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3.1 Integrals. 3.2 Efficient infinite series. 3.3 Other infinite series. 3.4 Machin-like formulae. 3.5 Infinite products. 3.6 Arctangent formulas. 3.7 Complex functions.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic. [2] The minimal polynomials of trigonometric numbers can be explicitly enumerated. [3]
cis is a mathematical notation defined by cis x = cos x + i sin x, [nb 1] where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form).
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. [1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S 1 because it is a one-dimensional unit n-sphere ...
In other words, the n th digit of this number is 1 only if n is one of 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the Liouville numbers ...
For example, a 7-simplex is (1,1) 8 = (1,2,1) 4 = (1,4,6,4,1) 2 = (1,8,28,56,70,56,28,8,1). The number of 1-faces (edges) of the n -simplex is the n -th triangle number , the number of 2-faces of the n -simplex is the ( n − 1) th tetrahedron number , the number of 3-faces of the n -simplex is the ( n − 2) th 5-cell number, and so on.