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Archimedean

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Archimedean property

From Wikipedia, the free encyclopedia.

The Archimedean property of any ordered algebraic structure, such as a linearly ordered group, and in particular of the system of real numbers, is the property of lacking (non-zero) infinitesimals. Such structures that lack infinitesimals are called Archimedean; those that possess infinitesimals are non-Archimedean. A number x would be infinitesimal if the inequality \left|x\right|+\cdots+\left|x\right|<1 continues to hold no matter how large the finite cardinal number n of terms in this sum.

The non-existence of nonzero infinitesimal real numbers follows from the least-upper-bound property of the real numbers, as follows. If nonzero infinitesimals exist, then the set of all of them has a least upper bound c. Either c is infinitesimal or it is not. If c is infinitesimal, then so is 2c, but that contradicts the fact that c is an upper bound of the set of all infinitesimals (unless c is 0, so that 2c is no bigger than c). If c is not infinitesimal, then neither is c/2, but that contradicts the fact that among all upper bounds, c is the least (unless c is 0, so that c/2 is no smaller than c).

Archimedes of Syracuse stated that for any two line segments, laying the shorter end-to-end only a finite number of times will always suffice to create a segment exceeding the longer of the two in length. Nonetheless, How Archimedes used infinitesimals in mathematical arguments, although he denied that those were finished mathematical proofs.

How Archimedes used infinitesimals

The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. His work with infinitesimals is found in the celebrated Archimedes Palimpsest. The palimpsest embodies Archimedes' account of his "mechanical method", so called because it relies on the concepts of torque exerted on a lever and of center of gravity. Both of those concepts were first introduced by Archimedes.

Ironically, Archimedes disbelieved in the existence of infinitesimals, and therefore said explicitly that his arguments fall short of being finished mathematical proofs.

The proof of the first proposition in the palimpsest is very beautiful, and appears below.

The first proposition in the palimpsest

The curve in this figure is a parabola. Archie1small.png

The points A and B are on the curve. The line AC is parallel to the axis of the parabola. The line BC is tangent to the parabola. The first proposition states:

  • The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB.

Proof: Let D be the midpoint of AC. The point D is the fulcrum of a lever, which is the line JB. The points J and B are equidistant from the fulcrum. As Archimedes had shown, the center of gravity of the interior of the triangle is at a point I on the "lever" so located that DI:DB = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I, and the whole weight of the section of the parabola at J, the lever is in equilibrium. If the whole weight of the triangle rests at I, it exerts the same torque on the lever as if the infinitely small weight of every cross-section EH parallel to the axis of the parabola rests at the point G where it intersects the lever. Therefore, it suffices to show that if the weight of that cross-section rests at G and the weight of the cross-section EF of the section of the parabola rests at J, then the lever is in equilibrium. In other words, it suffices to show that EF:GD = EH:JD. That is equivalent to EF:DG = EH:DB. And that is equivalent to EF:EH = AE:AB. But that is just the equation of the parabola. Q.E.D..

Other propositions in the palimpsest

A series of other propositions of geometry are proved in the palimpsest by similar arguments. Some of them have the location of a center of gravity as the conclusion. One of those states that the center of gravity of the interior of a hemisphere is located 5/8 of the way from the pole to the center of the sphere.

Q.E.D.

Q. E. D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "which was to be proved"). Q.E.D. may be written at the end of mathematical proofs to show that the result required for the proof to be complete has been obtained. It is not seen as frequently now as in earlier centuries.

End-of-proof symbolism in the present day is often the symbol ■ (solid black square) called the tombstone or halmos (after Paul Halmos who pioneered its use). The tombstone is sometimes open; □ (hollow black square).

In Hong Kong, students jokingly reinterpret Q.E.D. as "Question Easy Done" (in Chinglish) after they finish a "difficult" mathematical proof in their schoolwork.