By Georges Ifrah

"Georges Ifrah is the fellow. This publication, readily, principles. . . . it truly is extraordinary . . . a mind-boggling and enriching experience." –The parent (London) "Monumental. . . . a desirable trip taking us via many various cultures."–The instances (London)"Ifrah’s e-book amazes and fascinates via the scope of its scholarship. it really is not anything lower than the historical past of the human race instructed via figures." –International bring in Tribune Now in paperback, this is Georges Ifrah’s landmark overseas bestseller–the first whole, common learn of the discovery and evolution of numbers across the world. A riveting background of counting and calculating, from the time of the cave dwellers to the 20th century, this interesting quantity brings numbers to exciting existence, explaining their improvement in human phrases, the exciting events that made them precious, and the bright achievements in human idea that they made attainable. It takes us throughout the numbers tale from Europe to China, through historic Greece and Rome, Mesopotamia, Latin the US, India, and the Arabic nations. Exploring the numerous methods civilizations built and altered their mathematical platforms, Ifrah imparts a special perception into the character of human thought–and into how our knowing of numbers and the methods they form our lives have replaced and grown over millions of years. "Dazzling."–Kirkus stories "Sure to transfix readers."–PublishersWeekly

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**Extra resources for The Universal History of Numbers: From Prehistory to the Invention of the Computer**

**Sample text**

What about using a requirement of computational simplicity to determine the constant? 11) be used to determine the undetermined constant in (b)? 11. It is desirable that if the exact solution is monotonically increasing (or decreasing) then the numerical solution does the same thing. 3), where α > 0. A numerical method is said to be monotone if it produces a monotonically decreasing solution to this problem. 15. (a) Show that backward Euler is monotone. , they are monotone only if the step size is small enough) and that RK4 has the largest monotone interval of the three.

77) we lost energy conservation. 9, but it is also clear that the method does a respectable job in determining the energy. The reason is that velocity Verlet possesses a special property connected with preserving area, and orientation, in the phase plane. 10. Phase plane parallelograms used to introduce a symplectic approximation. Using the method, in one time step, a → A, b → B, and c → C. suppose that with the initial condition ya = (ya , va )T the method takes one time step and produces the value yA = (yA , vA )T .

You are to solve this problem using the backward Euler, trapezoidal, and RK4 methods. (a) Assuming h(0) = 0, show that y = h(t) + βe−αt is the solution of the problem. (b) Assume α = 20, β = 1, and h(t) = sin(t). On the same axes plot the exact and the three numerical solutions for 0 ≤ t ≤ 3 in the case M = 40. (c) Redo (b) for M = 80 and for M = 20. If one or more of the methods is unstable for either case you can exclude it from the plot (for that value of M ) but make sure to state this in your write-up.