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Since ancient Greece, people tried to draw a square with exactly the same area as a given circle using only a straight edge and compasses. In 1882 a 30-year-old German mathematician, Ferdinand von Lindemann, proved it cannot be done. This post explains the difference between irrational and transcendental numbers, why that difference decides the circle problem, the mistakes students make with these words, and what you can take from how the proof was found.
The MacTutor history of squaring the circle names Anaxagoras as the first mathematician on record to attempt it, and notes a joke about it in a play by Aristophanes from about 414 BC.
Who was Ferdinand von Lindemann?
According to his MacTutor biography at the University of St Andrews, Lindemann was born in Hannover on 12 April 1852 and died in Munich on 6 March 1939. He studied at Göttingen, Erlangen and Munich, took his doctorate under Felix Klein in 1873, and then visited mathematicians in England and France, including Charles Hermite in Paris.
He taught at Freiburg, then Königsberg from 1883, and Munich from 1893 until the end of his career. MacTutor notes he supervised more than sixty doctoral students, among them David Hilbert, whose famous list of problems we covered in David Hilbert and the problems that shaped a century.
What is the difference between irrational and transcendental?
You know from school that an irrational number cannot be written as p/q with integers p and q. √2 is irrational. But √2 is still a root of a simple equation, x² − 2 = 0, with whole-number coefficients.
A number that is a root of some polynomial equation with rational coefficients is called algebraic. √2 is algebraic. So is the golden ratio, a root of x² − x − 1 = 0. A transcendental number is one that is not algebraic: no polynomial with rational coefficients, of any degree, has it as a root.
MacTutor records that Johann Lambert proved π irrational in 1761. That ruled out π = p/q, but left open whether π might be a root of some polynomial. Lindemann closed that door.
Why does transcendence settle squaring the circle?
Take a circle of radius 1. Its area is π. A square with the same area needs side √π, which is about 1.7725.
With ruler and compasses you can start from a unit length and add, subtract, multiply, divide and take square roots of lengths you already have. Every length you can reach this way is a root of some polynomial with rational coefficients, so it is algebraic. As MacTutor puts it, Lambert's result alone did not decide the matter, because some irrational numbers, such as √2, can be constructed.
If you could construct √π, you could square it with the same tools and construct π. That would make π algebraic. Lindemann proved π is not algebraic, so √π cannot be constructed, and the circle cannot be squared.
How did Lindemann prove it?
MacTutor explains that Hermite had proved in 1873 that e is transcendental, and that Lindemann, after visiting Hermite, used similar methods. The key extra step was Euler's identity:
e^(iπ) = −1
Lindemann's paper, Über die Zahl π in Mathematische Annalen in 1882, rests on a stronger statement about e: raise e to any non-zero algebraic power and the answer is never algebraic. Now suppose π were algebraic. Then iπ would be algebraic too, since i is a root of x² + 1 = 0 and products of algebraic numbers are algebraic. So e^(iπ) would have to be non-algebraic. But e^(iπ) = −1, which is plainly algebraic. The contradiction means π cannot be algebraic.
MacTutor is frank that many historians feel Hermite did most of the hard work and missed the final step, while crediting Lindemann for spotting the trick Hermite did not see.
For earlier chapters in the story of π, read our posts on Zu Chongzhi's value of pi and Jamshid al-Kashi's calculation of pi.
What mistakes do students make with these ideas?
"π = 22/7." 22/7 is a convenient approximation. Lambert's proof shows no fraction equals π exactly.
"Irrational means it can't be drawn." √2 is the diagonal of a unit square. Irrational lengths can be constructed; transcendental ones cannot.
"Squaring the circle is unsolved." It is solved, in the negative. Proving something impossible is a complete answer.
"Transcendental numbers are rare." π and e are the famous ones, but Georg Cantor showed in 1874 that, in a certain sense, almost all numbers are transcendental: the algebraic numbers can be counted and the real numbers cannot.
What can a student learn from how Lindemann worked?
Lindemann did not invent a new method from nothing. He went to the person who had solved the nearest problem, studied the method, and found the one connection, e^(iπ) = −1, that carried it to a new target. When you are stuck, ask which problem you have already solved that looks most like this one, and what link would carry that method across.
If the words algebraic, irrational and transcendental still blur together, the Fermi Concept Library has a lesson for every concept in mathematics, each with its place in the map.
Write one example each of a rational, an irrational algebraic and a transcendental number, with the equation that proves the middle one is algebraic.
