# Jean-Robert Argand

Jean-Robert Argand , born the July 18th 1768 with Geneva and dead the August 13rd 1822 with Paris, is a Mathématicien Swiss amateur .

In 1806, whereas it holds a bookstore in Paris, it publishes a geometrical interpretation of the complex numbers like points in the plan, while making correspond to the number $a+i b$ (where I is the square root of -1) the point of coordinates (has, b). For this reason, the plan, seen like whole of the complex numbers, is sometimes called the plane of Argand .

## Complexes according to Argand

In its treaty Test on a manner of representing the imaginary quantities by geometrical constructions , Argand starts by associating with each positive number has a line $\ overline \left\{KA\right\}$, horizontal directed towards the line and length has . Then, he notices that he can associate with each negative number - B, a horizontal line $\ overline \left\{KB\text{'}\right\}$ directed towards the left length B. The sum consists with the setting end to end of lines. The operations of the product and the square root consists in working on the Proportionnalité:
(has; b) is proportional to (C; d) if the reports/ratios a: B and C: D are identical (even absolute value and even sign)

The product of has by B thus becomes the ab number such as (1; a) and (B; ab) is proportional. The geometrical construction of fourth proportional is a construction known for a long time. Therefore, Argand can build the line :

$\ overline \left\{KC\right\} = \ overline \left\{KA\right\} \ times \ overline \left\{KB\right\}$

The square root of X (positive) is the number there (positive) such as (1; there ) and ( there ; X ) is proportional. This construction is also realizable (see constructible Nombre). If $\ overline \left\{KA\right\}$ is associated with 1, $\ overline \left\{KP\right\}$ associated with there and $\ overline \left\{km\right\}$ associated with X, one will say that:

$\ overline \left\{km\right\}$ is à$\ overline \left\{KP\right\}$ what $\ overline \left\{KP\right\}$ is à$\ overline \left\{KA\right\}.$.

The difficulty which arises then is to build the square root of -1. If $\ overline \left\{KC\right\}$ is the number associated with -1, it is a question of finding a ligne$\ overline \left\{KB\right\}$ such as

$\ overline \left\{KB\right\}$ is with $\ overline \left\{KA\right\}$ what $\ overline \left\{KC\right\}$ is with $\ overline \left\{KB\right\}$.

This cannot be carried out while remaining on the line. Argand thus leaves the line and known as that

$\ overline \left\{KB\right\}$ is with $\ overline \left\{KA\right\}$ what $\ overline \left\{KC\right\}$ is with $\ overline \left\{KB\right\}$
when the report/ratio lengths are equal and angles AKB and BKC are equal .

What places the point B at the vertical of the point K at a distance from 1. The line $\ overline \left\{KB\right\}$ represents then imaginary the I (noted at the time $\ sqrt \left\{- 1\right\}.$

It creates then on the whole of the " lines dirigées" an addition (which is connected with what is called today the Relation of Chasles) and a product

The product:

$\ overline \left\{km\right\} \ times \ overline \left\{kN\right\}$
$\ overline \left\{KP\right\}$ such as $\ overline \left\{KP\right\}$ is with $\ overline \left\{kN\right\}$ is the line what $\ overline \left\{km\right\}$ is with $\ overline \left\{KA\right\}$.

With the definition of proportionality which it gives in the plan, that means that

• $\ frac \left\{KP\right\} \left\{kN\right\} = \ frac \left\{km\right\} \left\{KA\right\}$
• angles NKP and AKM is equal.
It shows whereas a product of directed lines corresponds to the product lengths and the sum of the angles.

It associates then with each complex, a directed line , and shows the correspondence between the operations. Each directed line thus has two possible representations

• by its coordinated Cartesian ( has; B ) which returns to the complex has + ib
• by its polar Coordonnées: length of the line and direction (or angle) of the line.
If the complex is has + ib , the length of the line is $\ sqrt \left\{a^2 + b^2\right\}$, length that Argand calls the module of the complex because it is the unit per which it should be divided to find its direction .

By proposing this representation of the complexes in geometrical form, the objective of Argand is double

1. to prove the reality complexes which people of the time still regard as imaginary and as simple artifice of calculation
2. to give a geometrical tool which can simplify the resolution of the algebraic problems largely.
He proposes even a demonstration of the fundamental Théorème of the algebra (partially false) thanks to this tool.

## Consequences of its test

Appeared in 1806, published by a person of obscure repute, this test falls very quickly into the lapse of memory. The idea is taken up then by Jacques then French François, professor at the imperial school of Artillery and of the Genius, which develops the same concept and adds to it an exploitable notation. He recognizes that the idea is not him and seeks its author of it. He follows then a correspondence between the two men, Argand seeking in vain to give an algebraic representation of the space of dimension three.

However this geometrical design of an algebraic tool runs up against the logical direction of certain mathematicians of the time who see only one artifice of calculation there. Meanwhile of other mathematicians develop in an independent way the same idea. It is only when Gauss and especially Cauchy, seizes this idea that this design acquires its noble letters and becomes a springboard which makes it possible Hamilton to create its Quaternion S.

## Sources

• Robert Argand, Test on a manner of representing imaginary quantities in the geometrical construction industries , 2nd edition, Gauthier Villars, Paris (1874) HTTP: //gallica.bnf.fr/ BNF
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