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By Yoshihide Igarashi

Exploring an unlimited array of themes concerning computation, Computing: A ancient and Technical Perspective covers the historic and technical starting place of historic and modern day computing. The e-book begins with the earliest references to counting by way of people, introduces a number of quantity structures, and discusses arithmetic in early civilizations. It publications readers all through the most recent advances in desktop technology, corresponding to the layout and research of desktop algorithms.

Through ancient bills, short technical factors, and examples, the booklet solutions a number of questions, including:

  • Why do people count number another way from the best way present digital desktops do?
  • Why are there 24 hours in an afternoon, 60 mins in an hour, etc.?
  • Who invented numbers, while have been they invented, and why are there diverse kinds?
  • How do mystery writings and cryptography date again to historic civilizations?

Innumerable members from many cultures have contributed their skills and creativity to formulate what has turn into our mathematical and computing history. through bringing jointly the historic and technical facets of computing, this e-book permits readers to achieve a deep appreciation of the lengthy evolutionary techniques of the sector built over hundreds of thousands of years. compatible as a complement in undergraduate classes, it offers a self-contained old reference resource for somebody attracted to this crucial and evolving field.

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Extra resources for Computing: A Historical and Technical Perspective

Sample text

Consequently, the fractions 1/2, 1/4, 1/8, 1/16, and so on became commonly used numbers in addition to natural numbers. Furthermore, the use of reciprocals of integers, such as 1/2, 1/3, 1/4, 1/5, 1/6, and so on, also became common and of great importance. The reciprocal of each nonzero natural number is called a unit fraction. They noticed that multiplications of unit fractions by integers were also useful for calculations. In this way, fractions appeared in ancient civilizations [7]. , 2/3). These symbols appeared in several mathematical tablets and papyri found in Egypt [10].

It follows that (mm)/(nn) = 2. Therefore, both m and n cannot be even numbers—at least one of them must be odd Rational and Irrational Numbers ◾ 21 since m/n is irreducible. From the equation, we can derive n2 = 2m2, which is an even number. , m = 2k, where k is one-half of m. Now, let us substitute for m. We have (mm)/(nn) = 2 = (2k)2/(nn) = 4k2/(nn). So, 2nn = 4k2, or equivalently, nn = 2k2 . Therefore, nn must be even, which makes n even. But both m and n could not be even (since m/n is irreducible), and so we have a contradiction.

1 Three different proofs of the Pythagorean theorem. Hippasus thrown overboard and drowned! An unfortunate footnote to this is that 2 is often called Pythagoras’s constant. Let us present a simple (non-constructive [6]) proof that there can exist no rational number x whose square is 2. , m/n is an ­irreducible fraction), and x2 = 2. It follows that (mm)/(nn) = 2. Therefore, both m and n cannot be even numbers—at least one of them must be odd Rational and Irrational Numbers ◾ 21 since m/n is irreducible.

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