Examples of multiplicity in the following topics:
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- Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule i2=−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+bdi2+(bc+ad)i (by the commutative law of multiplication)
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- 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.
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- If A is an n×m matrix and B is an m×p matrix, the result AB of their multiplication is an n×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.
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- 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:
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- 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:
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- 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).
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- 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.
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- 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:
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- .Wesaythatarootx_0$ has multiplicity m if (x−x0)m divides f(x) but (x−x0)m+1 does not.
- admits one complex root of multiplicity 4, namely x0=0, one complex root of multiplicity 3, namely x1=i, and one complex root of multiplicity 1, namely x2=−π.
- The sum of the multiplicity of the roots equals the degree of the polynomial, 8.
- where f1(x) is a non-zero polynomial of degree n−1. So if the multiplicities of the roots of f1(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.
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- (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.