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Calculating Π and Other Mathematical Tales
Published in José Guillermo Sánchez León, ® Beyond Mathematics, 2017
π is a transcendental number. This means that a finite polynomial of the form a0 + a1x + a2x2+ ... + an xn = 0, with the coefficients a0 ... an being rational numbers (or integers), in which π is one of its roots (solutions) doesn’t exist. As a result, it can be proven that constructing a square with the same area of a given circle by a finite number of steps only using compass and straightedge, also known as squaring the circle, is not possible. There are other transcendental numbers with e being probably the second best-known one after π. N[e, 100] 2.71828182845904523536028747135266249775724709369995957496696762772407663 0353547594571382178525166427
Error Analysis
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
The most important effect of this is that certain numbers cannot be represented precisely within a floating point system. It is obvious that transcendental numbers, such as π $ \pi $ or 2 $ \sqrt{2} $ , require infinite precision. Since infinite precision requires infinite storage capacity, we can quickly assume these numbers must be represented by approximations rather than true values.
Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
A function is said to be transcendental if it is not algebraic. A simple example is y = x + sin x, which is defined for all x but is obviously not algebraic. A transcendental number is one which is the root of an equation that is not algebraic, such as a root of x + sin x = 0.
Middle-school mathematics teachers’ provision of non-examples and explanations in rational number instruction
Published in International Journal of Mathematical Education in Science and Technology, 2022
While organizing the non-examples and linking them to each other, the coders also examined repeatedly the instructional explanations that accompanied these non-examples and discovered the key analytic categories that best represented the entire data set. For instance, the four teachers first explained the definition of a rational number as any number that can be written in the form of a/b, where a and b are integers but b is not equal to 0 and later presented 1/0, 2/0, 3/0, 5/0, and 7/0 as non-examples of a rational number. Thus, these non-examples were assigned the category ‘violation of a constraint in a definition’. T1 and T4 recalled how to find the circumference of a circle and provided the number π as a non-example of a rational number. However, they did not present the students with its decimal number form. Thus, the number π was assigned the category ‘irrationals in transcendental number form’. T2 explained that rational numbers do not fill the number line completely and asked whether the students knew irrational numbers such as . However, he did not introduce its decimal number form. Thus, this non-example referred to the category ‘irrationals in square root number form’. T4 attempted to explain the difference between rational and irrational numbers with the help of 0.257843 … and expressed that infinite non-repeating decimals, but not infinite repeating decimals, are irrational numbers. Thus, the non-example 0.257843 … was assigned the category ‘irrationals in infinite non-repeating decimal form’.