Prove e raised to i*pi = -1
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Step-by-step explanation:
This week’s post is going to be about a problem called squaring the circle. The problem dates back to the time of the Greeks. The problem itself is not the difficult part, the difficult part of the problem is that one of the restrictions of the problem is to solve it only using the technology available to the Greeks. First, this post will cover the problem itself. Secondly, the post will cover how pi relates to the problem. Finally, the post will cover how pi was proven to be transcendental.
Squaring the circle is a problem about finding a circle with the same area as a square. At first this problem may not seem hard, but not impossible. After a few minutes of thinking, it might become apparent that a circle of radius one, and a square of side length root pi would have the same are. That would be correct, but the Greeks did not have the square root of pi. The difficult part of the problem is that, the problem can only be solved with the abilities of the Greeks. That means there is no algebra, there are straight edges, and compasses. Straight edges, and compasses limit the ability of available math. With this Greek Technology the four basic functions of mathematics, addition, subtraction, multiplication, and division, can be performed. Addition and subtraction are straight forward. Adding is putting two line segments together, and subtraction is removing a line segment. Multiplication, and division require some scaling with similar triangles, but are also possible. There is one more operation that can be performed with a straight edge and a compass, and that is square rooting. Imagine a line of length x, and then a distance of one is added to it, then a semi-circle is drawn to the end of the line. If a line is drawn vertically from where the line x and line of length one meet, up to the semi-circle, the length of the line is the square root of x. Looking back at one of the solution to the problem, one might ask then why could the Greeks not square root pi? The answer is that pi is not a constructible number, a constructible number refers to a number that can made with a compass, and a straight edge.
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