## Lesson 12: Functions. The Last Joke (III)

June 24, 2012

First some notes to appreciate the joke:

Log means  tronco but it also represents the abbreviation of the logarithmic function to the base 10 you will study in your fourth year.

Timber  means viga madero para construcción.

To hesitate means vacilar, dudar

To aid and abet somebody  means instigar y secundar a alguien (en la comisión de un delito)

This is the joke:

A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it’s wrong, but also because he doesn’t want to be sanctioned for aiding and abetting.
His classmate calms him down: “Nobody will be able to trace my homework to you: I’ll be changing the names of all the constants and variables: a to b, x to y, and so on.”
Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying.
After the deadline, the student asks: “Did you really change the names of all the variables?”
“Sure!” the classmate replies. “When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x+1, I called it the timber of x+1…”

April 25, 2012

## A bit of History about Pi

Mainstream historians believe that ancient Egyptians had no concept of π

As early as the 19th century BCE, Babylonian mathematicians were using π ≈ 25/8, which is about 0.5 percent below the exact value

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BCE) that led to a fractional approximation of π ≈ 339/108 (which equals 3.13888…, which is correct to two decimal places when rounded, or 0.09 percent below the exact value).

Recall an n-gon is a polygon with n sides.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons which was used around 250 BC by Greek mathematician Archimedes.

Archimedes of Syracuse

(Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC)

This polygonal algorithm remained the primary approach for computing π for over 1,000 years. Archimedes computed upper and lower bounds of π by drawing regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons as you can see below.

By using the equivalent of 96-sided polygons, he proved that :

223/71 < π < 22/7.

Archimedes’ upper bound of 22/7 may have led to widespread belief that π was equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his book Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.

(From Wikipedia, several articles)

Nowadays we know hundred of thousands of decimal figures of PI, but using computers!

## A  joke

Time to relax

See you tomorrow!

## Lesson 3: Curiosities and Jokes (IV)

November 19, 2010

### The number Pi

The mathematical constant pi (π) is defined as the ratio of a circle’s circumference to its diameter. It is also the ratio between a circle’s area and its radius squared. It is approximately equal to 3.14159, but has been calculated to over one trillion digits.

### A Pythagoras’s student

The ancient Greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction), but one of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus’ “irrational numbers” and so Hippasus was thrown overboard and drowned!

# Jokes

We will finish with

Popular Pi Jokes

• Q: What do you get when you take the sun and divide its circumference by its diameter?
• A: Pi in the sky.
• Mathematician: Pi r squared”
• Baker: No! Pies are round, cakes are square!

Click on The “Pi” Joke and you will watch a funny animation.

Here you are pi song

If you are in the mood you can listen the complete version here .

See you!!

## Lesson 1: Jokes And Games On integers (IV)

October 1, 2010

### Now time to relax. So, I will tell you a joke.

Several students were asked the following problem:

Prove that all odd integers are prime.

Well, the first student to try to do this was a math student. Hey says “hmmm… Well, 1 is prime, 3 is prime, 5 is prime, and by induction, we have that all the odd integers are prime.”

Of course, there are some jeers from some of his friends. The physics student then said, “I’m not sure of the validity of your proof, but I think I’ll try to prove it by experiment.” He continues, “Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is an experimental error, 11 is prime, 13 is prime… Well, it seems that you’re right.”

The third student to try it was the engineering student, who responded, “Well, actually, I’m not sure of your answer either. Let’s see… 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is …, 9 is …, well if you approximate, 9 is prime, 11 is prime, 13 is prime… Well, it does seem right.”

Not to be outdone, the computer science student comes along and says “Well, you two sort’ve got the right idea, but you’d end up taking too long doing it. I’ve just whipped up a program to REALLY go and prove it…” He goes over to his terminal and runs his program. Reading the output on the screen he says, “1 is prime, 1 is prime, 1 is prime, 1 is prime….”

Do you understand every single word or expression?

### Here you are a couple of games, practise and have fun at the same time!

Ordering numbers

Integers Jeopardy Game – This game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. You can play it alone or in teams.