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Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in

7x2 − 3xy + 1.5 + y

the first three terms respectively have the coefficients 7, −3, and 1.5 (in the third term the variables are hidden (raised to the 0 power), so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression). The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c in

ax2 + bx + c

when it is understood that these are not considered as variables.

Thus a polynomial in one variable x can be written as

$a_k x^k + \cdots + a_1 x^1 + a_0$

for some integer k, where ak, ... a1, a0 are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest i with ai ≠ 0 (if any), ai is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

$\, 4x^5 + x^3 + 2x^2$

is 4.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal's triangle.

Linear algebra

In linear algebra, the leading coefficient of a row in a matrix is the first nonzero entry in that row. So, for example, given

$M = \begin{pmatrix} 1 & 2 & 0 & 6 \\ 0 & 2 & 9 & 4 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates (x1,x2,...,xn) of a vector v in a vector space with basis {e1,e2,...,en}, are the coefficients of the basis vectors in the expression

v = x1e1 + x2e2 + ... + xnen.

Coefficient is just the fancy name for the numbers multiplied by variables.

Examples of physical coefficients

1. Coefficient of Thermal Expansion (thermodynamics) (dimensionless) - Relates the change in temperature to the change in a material's dimensions.
2. Partition Coefficient (KD) (chemistry) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium.
3. Hall coefficient (electrical physics) - Relates a magnetic field applied to an element to the voltage created, the amount of current and the element thickness. It is a characteristic of the material from which the conductor is made.
4. Lift coefficient (CL or CZ) (Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the planform area of the airfoil.
5. Ballistic coefficient (BC) (Aerodynamics) (units of kg/m2) - A measure of a body's ability to overcome air resistance in flight. BC is a function of mass, diameter, and drag coefficient.
6. Transmission Coefficient (quantum mechanics) (dimensionless) - Represents the probability flux of a transmitted wave relative to that of an incident wave. It is often used to describe the probability of a particle tunnelling through a barrier.
7. Damping Factor a.k.a. viscous damping coefficient (Physical Engineering) (units of newton-seconds per meter) - relates a damping force with the velocity of the object whose motion is being

Chemistry

A coefficient is a number placed in front of a term in a chemical equation to indicate how many molecules (or atoms) take part in the reaction. For example, in the formula $2H_2 + O_2 \rarr 2H_2O$, the number 2's in front of H2 and H2O are stoichiometric coefficients.

References

• Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0876261403 .
• Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0534011381 .
• Steven Schwartzman (1994) The Words of Mathematics: an etymological dictionary of mathematical terms used in English, page 48, Mathematics Association of America, ISBN 0883855119.

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• coefficient — [ kɔefisjɑ̃ ] n. m. • déb. XVIIe; de co et efficient 1 ♦ Math. Nombre par lequel est multipliée une grandeur. Coefficient de proportionnalité. Valeur affectée d un coefficient. Chacun des nombres par lesquels sont multipliées les puissances de la …   Encyclopédie Universelle

• coefficient — of expansion coefficient of friction coefficient of rolling resistance aligning torque coefficient aligning stiffness coefficient braking force coefficient braking stiffness coefficient braking traction coefficient camber stiffness coefficient …   Mechanics glossary

• Coefficient — Co ef*fi cient, n. 1. That which unites in action with something else to produce the same effect. [1913 Webster] 2. [Cf. F. coefficient.] (Math.) A number or letter put before a letter or quantity, known or unknown, to show how many times the… …   The Collaborative International Dictionary of English

• coefficient — UK US /ˌkəʊɪˈfɪʃənt/ noun [C] ► in mathematics, a value that appears in front of and multiplies another value: »In 2x + 4y = 7, 2 is the coefficient of x. ► a number or a method of calculating a number that measures the characteristics of a… …   Financial and business terms

• Coëfficient — (lat.), jeder Ausdruck, in welchem nichts von dem, was als Hauptgröße in einem Gliede angesehen wird, vorkommt, u. womit diese Hauptgröße multiplicirt ist. Wo die letztere keinen C. hat, ist 1 als solcher anzusehen. Je nachdem in einem Gliede die …   Pierer's Universal-Lexikon

• coefficient — [kō΄ə fish′ənt] n. [ CO + EFFICIENT: orig. an adj. meaning “cooperating”] 1. a factor that contributes to produce a result 2. Math. a number or algebraic symbol prefixed as a multiplier to a variable or unknown quantity (Ex.: x in x(y+z),6 in… …   English World dictionary

• Coefficient — Co ef*fi cient, a. Co[ o]perating; acting together to produce an effect. {Co ef*fi cient*ly}, adv. [1913 Webster] …   The Collaborative International Dictionary of English

• Coëfficient — Coëfficient, in der Mathematik eine Größe, womit eine andere multiplicirt wird …   Herders Conversations-Lexikon

• coefficient — COEFFICIENT. s. m. Terme d Algèbre. On appelle ainsi Le nombre ou la quantité connue qui est au devant d une quantité algébrique, et qui la multiplie …   Dictionnaire de l'Académie Française 1798

• coefficient — also co efficient, 1660s (adj.), from CO (Cf. co ) + EFFICIENT (Cf. efficient). As a noun, from c.1600, probably influenced by Mod.L. coefficiens, which was used in mathematics in 16c., introduced by French mathematician François Viète (1540… …   Etymology dictionary

• coefficient — ► NOUN 1) Mathematics a quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x2). 2) Physics a multiplier or factor that measures some property …   English terms dictionary