Maths for all

Written by: GeorgeRT

Published On: September 26, 2024

1. Area in a plane

Two of the fundamental figures in a Euclidean plane are the triangle and the circle. However, area is a concept based on a rectangle. Its area is defined thus:
Rectangle area = lw, where l is length and w is width of the rectangle.

Why? That is the definition! It is the universal assumption in this world. If it were a different formula, the numerical value of the area would be different but then all areas will be correspondingly different by a constant multiplication factor.

Anyway, assume that the area of a rectangle is lw.

Proposition
If rectangular area is length times width, then the area of any triangle is ab/2 where b is the base and a is the altitude of the triangle.
(Our high school students would benefit from trying to prove this!)

The video below shows that the area of a circle is (pi)R^2, where R is the radius of the circle.
https://www.youtube.com/watch?v=whYqhpc6S6g

It gives the area formula, because the height of the triangle in the video is R and the base length is the circumference of the outermost circular chain. But this derivation assumes that the circumference is 2(pi)r.

Question: Cut the circumference of a circle of radius r into n equal segments and prove using the area formula above that the circumference is 2(pi)r as n goes to infinity.

Thus, assuming the area, we can find the circumference and conversely. So it is natural to ask if we can find their formulas independently. Archimedes was the first to do that. His famous proof involved comparing the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius.

2. LOGARITHM
Sound and earthquakes are measured in a scale that uses the concept of logarithms.

A 20 dB noise is ten times louder than a 10 dB noise. A noisy auditorium is 70 dB. A chainsaw is 80 dB. A jet airliner taking off is 90 dB. Krakatoa volcano eruption heard from 100 miles away was about one billion times louder than that.

When the Indonesian volcano Krakatoa violently erupted in 1883 it was heard on the island of Rodrigues near Madagascar over 4800 km (3000 mi) away, and the pressure wave traveled around the world at least 3 times. This is believed to be the loudest sound in recorded history. Yet, the eruption of Krakatoa has been computed at about 0.0003% as explosive as the Chixculub meteor impact that killed the non-avian dinosaurs.

So what is a logarithm?
We will continue this.