By Irving Adler
More than a hundred workouts with solutions and two hundred diagrams remove darkness from the textual content. academics, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will enjoy this remarkable exploration of the position of geometry within the improvement of Western clinical thought.
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Extra info for A New Look at Geometry
Let p be the number of n-gons that occur at each vertex. Then there will be p angles at each vertex, and since they fill out the space in the plane around the vertex, their sum is 360 degrees. Consequently the number of degrees in each of them is But we know already that the number of degrees in each angle of a regular n-gon is Equating these two expressions, and dividing by 180, we get the equation; If we multiply equation (2) by np, we get If we add 4 to both sides of equation (3) we get Factoring the left-hand side of equation (4) we get Since n – 2 and p – 2 are whole numbers, and their product is 4, we get all possible values of n and p by equating the pair n – 2, p – 2 to all possible pairs of whole numbers whose product is 4.
That is, the real number a occurs in the complex number system as the complex number a + 0i. To obtain a pictorial representation of the complex number system, we use the set of all points in a plane. First draw a horizontal line in the plane, and associate the real number system with the points on the line, as we have already done. We call this line the axis of real numbers. To associate a point of the plane with every complex number a + bi, we use this procedure: At the point a on the axis of real numbers draw the line that is perpendicular to the axis of real numbers.
The first pair in which the coefficients differ is 2x and 5x. Since 5 > 2, we say that 3x2 + 5x −4 > 3x2 + 2x −1. In this system the element x is “infinitesimal” compared to x2, because no matter how large a positive integer n may be, nx < x2 (that is, Ox2 + nx < 1x2). The assumption that lengths of line segments satisfy condition I is commonly called the axiom of Archimedes. It really should be called the axiom of Eudoxus, since Eudoxus was the first to state it explicitly. An ordered system is called Archimedean or non-Archimedean according as it does or does not satisfy this axiom.
A New Look at Geometry by Irving Adler