## Continue Learning.

We are **surrounded by technology** and you need **mathematics and science** to master it. Many top colleges and universities demand **good scores **to get in, and **top paying jobs** demand great math and science skills.

**And Practice!
**Here’s a site that has practice problems for standardized tests and AP exams. Go to:

*http://www.varsitytutors.com/practice-tests*

**Fun Questions:
**

Mathematics says a lot in a short space. Here are some fun facts, which anyone who knows the fundamentals of math can understand. *Can you answer some of the questions below?*

**Proofs without words: **Here’s a proof of the Pythagorean Theorem without words – **can you see why it works?**

** More advanced: **Can you see why this diagram proves the irrationality of the square root of two?

**How about a Klein Bottle?
**It has no edge, and therefore no inside or outside!

**Why?**

**How about a Sierpinski Triangle?**

You might think that this is a 2-dimensional figure, but actually, it has a dimension of 1.585, and in the limit, the area of the dark area of the triangle is zero. **Why?**

**Euclid’s proof of the infinitude of primes **– More than 2000 years ago, Euclid proved that the number of primes is infinite. Of course he didn’t count them all, **so how did he know?**

*Update – *February 2013: Curtis Cooper of the University of CentralMissouri just found the new largest prime number: 2^{57885161}-1 It’s only about 17 million digits long! If you want to see it, here it is.

**Existence of irrational numbers** – It makes no sense that irrational numbers like √2 and √3 should even exist, but they do! **What make them so strange, and why are they so important?**

**Here’s Van Koch’s snowflake curve: **As this is process is carried out indefinitely, some strange things happen:

- The length of the line becomes infinite, even though the area it encloses is finite.
- The line remains continuous at every point, but not differentiable at any point – that is, it’s just a bunch of sharp corners, with no “smooth” parts in between.
- If you look closely, you will find that this exhibits self-similarity, that is, the smaller parts have the same geometry and shape as the whole, no matter how small the part you are looking at.

By the way, for this reason, you can say that the length of the coastline of any country is infinite.

**Can you figure out why?
**

**Here’s another proof without words – see if you can figure out why this drawing proves that tan^{-1}1 + tan^{-1}2 + tan^{-1}3 = π
**

**See if you can figure out why this
drawing proves that
**

*tan*^{-1}(^{1}⁄_{3}) +*tan*^{-1}(^{1}⁄_{2}) =*tan*^{-1}(1)

**Did you know that all perfect squares are the
sums of odd numbers? Why?**

1+3 = 4

1+3+5 = 9

1+3+5+7 = 16

1+3+5+7+9 = 25

1+3+5+7+9+11 = 36

**Infinite numbers:**

- Did you know that there are different types of infinity?
- Did you know that the number of points on a line is the same as the number of points in a plane? In fact, it’s the same as the number of points in any
*n*-dimensional space (for finite*n*). - Did you know that most of the real numbers are irrational?

*Math problems for younger children:*

*Math problems for younger children:*