Adding 1 to each side we get

In my books I get this same statement using

Ian played with his calculator and powers of powers, during physics class. He came up with the following expression. If x goes to infinity what does this go to? Click here to see how Ian figured this out.

Continue the graph. What do you notice?

Where is the point for

Notice we get another spiral! (See chapter 6 for other spirals). If you double the angle, what happens to the length of the radius vector?

**12. Back to Ian's problem.**

Ways young people have solved these kinds of problems

To download Don's materials

To choose sample problems from other chapters

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