# Fundamental Mathematics/Arithmetic/Multiplication

Four bags of three marbles gives twelve marbles. There are also 3 sets consisting of 4 marbles of the same colour.
Multiplication can also be thought of as scaling. In the above animation, we see 3 being multiplied by 2, giving 6 as a result
Area of a cloth 4.5m × 2.5m = 11.25m2; 4½ × 2½ = 11¼

Multiplication (often denoted by the cross symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division). Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:

${\displaystyle 3\times 4=3+3+3+3=12.\!\,}$

Here 3 and 4 are the "factors" and 12 is the "product". There are differences amongst educators as to which number should normally be considered as the number of copies and whether multiplication should even be introduced as repeated addition. For example 3 multiplied by 4 can also be calculated by adding 3 copies of 4 together:

${\displaystyle 3\times 4=4+4+4=12.\!\,}$

Multiplication of rational numbers (fractions) and real numbers is defined by systematic generalization of this basic idea. Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths (for numbers generally). The area of a rectangle does not depend on which side is measured first which illustrates that the order in which numbers are multiplied together does not matter. In general the result of multiplying two measurements gives a result of a new type depending on the measurements. For instance:

${\displaystyle 2.5{\mbox{ meters}}\times 4.5{\mbox{ meters}}=11.25{\mbox{ square meters}},\!\,}$
${\displaystyle 11{\mbox{ meters/second}}\times 9{\mbox{ seconds}}=99{\mbox{ meters}}.\!\,}$

The inverse operation of multiplication is division. For example, 4 multiplied by 3 equals 12. Then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is also defined for other types of numbers (such as complex numbers), and for more abstract constructs such as matrices. For these more abstract constructs, the order in which the operands are multiplied sometimes does matter.

## Operation

${\displaystyle A\times B=C}$

Where

${\displaystyle A}$ = Multiplicand Factor
${\displaystyle B}$ = Multiplier Factor
${\displaystyle C}$ = Product

## Multiplication

• Commutative law: ${\displaystyle a\times b=b\times a}$ .
• Associative law: ${\displaystyle (a\times b)\times c=a\times (b\times c)}$ .
• Multiplicative identity: ${\displaystyle a\times 1=a}$ .
• Multiplicative inverse: ${\displaystyle a\times {\frac {1}{a}}=1}$ , whenever ${\displaystyle a\neq 0}$
• Distributive law: ${\displaystyle a\times (b+c)=(a\times b)+(a\times c)}$ .

## Multiplication table

*  1   2   3   4   5   6   7   8   9  10  11  12
1  1   2   3   4   5   6   7   8   9  10  11  12
2  2   4   6   8  10  12  14  16  18  20  22  24
3  3   6   9  12  15  18  21  24  27  30  33  36
4  4   8  12  16  20  24  28  32  36  40  44  48
5  5  10  15  20  25  30  35  40  45  50  55  60
6  6  12  18  24  30  36  42  48  54  60  66  72
7  7  14  21  28  35  42  49  56  63  70  77  84
8  8  16  24  32  40  48  56  64  72  80  88  96
9  9  18  27  36  45  54  63  72  81  90  99 108


1

 9*12=

2

 4*12=

3

 10*11=

4

 11*8=

5

 4*10=

6

 3*4=

7

 9*2=

8

 11*12=

9

 2*3=

10

 10*6=

11

 8*7=

12

 10*4=

13

 7*8=

14

 10*2=

15

 9*9=

16

 8*4=

17

 3*11=

18

 8*9=

19

 12*3=

20

 10*10=