fermat
Pierre Fermat

Pierre Fermat

1600. We are still under the reign of good King Henry IV. Whoever became Catholic to the throne of the most Christian nations of Europe, France, has just succeeded to extinguish, for a time, the embers of religious wars that bloodied France.

The Edict of Nantes is registered in all the courts of France and Navarre.

Pierre Fermat’s Biography

Pierre de Fermat was born in Toulouse in 1601, shortly after the outbreak of the century, so far the drivers of mathematical thinking that stir the European capitals. This is the law that draws first. He became a magistrate. Yet his interest in mathematics is urgent. Like Newton, a few years later, he discovered the mathematics in books and especially in the works of scholars Greek as Diophantus had developed methods of resolutions of equations with integer coefficients.

Diophantine equations

Fermat decided to pursue this branch of mathematics: the Diophantine equations. He proved the theorem, now known as the « Fermat’s little theorem »:

If p is a prime number and if a is an integer not divisible by p , then

a  p -1  – 1 is a multiple of p .

Pierre Fermat

But he is best known for his last theorem or « Fermat’s Last Theorem, » he stated, but he seems despite contrary annotation he left, never demonstrated:

There is no non-zero integers x , y and z such that:

x n + y n = z n ,

when n is an integer strictly greater than 2.

Pierre Fermat

Indeed, for n = 2, we fall back on the Pythagorean theorem. It was not until 1994 that Andrew Wiles brings in the demonstration.

The beginnings of calculus.

It is obvious that neither Newton nor Leibniz are the inventors of calculus . Archimedes before them (method of exhaustion) and especially Pierre Fermat (tangent method) had laid the foundation.

tangent

Pierre Fermat, it was to give the equation of a tangent (or at least its director coefficient) in any point of a curve.

For example at point A (5, -3) belonging to the parabola y = x²-8x + 12.

How to do ?

Fermat defines:

  • a first point A [a, f (a)] and
  • a second point A ‘[a + h, f (a + h)], not far (h on the abscissa axis).
tangeantefermat2
the tangents of the method of Fermat

The slope of the line A-A ‘is:

  • [F (a + h) – f (a)] / [(a + h) -a]

= [F (a + h) – f (a)] / h

If you then close A ‘of A, h will decrease. If we h tend to 0:

  • A ‘will merge with and A
  • [F (a + h) – f (a)] / h is the slope of the tangent famous A.

Fermat took for example the curve y = f (x) = x² is the expression of the simple parables, which passes through the origin. He takes notes and writes the slope of the tangent at A [a, f (a)]:

  • [F (a + h) – f (a)] / [(a + h) -a]
  • = (A + h) ² – a² / h
  • A² + = + 2ah h² – a² / h
  • + = 2ah h² / h
  • = 2a + h

He then made as promised h tend to 0. It left him more than 2a. He concludes that the slope of the tangent at A [a, f (a)] of the curve f (a) = a² is 2a.

Originally O (0.0) shows that the parable exchange slope. It is negative and positive right to left. The slope is 2 x 0 = 0. The tangent is horizontal.

This is even though the term did not yet exist, the derivative of the function f (x) = x² to be (re) discovered by Newton in the form of its inflammations.

 

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