1/8/2024 0 Comments Calculus with infinitesimalsWhen faced with an equation such as 2 x + 10 = 6, they recognized no possible solution. < r for every real r, and similarly, since – r < 0 for every real r, and so 2 must be our desired slope.Īnd π. So, what is the slope of our desired tangent? Well, the slope of the tangent must be a real are not real numbers), and it must fall between the slope of the chords BA and AC. We can certainly find the slope of the line going through these two points: How can we approach this problem? We know how to find the slope of a line given two points on the line, but here we are given only one point plus the information that the line is tangent to y = x be an infinitesimal (positive, say) and consider two points on the curve of y = x² which are infinitely close )²): We shall find the slope of the tangent to y = x² at the point (1, 1). These new infinitesimals, once suitably defined, will enable us to solve general problems of slopes of tangents and areas of regions with extraordinary ease. Negative numbers and imaginary numbers have no direct physical presence in the real world, yet both serve an essential role in solving problems about the real world. Of course our infinitesimals cannot themselves be real numbers, but so what? This sort of expansion of a number system through the introduction of new numbers which themselves correspond to nothing in the real world is common in mathematics. These new numbers will have the property that although different from 0, each is smaller than every positive real number and larger than every negative real number. Simply stated, our approach will involve expanding the real number system by introducing new numbers called infinitesimals. The approach to the calculus we shall employ is based on Leibniz’s ideas as formalized by Abraham Robinson in 1961 under the name of nonstandard analysis. Newton thought in terms of limits whereas Leibniz thought in terms of infinitesimals, and although Newton’s theory was formalized long before Leibniz’s, it is far easier to work with Leibniz’s techniques. Remarkably, the powerful methods developed by these two men solved the same class of problems and proved many of the same theorems yet were based on different theories. Their work was certainly built on foundations laid by others, but their penetrating insights represented what is easily the most significant mathematical breakthrough since the Greeks. The general idea of the calculus, its fundamental theorem, and its first applications to the outstanding problems of mathematics and the natural sciences are due independently to Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716). Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases. The first had been developed to determine the slopes of tangents to certain curves, the second to determine the areas of certain regions bounded by curves. In the beginning there were two calculi, the differential and the integral. The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas. The questions it answered and the questions it raised lay at the heart of man’s understanding of not only geometry and number, but also space and time and mathematical truth. The calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The history of modern mathematics is to an astonishing degree the history of the calculus. God created infinity, and man, unable to understand infinity, had to invent finite sets. God made the integers, all else is the work of man.
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