Organizing our work into an ordered table with


rows (r) of numerators (n) and denominators (d): r


1 2 3 4 5


= nr−1


n 1 3 7


17 41


+ 2dr−1


d 1 2 5


12 29


we see that by the fourth and fiſth rows, the fractions are getting quite close to √2. To see why, notice that for any row r > 1, nr


, and dr


Furthermore, for a given row r, if (nr then (nr+1


)2 − 2(dr+1 −1, then (nr+1 )2 − 2(dr+1 )2 − 2(dr


dd 2


n 22 −=±


 − 


 


 +  


n d


2 2 )2 = ±1. 1 = nr−1 )2 = −1. Similarly, if (nr + dr−1 )2 − 2(dr . )2 = 1, )2 − 2(dr )2 = )2 = 1. So, this table provides a


quick and efficient way to produce lots of values of n and d satisfying (nr


To see what this has to do with finding a ratio close


to √2, note that we can rewrite this last equation as .


Te leſt side of this equation can be factored to give n d


Since n d 21


 22 1 d


 


n d


−< 2 d


1  =± . 2


+> (it is close to 3), this means that .


If √2 were a rational number, say √2 = p/q, then the in- equality above makes no sense once d > q, unless n/d = p/q, but this is not true for our choice of n/d. Somehow the fact that there are lots of rational numbers very close to √2 tells us that it is an irrational number.


Participants model the spinning of a gear. American Institute of Mathematics · Summer/Autumn 2016 · MTCircular 15 ⊇


Some numbers have extraordinarily good rational


approximations, much better than √2. One example is π . Many people are familiar with 22/7, which is within just over one thousandth of π . Even more extraordinary is the fraction 355/113, which is within a few ten millionths of π . Te subject of how close a number can be approximated with fractions of limited-size denominator has a very rich history and has occupied some of the best mathematical minds for centuries. Other interesting observations and ideas that came up during the Circle session included:


• Participants can model the spinning of gears by forming two circles.


• Determining what speeds will be comfortable on a bike is a multi-layered ratio problem: You need to consider the ratio of front teeth to rear teeth, but also the ratio of the circumference of the tire to its diameter, and then your units of distance and time in order to find an answer.


• Te technical details of how a bike works, though accessible, are not intuitive. (For example, it gets easier to pedal when you go to fewer teeth in the front but to more teeth in the rear!) Understanding how a bike works was, in a lot of ways, the biggest achievement of the session! ⊆


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