Download PDF by Irving Adler: A New Look at Geometry
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 delight in this extraordinary exploration of the position of geometry within the improvement of Western clinical thought.
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This is equivalent to the rule in arithmetic that we never divide by zero, c) It excludes from consideration systems of ordered magnitudes in which one element may be “infinitesimal” compared to another, that is, systems in which a may be so small that no finite multiple of a is greater than b. An example of such an ordered system is the set of all polynomials anxn + ··· + a1x + a0 with real coefficients, in which the order relation > is defined as follows: To determine which of two polynomials is the greater, first write them both as polynomials of the same degree by supplying some zero coefficients where necessary.
It is shown in the theory of fields that among all the fields that lie within a given field there is a smallest field that is contained in every one of them. Moreover, this smallest field is either the rational number system or one of the fields Fp. These smallest possible fields are called prime fields. The prime field contained in a finite field cannot be the rational number system, since the rational number system contains infinitely many members. Therefore the prime field in any finite field must be one of the fields Fp.
This implies that m is an even number, because if m were odd, m2 would also be odd. Consequently there exists a whole number k such that m = 2k. Then m2 = 4k2. But m2 = 2n2. Equating these two expressions for m2, we see that 2n2 = 4k2, or n2 = 2k2. That is, n2 must be even, and therefore n must be even. Notice that while m and n are not both even, we have been compelled to say that they are both even. We were led into this absurdity as a result of assuming that the length s may be expressed as a ratio of whole numbers.