Multiplication

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Multiplication of long numbers mentally

The sign "×" (the cross of Saint Andrew) used in arithmetic to indicate multiplication

Commutative property:
3×4 = 12 = 4×3
twelve elements can be ordered in three rows of four, or four columns of three.

Animation to represent multiplication 2 × 3 = 6.

Multiplication is a binary operation and derivative of addition that is established on a set of numbers. In arithmetic, it is one of the four elementary operations, along with addition, subtraction, and addition. division, and is the inverse operation of the latter. This means that for every multiplication there is a division, for example for "5 times 2 equals 10" the equivalent division is "10 divided by 2 equals 5", or "10 divided by 5 equals 2".

There are two signs to indicate this operation between natural numbers: the cross "×" and the fat dot at half height (•). In the case of variables represented by letters (only letters or a mixture) the point is used (not the cross) but it can be dispensed with, for example 3ab (it is read “three a b”) xy + 2y (it is read “equis i plus two i's)

Multiplying a quantity by a number consists of adding said quantity as many times as the number indicates. Thus, 4×3 (read “four multiplied by three” or simply “four times three”) ») is equal to adding three times the number 4 (4+4+4)^(note) It can also be interpreted as 3 rows of 4 objects, or 4 rows of 3 (see drawing). 4 and 3 are the factors, and 12, the result of the operation, is the product. Multiplication is associated with the concept of geometric area: it is easy to see that the area of a rectangle is obtained by multiplying the length of both sides, just by imagining the surface covered with square tiles. We can multiply two numbers or more, and it doesn't matter in what order we carry out the operation or how we group the numbers; the same result will always be obtained:

3 • 4 • 5 = 5 • 3 • 4 = 4 • 5 • 3 = 12 • 5 = 15 • 4 = 20 • 3 = 60

The result of multiplying two or more numbers is called the product. The numbers that are multiplied are called factors or coefficients, and individually: multiplying (number to be added or number being multiplied) and multiplier (times the multiplicand is added). This differentiation makes little sense when, in the set where the product is defined, the commutative property of multiplication is given (for example, in numerical sets: 3×7 = 7×3, that is, the order of factors does not alter the product). However, it can be useful if used to refer to the multiplier of an algebraic expression (eg: in

${displaystyle a^{2}b+a^{2}b+a^{2}b}$ or ${displaystyle 3a^{2}b}$ ,

3 is the multiplier or coefficient, while the monomio ${displaystyle a^{2}b}$ is multiplying).

The potentiation is a particular case of multiplication where the exponent indicates the number of times a number must be multiplied by itself. Example: 2 • 2 • 2 • 2 • 2 • 2 • = 2 ⁶ = 64

Here, 6 is the exponent, and 2 is the base.

In modern algebra, the name «quotient» or «multiplication» is usually used with its usual notation «·» to designate the external operation in a module, to also designate the second operation that is defined in a ring (the one for the that the inverse element of 0 is not defined), or to designate the operation that endows a set with group structure. The inverse operation of multiplication is division.

Notation

Multiplication is indicated by a cross (×) or a dot (∙). In the absence of these characters, the asterisk (*) is usually used, especially in computing (this use has its origin in FORTRAN), but it is discouraged in other areas and should only be used when there is no other alternative. The letter xes (X x) is sometimes used, but this is discouraged because it creates unnecessary confusion with the letter normally assigned to an unknown in an equation. Finally, the multiplication sign can be omitted unless numbers are being multiplied or confusion about the names of the unknowns, constants, or functions may be caused (for example, when the name of an unknown has more than one letter and could be confused with the product of two others). Grouping signs such as parentheses (), brackets [ ], braces { } or slash | |. This is mostly used to multiply negative numbers by each other or by positive numbers.

If the factors are not written individually but belong to a list of elements with some regularity, the product can be written using an ellipsis, that is, explicitly write the first and last terms, (or in the case of a product of infinite terms only the first ones), and replace the others with an ellipsis. This is analogous to what is done with other operations applied to infinite numbers (such as addition).

Thus, the product of all natural numbers from 1 to 100 can be written:

{displaystyle 1cdot 2cdot ldots cdot 99cdot 100}

while the product of the even numbers between 1 and 100 would be written:

{displaystyle 2cdot 4cdot 6cdots 100}

This can also be denoted by writing the ellipsis in the middle of the line of text:

{displaystyle 1cdot 2cdot cdots cdot 99cdot 100}

In any case, it should be clear which terms are omitted.

Lastly, the product can be denoted by the product symbol, which comes from the Greek letter Π (capital Pi).

This is defined as:

{displaystyle prod _{i=m}^{n}x_{i}=x_{m}cdot x_{m+1}cdot x_{m+2}cdot cdots cdot x_{n-1}cdot x_{n}.}

The subscript ${displaystyle i,}$ indicates a variable that runs through the integers from a minimum value ( ${displaystyle m,}$ indicated in the subscript) and a maximum value. ( $n,$ indicated in the superscript.

Definition

Four bags of three balloons give a total of twelve balloons (3×4=12).

The multiplication of two integers n and m is expressed as:

{displaystyle sum _{k=1}^{n}m=mn}

This is just a way of symbolizing the expression “add m to itself n times”. You can make it easier to understand by expanding the above expression:

${displaystyle mn=underbrace {m+cdots +m} _{n}}$ ,

such that there are n addends. For example:

${displaystyle 5times 2=5+5=10}$
${displaystyle 2times 5=2+2+2+2+2=10}$
${displaystyle 4times 3=4+4+4=12}$
${displaystyle mtimes 6=m+m+m+m+m+m=6m}$
${displaystyle mtimes 5=m+m+m+m+m=5m}$

The product of infinite terms is defined as the limit of the product of the first n terms when n increases indefinitely.

Recursive definition

In the case of the multiplication of natural numbers ${displaystyle mathbb {N} ={0,1,2,3....,n,...}}$ can be applied Recursive definition of multiplication that includes these two steps:

{displaystyle mtimes 0=0}

{displaystyle m(n+1)=(mn)+m}

Where m and n are natural numbers, the induction principle is applied on the number n, which is initially n = 0, then assuming it is true for n, it follows that it also holds for n+1.

The following basic propositions are deduced:

Existence of the identity element,

{displaystyle ncdot 1=n}

all natural number n.

Associative property,

{displaystyle (mcdot n)cdot p=mcdot (ncdot p)}

for any m, n, p natural numbers

Commutative property:

{displaystyle mcdot n=ncdot m}

for any natural number.

Distributive property with respect to the addition:

{displaystyle mcdot (l+n)=mcdot l+mcdot n=(l+n)cdot m}

There are no divisors of zero:

{displaystyle mcdot n=0}

implies that at least one of the factors is equal to zero.

To indicate the product of two natural numbers, a point is used between the two factors, a cross between them, the simple juxtaposition of the literal factors or, one factor and the other in parentheses or the two factors in parentheses

Product of integers

It's an integer. $m$ calculated as follows:

Yeah.

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and

0}" xmlns="http://www.w3.org/1998/Math/MathML">p▪0{displaystyle p 2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;"/>

then.

{displaystyle m=ncdot p}

positive factors.

Yeah.

<math alttext="{displaystyle nn.0{displaystyle n vis0} <img alt="{displaystyle n

and

<math alttext="{displaystyle pp.0{displaystyle p vis0} <img alt="{displaystyle p

then. m = MINUSINES LINKnegative factors.

Yeah.

0}" xmlns="http://www.w3.org/1998/Math/MathML">n▪0{displaystyle n/2003/0}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;"/>

and

<math alttext="{displaystyle pp.0{displaystyle p vis0} <img alt="{displaystyle p

<math alttext="{displaystyle nn.0{displaystyle n vis0} <img alt="{displaystyle n

and

0}" xmlns="http://www.w3.org/1998/Math/MathML">p▪0{displaystyle p 2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;"/>

then. m = - Cambridge. a positive factor and the other negative.

Yeah.

{displaystyle n=0}

and

{displaystyle p=0}

then.

{displaystyle m=0=ncdot p}

. At least one zero factor.

The product of the integers is based on the product of the natural numbers and the absolute value is taken into account.

Product of Fractions

The fraction ${displaystyle {frac {p_{1}cdot p_{2}}{q_{1}cdot q_{2}}}}$ is the product of the fractions ${displaystyle {frac {p_{1}}{q_{1}}}}$ and ${displaystyle {frac {p_{2}}{q_{2}}}}$ of equality

{displaystyle {frac {p_{1}}{q_{1}}}cdot {frac {p_{2}}{q_{2}}}={frac {p_{1}cdot p_{2}}{q_{1}cdot q_{2}}}}

. It is assumed that ${displaystyle q_{1}neq 0,q_{2}neq 0}$ .

Root product

The following root product property holds:

The root of a product is equal to the product of the roots of the factors named above.
${displaystyle {sqrt[{n}]{{a}cdot {b}}}={sqrt[{n}]{a}}cdot {sqrt[{n}]{b}}}$

Properties

Multiplication of numbers from 0 to 10. Each line traced represents a multiplier. Axis x = multipliers. Axis and = products.

For whole numbers, integers, fractions, and real and complex numbers, multiplication has certain properties:

Locking property

The multiplication of two or more natural numbers gives us another natural number as a result example: 33*2=66

Commutative property

The order of the factors does not alter the product.

{displaystyle xcdot y=ycdot x}

Associative property

Only expressions of multiplication or addition are invariant with respect to the order of operations.

{displaystyle (xcdot y)cdot z=xcdot (ycdot z)}

Distributive property

The total sum of two numbers multiplied by a third number is equal to the sum of the products between the third number and each adding.

{displaystyle xcdot (y+z)=(xcdot y)+(xcdot z)}

Element of identity (neutro)

Multiplicative identity is 1; the product of every number multiplied by 1 is itself. This is known as the identity property.

{displaystyle xcdot 1=x}

Element zero (absorbing)

Any number multiplied by zero gives as zero product. This is known as the zero of multiplication.

{displaystyle xcdot 0=0}

{displaystyle 0cdot x=0}

Negative

Less one multiplied by any number is equal to the opposite of that number.

{displaystyle (-1)cdot x=(-x)}

Less one multiplied by less one is one.

{displaystyle (-1)cdot (-1)=1}

The product of natural numbers does not include negative numbers.
Inverse element: All number x, except zero, has a inverse multiplier, ${frac {1}{x}}$ , such that ${displaystyle xcdot left({frac {1}{x}}right)=1}$ .

Product of negative numbers

The product of negative numbers also requires some thought. First, consider the number —1. For any positive integer m:

{displaystyle (-1)m=(-1)+(-1)+...+(-1)=-m}

This is an interesting result that shows that any negative number is just a positive number times –1. So the multiplication of any integers can be represented by the multiplication of positive integers and factors –1. The only thing left to define is the product of (–1)(–1):

{displaystyle (-1)(-1)=-(-1)=1}

In this way, the multiplication of two integers is defined. The definitions can be extended to larger and larger sets of numbers: first the set of fractions or rational numbers, then all the real numbers, and finally the complex numbers and other extensions of the real numbers.

Connection with geometry

From a purely geometric point of view, multiplication between 2 values produces an area that is representable. Similarly the product of 3 values produces an equally representable volume.

Extensions

In mathematics, product is synonymous with multiplication.

Certain binary operations performed in specialized contexts are also called product.

Output scale is a binary operation between elements of a vector space that results in an element of the underlying field. The most relevant case is product point.
Vector product or product cross is an operation between vectors of a three-dimensional euclidian space that results in another vector.
Mixed output or triple scale product is a product that combines the vector product and the scaler.
Marriage It's a binary operation between matrices.
Cartesian product is an operation between sets whose result are ordered pairs of respective elements.
Topology product is a topology built in a cartesian product of topological spaces.
- Topology box is another topology built in a cartesian product of topological spaces that matches the previous one in finite products.
Output is a generalization of the vector product.
Direct output is an abstraction that allows to define algebraic structures in products of other algebraic (usually Cartesian products)
Producer Notation to denote an arbitrary product of terms.
Output (category theory) is an abstract generalization of the products found in various algebraic structures.

The term product is also related to

Product Rule, a method for calculating the derivative of a product of functions.

Product principleone of the fundamental principles of counting.
Product vacuum is the product of zero factors.

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Multiplication

Notation

Definition

Recursive definition

Product of integers

Product of Fractions

Root product

Properties

Product of negative numbers

Connection with geometry

Extensions

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