About numbers, episode 4 : infinite numbers

Medium : for amateurs

« Eternity is really long, especially near the end. »

Woody Allen (1935 - )

The lemniscate, symbol of infinity
Source favim.com

What is infinity ? How to grasp such a concept, by definition so immense that nothing can contain it ?
As surprising as it may seem, the answer lies not in philosophy, but really in mathematics.
We will see that it is possible to define infinity in a rigorous and coherent manner.
There even exists numerous different infinities : precisely an infinity of them !
Three separate notions of infinity play a major role in mathematics :
The ordinal infinity, defined as a number greater than any natural number, the cardinal infinity as the number of elements of an infinite set, and the infinity used in calculus and especially limits as an unreachable element.

However, the latter of the three is misleading, being more representative of the concept of unboundedness than that of infinity, as was explained to us by Riemann and Einstein.
Of course it doesn't stop there, and the mathematicians' imagination can give us a lot more definitions of infinity, which shows us how limitless the human mind can be.

Tweeter

Lire l'article

About numbers, episode 3 : rational numbers

Medium : for amateurs

The Vitruvian man, by Leonardo Da Vinci
Source Luc Viatour

What is a fraction ?
It seems easy to answer this question with the concept of proportion, but we will see that rational numbers possess some quite extraordinary properties.
Even though possible, it would not be very enlightening to make the same distinction we made with natural numbers and integers between ordinality and cardinality. But cardinality raises a few questions : what is a fractional quantity ? Worse, what is a negative fractional quantity ?
Once those interrogations answered, we will see a few examples of irrational numbers, such as $\pi$ and $\sqrt 2$.
Then we'll tackle a few structural and topological properties of the set of rational numbers.
Take an aspirin, it is going to be a bit more complicated than the previous posts of the series.

Tweeter

Lire l'article

About numbers, episode 2 : Integers

Easy : for the curious

Negative numbers
(Source Wikipedia commons)

A quantity of -2 objects, a marathon runner finishing -3rd, numbers smaller than nothing ?! What do negative numbers mean ?

The answer to that question is easy only to those who are used to them. And for most of us, it doesn't matter what they really are or what they mean, because the thing is these numbers are quite useful !

However, for the ancient Greeks as well as for the Arab and Indian mathematicians - who already understood the meaning of the zero and the negative numbers -, the negative roots of an equation are thought of being unnatural, absurd even according to Nicolas Chuquet (~1450-1488) and would not be recognized until the XIXth and XXth century ! So useful, but also very problematic...

We will see that their meaning is more subtle than we would think, and that it is illuminating to base it on the distinction between ordinals and cardinals that we made in the preceding chapter about natural numbers

Tweeter

Lire l'article

About numbers, episode 1 : Natural numbers

Easy : for the curious

« God made the integers, all the rest is the work of man » Léopold Kronecker (1823-1891) ¹

There is no satisfying definition of the general concept of number. However, lots of particular numbers can be rigorously defined. Natural numbers, integers, imaginary, transcendent, algebraic, computable numbers, etc. In this "story", we will see examples of numbers and will try to understand them intuitively and visually. This first episode is about natural numbers.

The "natural number" is a concept that fulfills two needs : that of ordering, and that of comparing sets "in power", i.e. by counting.

For the first need, one defines the ordinal numbers, for the second the cardinal numbers. These two notions seem at first hand to be different faces of the same objects...

Tweeter

Lire l'article

Groups in everyday life

Easy : for the curious

The group structure is a fundamental concept, used widely and outside mathematics. It is very often used in physics, and we shall see an example of an important group of the theory of special relativity, in a forthcoming article .
Meanwhile, we will simply give the definition and a few simple examples.

Tweeter

Lire l'article