sorting pi, e, and root 2

Mathematicians, always economical,   love to categorize numbers according to their properties. This is because numbers belonging to the same category behave in the same way. You don’t have to deal with each one! That’s an economical way of preserving the energy demand of brain cells.  In the grades we give pupils tasks that involve sorting numbers. Whole numbers  can be sorted out as odd or even, prime or composite, for example. This is a very good way of giving them a sense of how strict definitions are in mathematics and in understanding the nature of numbers. In the higher grades they meet other numbers which they can categorize as imaginary or real, transcendental or algebraic. The same mathematical thinking is used.

\pi is one of the most widely known irrational number. Ask a student or a teacher to give an example of an irrational number, the chances are they will give \pi as the first example or the second one, after square root of 2.  And of course at a distance third is the number e. Now, although they belong to the same set of numbers, the irrationals, they don’t really belong to the same category. For example, \pi and e are both irrationals but pi is transcendental and square root of 2 is algebraic.  The number e is also transcendental. Here’s a short and simple explanation. Click quote for source.

A transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+…+cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic.

It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental. This feat was finally accomplished for π by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler.

In 1873, Hermite proved that the constant e was transcendental. Combining this with Euler’s famous equation e^(i*π)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, π had to be transcendental in order to make i*π transcendental.

Of course understanding the proof of pi as a transcendental number is beyond the level of basic mathematics and hey, we don’t even talk about transcendental numbers before Grade 10. But students at this level can understand the expression ax^n+bx^(n-1)+…+cx^0=0 where all coefficients are integers and n is finite. With proper scaffolding or if they have been exposed to similar task of sorting numbers before  students can make sense of the logic and reasoning shown above which characterizes most of the thinking in mathematics.


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