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By Yair Shapira
Emphasizing the relationship among mathematical items and their functional C++ implementation, this booklet offers a complete creation to either the speculation in the back of the items and the C and C++ programming. Object-oriented implementation of third-dimensional meshes allows knowing in their mathematical nature. Requiring no necessities, the textual content covers discrete arithmetic, information constructions, and computational physics, together with high-order discretization of nonlinear equations. routines and recommendations make the publication appropriate for school room use and a helping web site offers downloadable code.
Read or Download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series) PDF
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Extra info for Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)
Example text
142857 -------1 | 7 10 7 30 28 -20 14 -60 56 -40 35 -- © 2009 by Taylor and Francis Group, LLC 30 CHAPTER 3. RATIONAL NUMBERS 50 49 -1 and the process repeats again and again. 142857142857142857 . . The six digits in the above period, 142857, are obtained one by one in the division algorithm in a minimal residual approach. Indeed, the rational number 1/7 is the solution of the equation 7x = 1. In other words, 1/7 minimizes the residual 1 − 7x. 3. 02. ) The process continues, producing uniquely the six digits in the period, and restarts all over again to produce the same period over and over again in the decimal representation of 1/7.
5. Write the recursive algorithm that produces the sum of two arbitrarily large natural numbers with arbitrarily long decimal representations. 6. Use mathematical induction on the length of the above decimal representations to show that the above algorithm indeed works and produces the correct sum. 7. Write the recursive algorithm that produces the product of two arbitrarily large natural numbers with arbitrarily long decimal representations. 8. Use mathematical induction on the length of the above decimal representations to show that the above algorithm indeed works and produces the correct product.
In other words, 1/7 minimizes the residual 1 − 7x. 3. 02. ) The process continues, producing uniquely the six digits in the period, and restarts all over again to produce the same period over and over again in the decimal representation of 1/7. Thus, the above division algorithm not only proves the existence of the periodic decimal representation of 1/7, but actually produces it uniquely. Just like 1/7, every rational number of the form n/m can be written as a periodic infinite decimal fraction.