Multiplication Stuff

Multiplicand

The original quantity or number that is to be multiplied by another.

Multiplier

The factor that indicates how many times another number is to be multiplied.

Product

The result obtained from performing multiplication on two or more numbers.

Factor

A number or algebraic expression that is multiplied with another to yield a product.

Commutative Property of Multiplication

The principle stating that changing the order of factors does not change the result: $a \times b = b \times a$.

Associative Property of Multiplication

The principle stating that changing the grouping of factors does not change the result: $(a \times b) \times c = a \times (b \times c)$.

Distributive Property

The principle stating that multiplying a sum by a number is equivalent to multiplying each addend separately and summing the products: $a \times (b + c) = (a \times b) + (a \times c)$.

Identity Property of Multiplication

The principle stating that the product of any number and $1$ is equal to the original number: $a \times 1 = a$.

Zero Property of Multiplication

The principle stating that the product of any number and $0$ is always $0$: $a \times 0 = 0$.

Multiplicative Inverse

The number that, when multiplied by a given non-zero number $a$, yields $1$, represented as $\frac{1}{a}$ or $a^{-1}$.

Repeated Addition

A conceptual model of multiplication where a quantity is added to itself a specified number of times.

Partial Products

The intermediate results formed by multiplying a multi-digit number by each digit of another factor before adding them together.

Lattice Multiplication

A method that uses a grid to break down multi-digit factor operations into smaller single-digit operations and diagonal additions.

Cross-Multiplication

A technique used to simplify or solve an equation between two ratios by setting the product of the numerator of one side and the denominator of the other side equal: if $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.

Multiplication of Signed Numbers

The rule stating that multiplying two numbers with the same sign yields a positive result, while multiplying numbers with opposite signs yields a negative result.

Product Rule for Exponents

The algebraic rule stating that when multiplying powers with the same base, the exponents are added: $x^a \times x^b = x^{a+b}$.

Multiplying Fractions

The process of calculating $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ by multiplying the numerators together and the denominators together.

Multiplying Decimals

The method of evaluating decimal factors as whole numbers and placing the decimal point in the final result so it has as many decimal places as the sum of decimal places in the factors.

Dot Product

An algebraic operation taking two equal-length sequences of vectors and returning a single scalar: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta)$.

Cross Product

A binary operation on two vectors in three-dimensional space resulting in a vector perpendicular to both input vectors.

Matrix Multiplication

A mathematical operation producing a single matrix from two matrices $A$ and $B$, where each element $c_{i,j}$ is calculated as the dot product of row $i$ of $A$ and column $j$ of $B$.

Polynomial Multiplication

The process of multiplying algebraic expressions containing terms with variables using the distributive property.

FOIL Method

An acronym for First, Outer, Inner, Last, describing the standard order for expanding two binomials: $(a + b)(c + d) = a \times c + a \times d + b \times c + b \times d$.

Multiplicative Comparison

A conceptual framework interpreting multiplication as comparing two quantities where one is a specified number of times as large as the other.

Scientific Notation Multiplication

The process of combining exponential expressions in base 10: $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$.

Commutative Property of Multiplication

The principle stating that changing the order of factors does not change the result: $a \times b = b \times a$.

Associative Property of Multiplication

The principle stating that changing the grouping of factors does not change the result: $(a \times b) \times c = a \times (b \times c)$.

Distributive Property

The principle stating that multiplying a sum by a number is equivalent to multiplying each addend separately and summing the products: $a \times (b + c) = (a \times b) + (a \times c)$.

Identity Property of Multiplication

The principle stating that the product of any number and $1$ is equal to the original number: $a \times 1 = a$.

Zero Property of Multiplication

The principle stating that the product of any number and $0$ is always $0$: $a \times 0 = 0$.

Multiplicative Inverse

The number that, when multiplied by a given non-zero number $a$, yields $1$, represented as $\frac{1}{a}$ or $a^{-1}$.

Repeated Addition

A conceptual model of multiplication where a quantity is added to itself a specified number of times.

Partial Products

The intermediate results formed by multiplying a multi-digit number by each digit of another factor before adding them together.

Lattice Multiplication

A method that uses a grid to break down multi-digit factor operations into smaller single-digit operations and diagonal additions.

Cross-Multiplication

A technique used to simplify or solve an equation between two ratios by setting the product of the numerator of one side and the denominator of the other side equal: if $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.

Multiplication of Signed Numbers

The rule stating that multiplying two numbers with the same sign yields a positive result, while multiplying numbers with opposite signs yields a negative result.

Product Rule for Exponents

The algebraic rule stating that when multiplying powers with the same base, the exponents are added: $x^a \times x^b = x^{a+b}$.

Multiplying Fractions

The process of calculating $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ by multiplying the numerators together and the denominators together.

Multiplying Decimals

The method of evaluating decimal factors as whole numbers and placing the decimal point in the final result so it has as many decimal places as the sum of decimal places in the factors.

Dot Product

An algebraic operation taking two equal-length sequences of vectors and returning a single scalar: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta)$.

Cross Product

A binary operation on two vectors in three-dimensional space resulting in a vector perpendicular to both input vectors.

Matrix Multiplication

A mathematical operation producing a single matrix from two matrices $A$ and $B$, where each element $c_{i,j}$ is calculated as the dot product of row $i$ of $A$ and column $j$ of $B$.

Polynomial Multiplication

The process of multiplying algebraic expressions containing terms with variables using the distributive property.

FOIL Method

An acronym for First, Outer, Inner, Last, describing the standard order for expanding two binomials: $(a + b)(c + d) = a \times c + a \times d + b \times c + b \times d$.

Multiplicative Comparison

A conceptual framework interpreting multiplication as comparing two quantities where one is a specified number of times as large as the other.

Scientific Notation Multiplication

The process of combining exponential expressions in base 10: $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$.