Coefficient

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

$7x^2-3xy+1.5+y$

the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. 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 or any expression in these parameters. In such a case one must clarify which symbols represent variables and which ones represent parameters. Following Descartes, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but it is not always the case. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1.5 + y.

When one writes

$ax^2+bx+c$,

it is generally supposed that x is the only variable and that a, b and c are parameters; thus the constant coefficient is c in this case.

Similarly, any polynomial in one variable x can be written as

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

for some integer $k$, where $a_k, \dotsc, a_1, a_0$ 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 $a_i \ne 0$ (if any), $a_i$ 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 (also Leading entry) 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 $(x_1, x_2, \dotsc, x_n)$ of a vector $v$ in a vector space with basis $\lbrace e_1, e_2, \dotsc, e_n \rbrace$, are the coefficients of the basis vectors in the expression

$v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .$

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 (K_D) (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 (C_L or C_Z) (Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the plan-form 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 damped.

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

${\displaystyle {\ce {2H2{+}O2\rightarrow 2H2O}}}$ ,

the number 2's in front of ${\displaystyle {\ce {H2}}}$ and ${\displaystyle {\ce {H2O}}}$ are stoichiometric coefficients.

See also

• Degree of a polynomial
• Monic polynomial

References

• Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0-87626-140-3 .
• Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0-534-01138-1 .
• Steven Schwartzman (1994) The Words of Mathematics: an etymological dictionary of mathematical terms used in English, page 48, Mathematics Association of America, ISBN 0-88385-511-9.