Using the model to find the solution: It is a simplified representation of the actual situation It need not be complete or exact in all respects It concentrates on the most essential relationships and ignores the less essential ones.

History of analysis The Greeks encounter continuous magnitudes Analysis consists of those parts of mathematics in which continuous change is important. These include the study of motion and the geometry of smooth curves and surfaces—in particular, the calculation of tangents, areas, and volumes.

Ancient Greek mathematicians made great progress in both the theory and practice of analysis. The Pythagoreans and irrational numbers Initially, the Pythagoreans believed that all things could be measured by the discrete natural numbers 1, 2, 3, … and their ratios ordinary fractions, or the rational numbers.

This belief was shaken, however, by the discovery that the diagonal of a unit square that is, a square whose sides have a length of 1 cannot be expressed as a rational number.

Against their own intentions, the Pythagoreans had thereby shown that rational numbers did not suffice for measuring even simple geometric objects. For the Greeks, line segments were more general than numbers, because they included continuous as well as discrete magnitudes. Pythagorean theoremVisual demonstration of the Pythagorean theorem.

In the box on the left, the green-shaded a2 and b2 represent the squares on the sides of any one of the identical right triangles. On the right, the four triangles are rearranged, leaving c2, the square on the hypotenuse, whose area by simple arithmetic equals the sum of a2 and b2. For the proof to work, one must only see that c2 is indeed a square.

This is done by demonstrating that each of its angles must be 90 degrees, since all the angles of a triangle must add up to degrees. This was realized by Euclid, who studied the arithmetic of both rational numbers and line segments. His famous Euclidean algorithmwhen applied to a pair of natural numbers, leads in a finite number of steps to their greatest common divisor.

Euclid even used this nontermination property as a criterion for irrationality. Thus, irrationality challenged the Greek concept of number by forcing them to deal with infinite processes.

In his Physics c. There is no motion because that which is moved must arrive at the middle [of the course] before it arrives at the end. Presumably, Zeno meant that, to get anywhere, one must first go half way and before that one-fourth of the way and before that one-eighth of the way and so on.

Still, despite their loathing of infinity, the Greeks found that the concept was indispensable in the mathematics of continuous magnitudes. So they reasoned about infinity as finitely as possible, in a logical framework called the theory of proportions and using the method of exhaustion.

It established an exact relationship between rational magnitudes and arbitrary magnitudes by defining two magnitudes to be equal if the rational magnitudes less than them were the same. In other words, two magnitudes were different only if there was a rational magnitude strictly between them.

This definition served mathematicians for two millennia and paved the way for the arithmetization of analysis in the 19th century, in which arbitrary numbers were rigorously defined in terms of the rational numbers.

The theory of proportions was the first rigorous treatment of the concept of limits, an idea that is at the core of modern analysis. The method of exhaustion The method of exhaustionalso due to Eudoxus, was a generalization of the theory of proportions. In this way, he could compute volumes and areas of many objects with the help of a few shapes, such as triangles and triangular prisms, of known dimensions.

Among his discoveries using exhaustion were the area of a parabolic segment, the volume of a paraboloid, the tangent to a spiral, and a proof that the volume of a sphere is two-thirds the volume of the circumscribing cylinder.

His calculation of the area of the parabolic segment involved the application of infinite series to geometry. For information on how he made his discoveries, see Sidebar: Models of motion in medieval Europe The ancient Greeks applied analysis only to static problems—either to pure geometry or to forces in equilibrium.

Analysis began its long and fruitful association with dynamics in the Middle Ageswhen mathematicians in England and France studied motion under constant acceleration. This result was discovered by mathematicians at Merton College, Oxford, in the s, and for that reason it is sometimes called the Merton acceleration theorem.

A very simple graphical proof was given about by the French bishop and Aristotelian scholar Nicholas Oresme.

He observed that the graph of velocity versus time is a straight line for constant acceleration and that the total displacement of an object is represented by the area under the line. This area equals the width length of the time interval times the height velocity at the middle of the interval.

Merton acceleration theoremDiscovered in the s by mathematicians at Merton College, Oxford, the Merton acceleration theorem asserts that the distance an object moves under uniform acceleration is equal to the width of the time interval multiplied by its velocity at the midpoint of the interval its mean speed.

In making this translation of dynamics into geometry, Oresme was probably the first to explicitly use coordinates outside of cartography. He also helped to demystify dynamics by showing that the geometric equivalent of motion could be quite familiar and tractable. For example, from the Merton acceleration theorem the distance traveled in time t by a body undergoing constant acceleration from rest is proportional to t2.

At the time, it was not known whether such motion occurs in nature, but in the Italian mathematician and physicist Galileo discovered that this model precisely fits free-falling bodies. Galileo also overthrew the mistaken dogma of Aristotle that motion requires the continual application of force by asserting the principle of inertia: From this he concluded that a projectile—which is subject to the vertical force of gravity but negligible horizontal forces—has constant horizontal velocity, with its horizontal displacement proportional to time t.

In the German astronomer Johannes Kepler took this idea to the cosmic level by showing that the planets orbit the Sun in ellipses.The Hundred Greatest Mathematicians of the Past. This is the long page, with list and biographies. (Click here for just the List, with links to the caninariojana.com Click here for a .

History of analysis The Greeks encounter continuous magnitudes. Analysis consists of those parts of mathematics in which continuous change is important. These include the study of motion and the geometry of smooth curves and surfaces—in particular, the calculation of tangents, areas, and volumes.

algebraic number. An algebraic number is a real number that is a root of a polynomial equation with integer coefficients. For example, any rational number a/b, where a and b are non-zero integers, is an algebraic number of degree one, because it is a root of the linear equation bx - a = 0.

The square root of two is an algebraic number of degree two because it is a root of the quadratic. Pierre de Fermat, (born August 17, , Beaumont-de-Lomagne, France—died January 12, , Castres), French mathematician who is often called the founder of the modern theory of caninariojana.comer with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th caninariojana.comndently of Descartes, Fermat discovered the fundamental principle of .

A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later. Please report any errors to me at [email protected] History of analysis The Greeks encounter continuous magnitudes.

Analysis consists of those parts of mathematics in which continuous change is important. These include the study of motion and the geometry of smooth curves and surfaces—in particular, the calculation of tangents, areas, and volumes.

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Pierre de Fermat - Wikipedia