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A few weeks ago I traveled with my son Aurélien to Liège in Belgium; at a coffee break at a gasoline station two students (I do not remember their names let’s call them Martin and Isabelle) asked us if we could take them on board to Liège. They told us that, that weekend all students entering engineering schools in France were involved in a competition; they should try without using public transport to travel as far as possible away from their school. Their journey was recorded through an I Phone application.
Normally I do not take any hitchhikers anymore I had difficulties at several occasions but this would be an exception.
We talked about their study and interest and at a given point in time I asked them which in their opinion is the most beautiful formula in mathematics?
Martin immediately said: Euler’s formula eiπ + 1 = 0
Isabelle didn’t remember that formula but when I said that the general formula is:
eix = cos(x) + i.sin(x) she vaguely remembered it.
As to her favorite formula she asked for a little more time to reflect.
I said that Euler’s formula is also my favorite one but I almost changed my mind after a period of passion followed a period of sleepiness nights and frustrations! The reason behind this was the “1” in the formula, the “1” for which I loved it!
When saying this I observed through the interior mirror the facial expressions and insofar possible the body language of both Martin and Isabelle. It was clear to me that they didn’t believed me and were thinking that I exaggerated by talking about frustrations etc. As a reaction I told them that they should read a study made on Experiences related to the brain’s reaction with respect to mathematical beauty. This study reveals that Euler’s formula had the highest and most consistently reaction of the “emotional brain” compared to any other formula submitted to the participants (by the way all mathematicians!).
I then asked Martin if he knew what exactly is meant by “1” in this formula. He first didn’t understand my question but when I insisted he said: well 1 is 1 it is thus just 1!
I told him that I also thought that 1 was just 1 and that was precisely the reason that I loved this formula above any other. I often said to people the beauty of this formula lies in the fact that eiπ is composed of two transcendental numbers: π (3,141 592 653 …) and e (2,718 28 …) and the imaginary number “i” a number that should not exist and that the combination of all this equals (-) 1. Just 1 nothing more or less! Amazing! No?
But what is the nature of 1 in this formula? To understand this we must find out to which set of numbers this “1” belongs.
At the origin there was the set of natural numbers 1, 2, 3, 4 etc these are whole numbers for example used for counting objects e.g. in my garden I have planted 4 threes. In such case writing 4.0 makes no sense it is just 4 nothing else!
Important to note is that the sum or the multiplication of natural numbers is also a natural number. This is, however, not the case for a division or a subtraction.
e.g. for a division ½ = 0.5 ; 0.5 is not a natural number and 5 – 6 = -1 minus 1 is not a natural number. In mathematics it is said that the set of natural numbers is closed for the addition and multiplication.
The next set of numbers are the Integers. This set is composed of the natural numbers and their additive inverses and zero.
…. -4, -3, -2, -1, 0, 1, 2, 3, 4 ….
The set of integers is closed for addition, subtraction and multiplication but not for division.
Next set is the set of rational numbers. A rational number is per definition a number that can be expressed as the ratio of two integers i.e. can be written as p/q in which p is an integer and q a non-zero integer.
The set of rational numbers is closed for the basic mathematical operation: addition, subtraction, multiplication and division.
The drawing illustrates that all natural numbers and integers are rational numbers but that not all rational numbers are natural numbers or integers.
It is evident that the “1” in the set of natural numbers and positive integers is “1” and nothing more or less. That is precisely the “1” I would have loved to see in Euler’s formula but is this really the case? The question is: to which set does “1” belongs?
To answer this question we must consider first of all other sets of numbers to which the other components of the formula belong and in particular the transcendental numbers.
Driving a car and at the same time explain the nature of transcendental numbers is not easy but luckily the passengers were all acquainted with mathematics.
I didn’t ask them the question “what are transcendental numbers?” but started immediately with explaining the set of irrational numbers.
Irrational numbers are numbers that cannot be expressed as a fraction of integers and do not terminate, nor do they repeat.
A typical example of an irrational number is 21/2 = 1.414213562373095….
Within the set of irrational numbers there is a sub-set called transcendental numbers. These numbers are not a solution of a non-zero polynomial equation. Written in a mathematical manner:
In which n ≥ 0 and at least one coefficient ≠ 0 (if all coefficients would be zero then the polynomial equation would be a zero equation)
As mentioned earlier 21/2 is an irrational number but is it also a transcendental number? The number looks similar to π which also when written as a decimal number, does not terminate, nor does it repeat. One would thus be tempted to classify it as a transcendental number, but this would be wrong. Indeed 21/2 is the solution of the polynomial equation x2 − 2 = 0 and thus not a transcendental number!
Using the above equation of a polynomial all =0 except for k=0 and k=2 where “a” is respectively: =-2 and =1 thus:
f(X)= -2 + X2
The next set is the one of Real Numbers. Real numbers are at the base of Mathematical analysis; one of the most important branches of mathematics dealing with series, limits, differentiation, integration, analytic functions etc.
The idea is that any real number can be represented on a line (the Real Number Line) where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as 0.9528469247 (lying on the Real Number Line between 0.94 and 0.96) where each consecutive digit is measured in units one tenth the size of the previous one.
The set of Real Numbers comprises as sub-sets the set of rational and the set of irrational numbers.
Given the definition of the set of Real Numbers there is no doubt that the “1”in Euler’s Formula is a Real Number and lies somewhere between:
0.999… ≤ ”1” ≤ 1.0001
Take for example the real number 0.999 which can be written also as the sum of 0.9 + 0.09 + 0.009; which can also be written as:
0.999 = 1 – 0.001
The number 0.999… can also be written as a series or sum of numbers starting with 0.9 and each next number is divided by 10 i.e. 0.09, 0.009, 0.0009 …
This series can be written mathematically as:
(∑ stands for the sum of the individual elements of a series for n going from 1 to k, k-1 being the number of zeros preceding the 9)
As shown above this can also be written as: 1 – 1/10n (in the example above with n = 3 this would be 1 – 1/103= 1 – 0.001). The larger n the more zeros are preceding the 1. It is easy to understand that when n tends to infinity the number of zeros preceding the 1 are also tending to infinity and that this number becomes smaller and smaller; thus closer and closer towards zero.
To calculate the value of 1 – 1/10n whereby n tends to infinity the theory of limits is used.
And this is precisely the “1” in Euler’s formula.
In conclusion in the field of mathematical analysis of which Euler’s formula forms an integral part the integer 1 and the 1 obtained as a value of a series tending to 1 as shown above are considered identical. Martin agreed but in reality he didn’t care; for him 1 is 1 and that is it!
Then there is in that equation this mysterious imaginary number “i” ! This number which is the square root of minus one (i = -11/2 ) should in fact not exist since the square of any number is by definition always positive but here it is negative i.e. -1 !
Some mathematicians decided, however, otherwise and accepted this contradiction and started working with it. This number was formalized in the sixteenth century and is now widely used in applied mathematics. The history and use of imaginary numbers is very exciting but as Kipling would have said “that’s another story”!
Just before arriving Isabelle remembered her favorite formula. In fact it was not about a formula but Euclid’s fifth postulate.
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