Pythagorean Theorem: Practical use #1

Long ago in Ancient Greece, an influential philosopher and mathematician named Pythagoras made an important discovery that would affect our perception of the world.

This discovery, known as the Pythagorean Theorem states that: The sum of the squares of two of a right triangle is equal to the square of the hypotenuse of that same right triangle. Or in equation form:

a2 + b2 = c2

By using this equation we can find the length of any side of a right triangle provided we have the lengths of the other two sides. For example:

Likewise, this same theorem can be used to find unknown lengths of diagonals and sides in many polygons such as squares and rectangles. See the example below:

This is just one of many ways that the Pythagorean Theorem can be used. How might someone use this mathematical principle in the real world?

Surface Area of Solids

Trying to wrap a present? Using just the right amount of wrapping paper requires a knowledge of solids and surface area. You need to know the shape of box, container, etc. that you are wrapping and the correct surface area formula in order to determine how much paper you need to cover your gift.

 A solid is essentially comprised (made up of) several smaller 2D shapes. When we take the total surface area of a solid we are actually adding up the areas of all the 2D shapes.

 

The cube above is composed of 6 identical squares. The area of each square (on the right) is the same, so total surface area of the cube (on the left) is sum of the areas of all six squares. So if the sides of the square were 2cm, the area of one square is 2 x 2 = 4 sq. cm. The total surface area would then be 6(2 x 2) or 2x2 +2x2 +2x2 +2x2 +2x2 +2x2, which equals 24 sq. cm.

Below is a chart of sample solid shapes:

Rube Goldberg: The man with the crazy machines

Rube Goldberg (1883-1970) was a Pulitzer Prize winning cartoonist, sculptor and author. 

Reuben Lucius Goldberg (Rube Goldberg) was born in San Francisco on July 4, 1883. After graduating from the University of California Berkeley with a degree in engineering, Rube went on to work as an engineer for the City of San Francisco Water and Sewers Department. 

After six months Rube shifted gears and left the Sewers Department to become an office boy in the sports department of a San Francisco newspaper. While there he began to submit drawings and cartoons to the editor until he was finally published. Rube soon moved from San Francisco to New York to work for the Evening Mail drawing daily cartoons. This led to syndication and a national presence – and the rest is history. (Read more about Rube Goldberg here: http://www.rubegoldberg.com/about).

He is most famous for his "machines" cartoons like the one below:

His cartoons would actually inspire engineers and other tech savvy people to invent the machines he created in his comics. Below is an example of a video that utilizes a Rube Goldberg Machine:

How does this work? Well, through our understanding of the laws of conservation of momentum and energy we know that what you start with is what you end with. Hence, each device transfers energy and momentum to next device in the system to ultimately accomplish one final task.

Number Systems and Binary Code

We know our numbers right? 0, 1, 2, 3, 4.. and so on. But did you know that there are other ways of representing numbers?

In ancient Rome, Roman numerals were used as the accepted number system. Combinations of the letters I, V, X and so on were used to represent the numbers 1, 5, 10, etc. (See below).

Can you see the pattern with Roman numerals? How about a test? What would the current year be in Roman numerals? How about the year this movie came out?

The system we use is called decimal form and it goes from 0 to 9. Another system known as Binary, uses groups of 0s and 1s to represent every known number. Ever wonder how numbers are transformed into binary form? Check it out here!

Decimal works with numerals 0-9. When the number exceed 9 the numerals move one place value to the left and begin counting up again. Binary uses the numerals 0 and 1. When the number exceeds 1 the numerals also move one place value to the left. Can you use the chart below to make comparisons between the number systems?