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The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.. An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
Prime number: A positive integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A positive integer that can be factored into a product of smaller positive integers. Every integer greater than one is either prime or composite.
Newton's method is one of many known methods of computing square roots. Given a positive number a, the problem of finding a number x such that x 2 = a is equivalent to finding a root of the function f(x) = x 2 − a. The Newton iteration defined by this function is given by
Taking the positive root, one finds = = / = /. A geometric way of deriving the sine or cosine of 45° is by considering an isosceles right triangle with leg length 1. Since two of the angles in an isosceles triangle are equal, if the remaining angle is 90° for a right triangle, then the two equal angles are each 45°.
The root system G 2 has 12 roots, which form the vertices of a hexagram. See the picture above. One choice of simple roots is (α 1, β = α 2 − α 1) where α i = e i − e i+1 for i = 1, 2 is the above choice of simple roots for A 2. The G 2 root lattice—that is, the lattice generated by the G 2 roots—is the same as the A 2 root lattice.
Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2 n members. Integer powers of 2 are important in computer science. The positive integer powers 2 n give the number of possible values for an n-bit integer binary number; for example, a byte may take 2 8 = 256 different values.
where p 0, p 1, …, p m, the coefficients of p, are elements of K, p m ≠ 0 if m > 0, and X, X 2, …, are symbols, which are considered as "powers" of X, and follow the usual rules of exponentiation: X 0 = 1, X 1 = X, and = + for any nonnegative integers k and l. The symbol X is called an indeterminate [5] or variable. [6]