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### Top 10 Greatest Moments in Math History: Infinity

Moment 8: Infinity
Just like zero, the concept of infinity was at first rejected [14]. But also like zero, infinity was eventually accepted as a crucial piece of mathematics in the 1600s. Infinity needed to be embraced before calculus could be done.

The irrational number sqrt(2), the diagonal length through a square with side length 1, was possibly the first time our mathematical ancestors were faced, and had to accept, infinity. sqrt(2) is irrational so its decimal neither terminates nor repeats. Greek philosopher Hippasus, possibly thrown overboard by the Pythagoreans who believed whole numbers ruled the universe [15], was the first to declare that sqrt(2) was irrational. Hippasus and the Pythagoreans lived long before calculus, but finding that some numbers did not terminate was a precursor to thinking in the infinite.

Before calculus, some creative and time consuming methods were used to work around another infinite number, pi, to find the area of circles. Mathematicians would scribe regular polygons onto the outside and inside of a circle and then take the average of the two polygons’ areas to approximate the area of the inner circle. Archimedes of Syracuse, during his lifetime 2200 years ago, was the first to employ this method [14] and eventually realized that the more sides his two polygons had the closer the two calculated areas were and therefore the more accurate the estimation of the circle’s area. He began to think about limits. If there were an infinite number of sides to his polygons the area the two enclosed would be zero and therefore the exact area of the circle [14]. Without this idea of “approaching zero”, our modern day calculus would have had no chance of coming into being.

Works Cited:

[14] Rooney, Ann, The Story of Mathematics, Arcturus Publishing Company, London, United Kingdom, 2009

[15] Olley, Robert H., “Was Hippasus Pushed? (and Other Mysteries Of Mathematics)”, blogged on August 26, 2008, http://www.scientificblogging.com/beamlines