Appendix A

Classification of Real NumbersDedekind and Cantor constructed the real numbers in 1872. Then in 1882 and 1884 Linder-mann and Weierstrass were able to classify certain important real numbers like logarithms,sines and cosines and π . This was an amazing achievement.

Recall that algebraic numbers are those which are roots of a polynomial with rationalor integer coefficients. (Note that if the coefficients are rational, you could simply multiplyby the product of the denominators and reduce to one which has all integer coefficients.)This of course includes many complex numbers. For example, x2 + 1 has roots ±i. Thealgebraic numbers include all rational numbers. For example the root of mx−n = 0 is n

m .Numbers which are not algebraic are called transcendental.

Most numbers are transcendental. This follows from Problem 14 on Page 51 and Prob-lem 16 on Page 51. However, it is very difficult to show that a particular number is tran-scendental. Lindermann and Weierstrass made some progress on this in 1882 and 1884. Inparticular, Lindermann showed that π is transcendental. This solved the ancient problemabout whether one could square the circle. If you start with the unit circle, its area is π andthe question was whether you could construct with compass and unmarked straight edgeonly, a square of area π .

You can’t do it because all constructible numbers are algebraic. In fact they all involveroots of quadratic polynomials and linear polynomials which is essentially why you cannottrisect an arbitrary angle either, such as a 60◦ angle. If you could square the circle, then youwould end up needing the sides of the square to be

√π which, if algebraic, would require

π to also be algebraic. This is explained below. It turns out that doing algebra to algebraicnumbers results in algebraic numbers.

This theorem of Lindermann is a very significant result and it seems to be neglectedthese days. This is why I am including a treatment of it which I hope will be somewhatunderstandable. It is very technical however. I have not seen the original proof of this the-orem. I suspect it is not what is about to be presented which depends on work of Steinbergand Redheffer dating from 1950. However, the use of the symmetric polynomial theoremused here seems an interesting way to proceed. This symmetric polynomial theorem is veryimportant for its own sake.

I will use the concept of a vector space and a basis for it in what follows. A beginninglinear algebra course which is not restricted to row operations should contain sufficientbackground. However, if you have not seen the notion of an abstract vector space andbasis, it would be better to learn this first. It is in any of my books on linear algebra.

A.1 Algebraic Numbersa is an algebraic number when there is a polynomial p(x)≡ xn +an−1xn−1 + · · ·+a0 witheach ak rational having a as a root. Out of all such polynomials, the one which has n assmall as possible is called the minimum polynomial for the algebraic number a and n iscalled the degree of this algebraic number. This minimum polynomial is unique. Indeed, ifp(x) and p̂(x) are two such, then by the division algorithm,

p̂(x) = p(x)q(x)+ r (x)

where r (x) has smaller degree than n or is 0. The first case cannot happen because r (a) = 0and so r (x) = 0. Now matching coefficients shows that p̂(x) is a multiple of p(x) and so

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