Encyclopaedie der mathematischen Wissenschaften und

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Extra resources for Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Algebra und Zahlentheorie

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5. 1. Sketch the curves w + v = - ( b / a ) , ~i = c / u for the following equations ux2+bx+c = 0. On the basis of the sketch, determine the number of' real solutions the equation has arid the approximate value of the roots. 2. Solve the equation x 2 2 2 2 = 0 in the field with five elements. Does it have any solutions in the field with three elements? 3. Which of the following two systems of three equations is determinate and which is indeterminate? z + 2 y - 3 2 = 2 , z + y + z = 5 , 22 - 3y 42 = 1 , x 2y 32 = 2 .

The law of inertia, stated by Descartes in the seventeenth century, says that any unaccelerated motion will be motion in a straight line at constant speed. Now in nature, besides straight-line motion at constant, 4. w 40 w ALGEBRA IS IMPORTANT FIGURE 8. Forces on a body in circular motion. speed, there are also many examples of motion in a circle at constant speed. To a very good approximation, for example, planetary orbits can be regarded as such a uniform circular motion. Since uniform circular motion is not in a straight line, it must be accelerated.

1. Diophantine equations. We shall also discuss indeterminate systems only briefly. Since an indeterminate system generally has a solution set that can be represented as a curve or surface in a higher-dimensional space, it requires methods from analysis (calculus) that can be omitted in a discussion of classical algebra. There is one important case, however, that should at least be mentioned. In order to reduce the solution set from a continuous curve or surface t o a discrete set of points, mathematicians somet’imes impose the additional requirement that the solutions be integers.

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