I know when I heard about continued fractions I was a little scared off by all the computations and propositions. It got me wondering (as with almost every other math topic!) why do continued fractions matter? What made people first start to study them and find them interesting???

The education minors all know about this: motivation! So, what was the motivation behind actually studying continued fractions?

Well, according to wikipedia:

"**The study of continued fractions is motivated by a desire to have a "mathematically pure" representation for the real numbers.**

**Most people are familiar with the decimal representation of real numbers**, which may be defined by

r = sumation of (a _{i} * $10^{-i})$,

where a _{0} may be any integer, and every other a _{i} is an element of {0, 1, 2, …, 9}. In this representation, the number π, for example, is represented by the sequence of integers (a _{i} ) = (3, 1, 4, 1, 5, 9, 2, …).

**This decimal representation has some problems.** One problem is that **many rational numbers lack finite representations** in this system. For example, the number 1/3 is represented by the infinite sequence (0, 3, 3, 3, 3, ….). Another problem is that the constant 10 is an essentially arbitrary choice, and one which biases the resulting representation toward numbers that have some relation to the integer 10. For example, 137/1600 has a finite decimal representation, while 1/3 does not, not because 137/1600 is simpler than 1/3, but because 1600 happens to be a factor of a power of 10, namely 106. **Continued fraction notation is a representation of the real numbers that avoids both these problems**."