## Two Little Circles

In the diagram shown we see a big circle of radius R that is tangent to both the x and y axes. What is the radius of the little circle to its southwestern corner? Express its radius in terms of R. (The little circle is also required to be tangent to the axes. The diagram is draw for a circle of radius 2 but we’re working with a general radius R.)

After having solved this problem, the next question will be this:

You can easily see that we can repeat the process by looking at the southwestern circles for ever and ever because there will always be a gap, and you get smaller and smaller circles with small radii. What if you add up all their radii? Do they add up? If so to what number do they add up, in terms of the original radius R. If they don’t add up, why don’t they?

## Comparing huge numbers

Comparing huge numbers is often times not easy since you practically cannot write them out to compare their digits. (By ‘compare’ here we mean telling which number is greater (or smaller).) So it can sometimes be a challenge to determine.

Notation: recall that N! stands for “N factorial,” which is defined to be the product of all positive whole numbers (starting with 1) up to and including N. (E.g., 5! = 120.) And as usual, M^{n} stands for M raised to the power of n (sometimes also written as M^n).

Here are a couple examples of huge numbers (which we won’t bother writing out!) that aren’t so easy to compare but one of which is larger, just not clear which. I don’t have a technique except maybe in an *ad hoc* manner.

In each case, which of the following pairs of numbers is larger?

(1) (58!)^{2} and 100!

(2) (281!)^{2} and 500!

(3) (555!)^{2} and 1000!

(4) 500! and 10^{1134}

(5) 399! + 400! and 401!

(6) 8^{200} and 9^{189}

(The last two of these are probably easiest.)

Have fun!

## Escher Math

You’ve all seen these Escher drawings that seem to make sense locally but from a global, larger scale, do not – or ones that are just downright strange. We’ll it’s still creative art and it’s fun looking at them. They make you think in ways you probably didn’t. That’s Art!

Now I’ve been thinking if you can have similar things in math (or even physics). How about Escher math or Escher algebra?

Here’s a simple one I came up with, and see if you can ‘figure’ it out! 😉

**(5 + {4 – 7)2 + 5}3**.

LOL! 🙂

How about Escher Logic!? Wonder what that would be like. Is it associative / commutative? Escher proof?

Okay, so now … what’s your Escher?

Have a great day!