### What is the square root of pi?

School: Maine-Endwell Middle School
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Riley Assistant Professor

Research area: Geometric group theory
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Before we talk about  √π, let's talk about π itself. Over three thousand years ago, many cultures discoved that the fraction of the circumference of a circle to its diameter was always the same, regardless of the size of the circle. That fraction is called π. The definition is easy enough to say, but how do we write π down? An ancient proglem is to write down π as a decimal. To estimate π, you could draw a large circle, measure the circumference and diameter of the circle and take their ratio. If you were extremely careful, you could write down a couple of decimals of π.

But getting even three or four decimals by measuring is difficult. Instead, using some deep mathematics (an area called calculus), scientists have derived formulas to help them write down π. The first person to find such a formulas was the famous Greek scientist Archimedes (also known for running through the streets naked yelling "Eureka!"). Over 2200 years ago, Archimedes used his formula to get the first four digist of π. Until 1600, mathematicians could only find the first 26 digits: 3.14159265358979323846264338327950288.  Today, with better formulas and using supercomputers, we know 10,000,000,000,000 digits of π!

But even knowing ten trillion digits is still just an approximation. You could spend your entire life writing down π, but you could never finish, because π cannot be written with finitely many decimals. In fact, most numbers are just like π: they are impossible to write down completely. So what is √π? Well, it is the only positive number which, when multiplied by itself, makes π. I can tell that √π is approximately 1.7724538509. But I cannot tell you exactly what √π is because, just like π, it cannot be written with finitely many decimals. In fact, √π is important for another reason. For thousands of years, mathematicians have tried to "square the circle," which means to draw a square with area equal to π, using just a compass and straightedge. A square with area π has side lengths √π. Many people tried and failed to square the circle. It wasn't until 1882 that mathematicians proved that squaring the circle is impossible. That didn't  stop a cooky amateur mathematician named Edwin Goodwin to write Indiana House Bill #246, which was a law declaring his 'proof' that the circle could be squared was correct. As part of that bill, Indiana tried to declare that π is officiall either 4 or 3.2! Fortunately, even though the Indiana House passed the bill, the Senate realized that no one could legislate mathematical truth, and killed the Indiana π Bill.

Last Updated: 9/18/13