Algorithm (mathematics)
An algorithm in mathematics refers to a defined set of steps designed to solve a particular mathematical problem. This concept can be likened to a recipe, where each step lays out a method to achieve a specific goal, such as solving equations or performing calculations. By breaking down complex problems into manageable steps, algorithms enable mathematicians and students alike to identify more efficient ways to reach solutions, often revealing opportunities to streamline processes by eliminating unnecessary actions.
The term "algorithm" itself has historical roots, originating from the Persian mathematician Al-Khwarizmi, whose work built upon earlier Indian mathematical concepts. Over time, the definition of algorithms has expanded from solving equations to encompass various strategies for addressing different types of problems. Algorithms are utilized in educational settings as they offer clear, structured methods for learners, making abstract concepts more accessible. Examples exist for fundamental operations like addition, subtraction, multiplication, and division, with specific techniques employed for each. Overall, algorithms play a crucial role not only in mathematics but also in computer science, facilitating automated problem-solving without the need for pre-programmed answers.
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Algorithm (mathematics)
An algorithm is a set of steps to be followed in order to solve a particular type of mathematical problem. As such, the concept has been analogized to a recipe for baking a cake; just as the recipe describes a method for accomplishing a goal (baking the cake) by listing each step that must be taken throughout the process, an algorithm is an explanation of how to solve a math problem that describes each step necessary in the calculations. Algorithms make it easier for mathematicians to think of better ways to solve certain types of problems, because looking at the steps needed to reach a solution sometimes helps them to see where an algorithm can be made more efficient by eliminating redundant steps or using different methods of calculation.
Algorithms are also important to computer scientists. For example, without algorithms, a computer would have to be programmed with the exact answer to every set of numbers that an equation could accept in order to solve an equation—an impossible task. By programming the computer with the appropriate algorithm, the computer can follow the instructions needed to solve the problem, regardless of which values are used as inputs.
Overview
The word algorithm originally came from the name of a Persian mathematician, Al-Khwarizmi, who lived in the ninth century and wrote a book about the ideas of an earlier mathematician from India, Brahmagupta. At first the word simply referred to the author’s description of how to solve equations using Brahmagupta’s number system, but as time passed it took on a more general meaning. First it was used to refer to the steps required to solve any mathematical problem, and later it broadened still further to include almost any kind of method for handling a particular situation.
Algorithms are often used in mathematical instruction because they provide students with concrete steps to follow, even before the underlying operations are fully comprehended. There are algorithms for most mathematical operations, including subtraction, addition, multiplication, and division.
For example, a well-known algorithm for performing subtraction is known as the left to right algorithm. As its name suggests, this algorithm requires one to first line up the two numbers one wishes to find the difference between so that the units digits are in one column, the tens digits in another column, and so forth. Next, one begins in the leftmost column and subtracts the lower number from the upper, writing the result below. This step is then repeated for the next column to the right, until the values in the units column have been subtracted from one another. At this point the results from the subtraction of each column, when read left to right, constitute the answer to the problem.
By following these steps, it is possible for a subtraction problem to be solved even by someone still in the process of learning the basics of subtraction. This demonstrates the power of algorithms both for performing calculations and for use as a source of instructional support.
Bibliography
Cormen, Thomas H. Algorithms Unlocked. Cambridge, MA: MIT P, 2013.
Cormen, Thomas H. Introduction to Algorithms. Cambridge, MA: MIT P, 2009.
MacCormick, John. Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today's Computers. Princeton: Princeton UP, 2012.
Parker, Matt. Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More. New York: Farrar, 2014.
Schapire, Robert E., and Yoav Freund. Boosting: Foundations and Algorithms. Cambridge, MA: MIT P, 2012.
Steiner, Christopher. Automate This: How Algorithms Came to Rule Our World. New York: Penguin, 2012.
Valiant, Leslie. Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World. New York: Basic, 2013.