High School Math

Solving Inequalities

Take a look at this interesting math problem:

Solve  (2x-9)(x-4) ≥ 0

For this inequality to have a value that is greater or equal to 0, it is also necessary for the product of both parts of the equation (the values in parenthesis) to be 0 or greater. This is possible in two cases:

1) Both sets of parentheses equate to 0 or more, as the product of the two values that are greater or equal to 0 is also always greater than or equal to 0, or -

2) Similarly, both sets of parentheses equate to less than or equal to 0, as the product of two negative or zero values is also always greater than or equal to 0.

Consequently the final result is the result of either case, but what is interesting here is that the final result is the union of the outcome of both cases, and at the same time (during the solving of the individual cases) both values must be greater or equal to 0 (per case 1) or less or equal to 0 (case 2).  Let's look at the above example:

Case 1:

2x-9 ≥ 0 and  x-4 ≥ 0

We have a value, x, that is greater than or equal to 9/2 and also a value of x that is greater than or equal to 4.  The final result therefore is x ≥ 9/2

Case 2:

2x-9 ≤ 0 and   x-4 ≤ 0

X is equal to or less than 9/2; x-4 is equal to less than 4.  Therefore the final result is  x ≤ 4

This means the solution to the problem example is either of these results: x ≤ 4 or x ≥ 9/2.

The Number Pi

Background

Those of you who have done maths at high school level or higher have encountered the special number called pi or π a few times. Whenever you want to calculate the circumference or area of a circle, it is used in the well-known formulas¹:

Circumference = pi * Diameter

Area = pi * Radius * Radius

It also appears in a number of other formulas, such as the volume of a sphere.

History

The earliest accounts of the number for pi appear to be from the Babylon culture, around 1900 – 1680 BC² . Scientists measuring the diameter and the circumference and the diameter realised that the quota between them was always the same, they got 3.125. The Egyptians and other early cultures then made advancements in the approximations, but it was not until the great Archimedes of Greece ∼287-212 BC, that the real breakthrough came. He used great trigonometric skills² to get it right within 99.9 % .

In 1761, it was proven by a Swiss mathematician that the number is irrational, i.e. the number of decimals are infinite³. This era also provided the name from the 16th letter in the Greek alphabet, π. The title was given by fellow great scientist and maths expert Leonhard Euler.

Number of digits

3,1415 and then, how many digits can you remember? Modern day computers have been able to calculate it up to 12 trillion digits, but the record, according to Guinness Book of Records, for a human stands at 67890, set by someone in China in 2005.  

Pronunciation

There is a bit of discussion as to how the letter pi should be pronounced. The Greek sound is like ”pee”, but the English language uses the i sound as in ”pie”. Most other languages have stuck to the original Greek version ⁴ .

Further reading:

If this has sparked your interest in the intricacies of pi, I would recommend A History of Pi from Czech-American Petr Beckmann⁶. That should give you a much more comprehensive understanding of the importance of this ”magical” number.

References:

[1] http://www.mathgoodies.com/lessons/vol2/circumference.html

[2] http://www.livescience.com/29197-what-is-pi.html

[3] http://en.wikipedia.org/wiki/Pi

[4] http://english.stackexchange.com/questions/11363/why-are-greek-letters-pronounced-incorrectly-in-scientific-english

[5] http://www.eveandersson.com/pi/books

[6] Beckmann, Petr: A history of Pi. Golam Press, 1970.

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