Coefficient

For other uses of this word, see coefficient (disambiguation).

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 a_k, ... 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 a_i ≠ 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 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,...,x_n) of a vector v in a vector space with basis {e₁,e₂,...,e_n}, are the coefficients of the basis vectors in the expression

v = x₁e₁ + x₂e₂ + ... + x_ne_n.

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 (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 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.

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 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.