Thread:Surds (1)

I find the introduction as to why we need to look at surds as unconvincing. In my opinion, a major reason is the following: Rational numbers can be looked upon in two different ways: as fractions and through their decimal expansions (or expansions in other bases). When we want to operate on rational numbers, it is easier to do it through their fraction representation. For example, 1/2 + 1/3 = 5/6 is easy, while doing 0.5 + 0.33333 = 0.833333 is almost as easy. However, doing 1/7 + 1/17 = 24/119 is very easy, doing 0.142857142857... + 0.05882352941176470588235294117647... = 0.20168067226890756302521008403361 is far more difficult. Even worse problems arise if we wish to multiply or divide two rational numbers using the decimal representation. (Try 1/7 by 1/17: the answer is 17/7 = 2.42857142857142857... What makes life easy for us with rational numbers is the representation as fractions.

With irrational numbers we are in trouble. The ONLY way to calculate $$\pi2$$ is to use the decimal expansion (or expansion in other bases or through continued fractions or as infinite products). In each case, the expansion involves an infinite number of digits or terms. Try for example to show that $$\cos(\pi)=1$$ using the decimal expansion of $$\pi$$ and the Taylor series expansion of $$\cos(x)$$!

There is, however, one group of irrational numbers which can be handled with ease, namely quadratic surds. We can handle surds without ever writing down infinite terms or digits. We can even conclude that $${{1}\over{1+\sqrt (2)}} = \sqrt(2)-1$$! Imagine establishing this if we did not know how to deal with surds!--B r sitaram 03:15, 1 May 2008 (UTC)