Today is a special day. You see, today marks the very first day of the rest of my life. In order to mark this special occasion, I am starting a blog. This blog post will forever be remembered (or forgotten) as the first among many.
I have decided to dive right into things, and begin with one of my favorite mathematical formulas. This formula is incredibly useful, the concept is simple, and it is easy to remember. It is known as the Pythagorean Theorem.
What is it?
The Pythagorean theorem is a formula which directly relates to right triangles. A right triangle is composed of a 90 degree angle, also known as a square angle, and two smaller angles. The theorem gives a relation for each of the three sides. Using the formula, if you know the length of two sides of the triangle, you can find the length of the third side.
Generally, the two sides which form the right angle are assigned the variables A and B, and the side opposite the right angle is assigned the variable C (look at the picture for a good visualization). The formula states that A squared plus B squared equals C squared.
Where did it come from?
Although named for Pythagoras, the Pythagorean theorem’s history is debated. In all likelihood, Pythagoras was not actually the first to discover this property, nor the first to write about it. Some historians believe the Babylonians may have known about the theorem, accounts of the Pythagorean Theorem in India may go as far back as 8000 BC.
The methods of proof for the theorem are numerous and vary widely. A great number of these are simple geometric proofs, and require nothing but some algebra and geometry to understand. If you’re interested, there is a long list of them at Wolfram MathWorld.
Why is it important?
The theorem comes up surprisingly often in higher math. You’ll see it all the time, especially in calculus and physics. One of the most common usages of it arises in vectors. A vector is merely a direction and a magnitude, usually visualized as an arrow or a line.
In the diagram below, you can see two vectors, A and B. When you put them end-to-end, they add to vector C. Each vector is composed of two components, one vertical and one horizontal component. If you add the horizontal length of A to the horizontal length of B you obtain the horizontal length of C, and the same applies to the vertical lengths. When you have the horizontal and vertical lengths of C, you simply plug them into the Pythagorean theorem to find the length of vector C. Thus, 152+92=C2. Solving (with a calculator) gives C=17.5. It’s a surprisingly simple process, with an immense number of applications.