multiplicity

(noun)

The number of values for which a given condition holds.

Related Terms

Examples of multiplicity in the following topics:

  • Addition, Subtraction, and Multiplication

    • Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule $i^2=-1$.
    • The multiplication of two complex numbers is defined by the following formula:
    • The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit.
    • Indeed, if i is treated as a number so that di means d time i, the above multiplication rule is identical to the usual rule for multiplying the sum of two terms.
    • = $ac + bdi^2 + (bc + ad)i$ (by the commutative law of multiplication)

  • Division of Complex Numbers

    • Division of complex numbers is accomplished by multiplying by the multiplicative inverse of the denominator.
    • The multiplicative inverse of $z$ is $\frac{\overline{z}}{\abs{z}^2}.$
    • The key is to think of division by a number $z$ as multiplying by the multiplicative inverse of $z$.
    • For complex numbers, the multiplicative inverse can be deduced using the complex conjugate.
    • So the multiplicative inverse of $z$ must be the complex conjugate of $z$ divided by its modulus squared.

  • Matrix Multiplication

    • If $A$ is an $n\times m $ matrix and $B$ is an $m \times p$ matrix, the result $AB$ of their multiplication is an $n \times p$ matrix defined only if the number of columns $m$ in $A$ is equal to the number of rows $m$ in $B$.  
    • Scalar multiplication is simply multiplying a value through all the elements of a matrix, whereas matrix multiplication is multiplying every element of each row of the first matrix times every element of each column in the second matrix.  
    • Scalar multiplication is much more simple than matrix multiplication; however, a pattern does exist.  

  • Basic Operations

    • The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
    • Multiplication also combines multiple quantities into a single quantity, called the product.
    • In fact, multiplication can be thought of as a consolidation of many additions.
    • Division is the inverse of multiplication.
    • Addition and multiplication are commutative operations:

  • Addition and Subtraction; Scalar Multiplication

    • Matrix addition, subtraction, and scalar multiplication are types of operations that can be applied to modify matrices.
    • There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication.
    • Matrix addition, subtraction and scalar multiplication can be used to find such things as: the sales of last month and the sales of this month, the average sales for each flavor and packaging of soda in the $2$-month period.
    • In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction.
    • Scalar multiplication has the following properties:

  • Multiplication of Complex Numbers

    • The key to performing the multiplication is to remember the acronym FOIL, which stands for First, Outer, Inner, Last.
    • Note that this last multiplication yields a real number, since:
    • Note that the FOIL algorithm produces two real terms (from the First and Last multiplications) and two imaginary terms (from the Outer and Inner multiplications).

  • The Order of Operations

    • The order of operations is an approach to evaluating expressions that involve multiple arithmetic operations.
    • Multiplication and division are of equal precedence (tier 3), as are addition and subtraction (tier 4).
    • Here we have an expression that involves subtraction, parentheses, multiplication, addition, and exponentiation.
    • It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
    • This mnemonic makes the equivalence of multiplication and division and of addition and subtraction clear.

  • Solving Equations: Addition and Multiplication Properties of Equality

    • The addition and multiplication properties of equalities are useful tools for solving equations.
    • Then undo the multiplication operation (using the division property) by dividing both sides of the equation by 34:
    • Second, use the multiplication property to multiply both sides of the equation by 8:

  • The Fundamental Theorem of Algebra

    • .$ We say that a root $x_0$ has multiplicity $m$ if $(x-x_0)^m$ divides $f(x)$ but $(x-x_0)^{m+1}$ does not.
    • admits one complex root of multiplicity $4$, namely $x_0 = 0$, one complex root of multiplicity $3$, namely $x_1 = i$, and one complex root of multiplicity $1$, namely $x_2 = - \pi$.
    • The sum of the multiplicity of the roots equals the degree of the polynomial, $8$.
    • where $f_1(x)$ is a non-zero polynomial of degree $n-1.$ So if the multiplicities of the roots of $f_1(x)$ add to $n-1$, the multiplicity of the roots of $f$ add to $n$.
    • The multiplicities of the complex roots of a nonzero polynomial with complex coefficients add to the degree of said polynomial.

  • Multiplying Algebraic Expressions

    • (Note that multiplying monomials is not the same as adding algebraic expressions—monomials do not have to involve "like terms" in order to be combined together through multiplication.)
    • Any negative sign on a term should be included in the multiplication of that term.
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.