History[ edit ] A Texas Instruments calculator that contains a computer algebra system Computer algebra systems began to appear in the s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martinus Veltmanwho designed a program for symbolic mathematics, especially high-energy physics, called Schoonschip Dutch for "clean ship" in Reduce became free software in Freely available alternatives include SageMath which can act as a front-end to several other free and nonfree CAS.
Time Life Books Gullberg, Jan. Hailstone Numbers Choose a positive integer. If your number is even, divide it by two. If your number is odd, multiply it by three and add 1. Take your new number as the starting number, and repeat until you can't go any farther. The process sounds easy enough, but how to predict what will happen may be less obvious.
Hailstone numbers are generated by the simple mathematical process described above. A small integer value of n can demonstrate the process. To compare hailstone sequences from different starting numbers, it is useful to note both the writing a computer algebra system value attained, and the number of computations needed to reach 1.
To compare, change n by a small amount and note what happens to the sequence. Here the largest value is again 16, but the number of steps is 8. How high will hailstone numbers go?
And just how many steps are needed? Some larger examples may show greater variation attained in hailstone number sequences. The pattern is 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
Here the maximum value is 88, and the length is This is where the surprises of hailstone numbers become more obvious. The sequence is 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The maximum value is 40, and the number of computations only The sequence has a maximum value of 9, and takes computations to reach 1!
So far, it appears that the maximum values and numbers of computations needed are in some way related to the size of the starting number. Perhaps more notable is the contrast between even and odd starting numbers.
There is, as yet, no complete understanding of the apparently chaotic nature of the sequences produced by this simple procedure. Also, another puzzling question remains. Will any starting number used for a hailstone sequence eventually reach 1?
From the examples above, it would appear so. Certainly the hailstones these numbers are named for have somewhat predictable behavior. Hailstones go up and down inside of clouds, gaining size and mass with each pass through. Then, when heavy enough, hailstones fall out of the clouds and tumble to earth.
It seems that our hailstone numbers have their own earth.
They increase and decrease, and eventually fall to the value of 1. Studies have been carried out to try larger and larger starting numbers, and so far each number tried has led to the repeating cycle of 4, 2, 1.
No one has been able to prove, however, that every number used as the starting number for a hailstone sequence will eventually reach 1. While the sequences can be used to study many patterns and comparisons, the final question may remain unanswered.
Contributed by Laurie Kiss References: Slicing pizzas, racing turtles, and further adventures in applied mathematics. Although the definition and uses of the term differ somewhat today, there is still a need for the concept.
Gottfried Leibniz applied the term to various aspects of a curve in Peter Dirichlet, inconceived of a function as a variable y having its value fixed or determined in some definite manner by the values assigned to a variable x or to several values of x.
At the time, the values for the variables were real or complex numbers. There was a need to develop ideas in the days of Descartes, Leibniz, and Dirichlet to better understand their surroundings. Dirichlet studied progressions, evaluating integrals, gravitational attraction, the solar system, harmonic motion, and trigonometric series.
The concept of function today compared to the times of Descartes, Leibniz, and Dirichlet has been altered. By definition, a function is a correspondence, or rule, that pairs each element of a set the domain with exactly one element of another set the range.Online homework and grading tools for instructors and students that reinforce student learning through practice and instant feedback.
The Solutions of a System of Equations. A system of equations refers to a number of equations with an equal number of variables.
We will only look at the case of two linear equations in two unknowns. Big Ideas Math® and Big Ideas Learning® are registered trademarks of Larson Texts, Inc. Do not duplicate or distribute without written permission from Big Ideas.
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A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later.
Please report any errors to me at [email protected] Introduction to computer algebra systems? [closed] Ask Question. Lisp-like (prefix) output for the Reduce/Redlog computer algebra system. 0.