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Frequently trees are assumed to have only one root. Trees in set theory are often defined to grow downward making the root the greatest node. [citation needed] Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph.
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
Microsoft Math contains features that are designed to assist in solving mathematics, science, and tech-related problems, as well as to educate the user. The application features such tools as a graphing calculator and a unit converter. It also includes a triangle solver and an equation solver that provides step-by-step solutions to each problem.
This was an open problem until 2007, when an efficient algorithm based on dynamic programming was published. [14] The minimum number of knife changes problem (for the one-dimensional problem): this is concerned with sequencing and permuting the patterns so as to minimise the number of times the slitting knives have to be moved.
the addition and multiplication of R restrict to give operations S × S → S making S a ring with the same multiplicative identity as R. 1 ∈ S; and for all x, y in S, the elements xy, x + y, and −x are in S. S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism.
In a fraction, the number of equal parts being described is the numerator (from Latin: numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin: dēnōminātor, "thing that names or designates"). [2] [3] As an example, the fraction 8 / 5 amounts to eight parts, each of which is of the ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
The square root of a quantity strongly related to the discriminant appears in the formulas for the roots of a cubic polynomial. Specifically, this quantity can be −3 times the discriminant, or its product with the square of a rational number; for example, the square of 1/18 in the case of Cardano formula.