Editing Elementary arithmetic (section)

==Multiplication==
{{Main|Multiplication}}[[Multiplication]] is a mathematical operation of repeated addition. When two numbers are multiplied, the resulting value is a product. The numbers being multiplied are multiplicands, multipliers, or factors. Multiplication can be expressed as "five times three equals fifteen," "five times three is fifteen," or "fifteen is the product of five and three."

Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅). The statement "five times three equals fifteen" can be written as "<math>5 \times 3 = 15</math>", "<math>5 \ast 3 = 15</math>", "<math>(5)(3) = 15</math>", or "<math>5 \cdot 3 = 15</math>".

In elementary arithmetic, multiplication satisfies the following properties{{efn|While elementary arithmetic mainly operates under the [[Set (mathematics)|set]] of [[Natural number|natural numbers]] (sometimes including 0), multiplication under other number sets can satisfy more or less properties than those listed here, such as having an [[inverse element]] in the [[Rational number|rational numbers]] and beyond, or lacking [[commutativity]] in the [[quaternions]] and higher order number sets.}}:
* [[Commutativity]]. Switching the order in a product does not change the result: <math>a \times b = b \times a</math>.
* [[Associativity]]. Rearranging the order of parentheses in a product does not change the result: <math>a \times (b \times c) = (a \times b) \times c</math>.
* [[Distributivity]]. Multiplication distributes over addition: <math>a \times (b + c) = a \times b + a \times c</math>.
* [[Identity element|Identity]]. Any number multiplied by 1 is itself: <math>a \times 1 = a</math>.
* [[Absorbing element|Zero]]. Any number multiplied by 0 is 0: <math>a \times 0 = 0</math>.

In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit".

=== Example of multiplication for a single-digit factor ===
Multiplying 729 and 3, starting on the ones column, the product of 9 and 3 is 27. 7 is written under the ones column and 2 is written above the tens column as a carry digit.

{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|Tens}}  
|{{verth|'''''Ones'''''}}
|-
| || ||'''2'''||
|-
| ||7 ||2 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
| || || ||7
|}

The product of 2 and 3 is 6, and the carry digit adds 2 to 6, so 8 is written under the tens column.

{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|'''''Tens'''''}}
|{{verth|Ones}}
|-
|
|
|2
|
|-
| ||7 ||2 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
| || ||8 ||7
|}

The product of 7 and 3 is 21, and since this is the last digit, 2 will not be written as a carry digit, but instead beside 1.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|'''''Hundreds'''''}}
|{{verth|Tens}}
|{{verth|Ones}}
|-
|
|
|2
|
|-
| || 7 || 2 || 9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
|2 ||1 ||8 ||7
|}

The result:
:<math>3 \times 729 = 2187</math>

=== Example of multiplication for multiple-digit factors ===
Multiplying 789 and 345, starting with the ones column, the product of 789 and 5 is 3945.
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| ||7 ||8 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
|3 ||9 ||4 ||5
|}

4 is in the tens digit. The multiplier is 40, not 4. The product of 789 and 40 is 31560.
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|-
|
| ||7 ||8 ||9
|-
|
| style="border-bottom: 1px solid black;" |× || style="border-bottom: 1px solid black;" |3 || style="border-bottom: 1px solid black;" |4 || style="border-bottom: 1px solid black;" |5
|-
|
|3 ||9 ||4 ||5
|-
|3
|1 ||5 ||6 ||0 
|}

3 is in the hundreds digits. The multiplier is 300. The product of 789 and 300 is 236700.
{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|
| ||7 ||8 ||9
|-
|
|
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
|
|
|3 ||9 ||4 ||5
|-
|
|3
|1 ||5 ||6 ||0 
|-
|2
|3
|6 ||7 ||0 ||0 
|}

Adding all the products,
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| || || || ||7 ||8 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
| || || ||3 ||9 ||4 ||5
|-
| || ||3 ||1 ||5 ||6 ||0
|-
|style="border-bottom: 1px solid black;"|+ ||style="border-bottom: 1px solid black;"|2 ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|6 ||style="border-bottom: 1px solid black;"|7 ||style="border-bottom: 1px solid black;"|0 ||style="border-bottom: 1px solid black;"|0
|-
| ||2 ||7 ||2 ||2 ||0 ||5
|}

The result:

:<math>789 \times 345 = 272205</math>