Table shows basic properties of mathematical operations such as commutativity of addition or distributive property of multiplication over addition.

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Basic number operations#

Operation name	Operation symbol	Example
Addition	Show source $+$	Show source $6 + 3 = 9$
Subtraction	Show source $-$	Show source $6 - 3 = 3$
Multiplication	Show source $\cdot$	Show source $6 \cdot 3 = 18$
Division	Show source $:$	Show source $6 : 3 = 2$

Properties of algebra operations#

Name	Formula
The commutativity of addition	Show source $a+\mathrm{b}=\mathrm{b}+a$
The commutativity of multiplication	Show source $a \cdot \mathrm{b}=\mathrm{b} \cdot a$
The associative of addition	Show source $a+\mathrm{b}+c=a+\mathrm{b}+c$
The associative of multiplication	Show source $a \cdot \mathrm{b} \cdot c=a \cdot \mathrm{b} \cdot c$
The distributive property of multiplication over addition	Show source $a \cdot \left(\mathrm{b}+c\right)=a \cdot \mathrm{b}+a \cdot c$
The addition of zero	Show source $a+0=a$
The multiplication by one	Show source $a \cdot 1=a$
The multiplication by zero	Show source $a \cdot 0=0$

Names of arguments (operands) and result#

Operation name	Name of the first argument	Colloquial operation name	Name of second argument	eqSymbol	Name of result
Addition	first summand	plus	second summand	=	sum
Subtraction	minuend	minus	subtrahend	=	difference
Multiplication	first factor	times	second factor	=	product
Division	dividend	per	divisor	=	product

Commutativity and associative laws#

Property	Addition	Subtraction	Multiplication	Division
Operation has commmutativity property (the order of terms does not matter)	yes	no	yes	no
Operation has associative property (it does not matter where the bracket stands, i.e. how terms are grouped)	yes	no	yes	no
Commmutativity example	$3 + 2 = 5$ $2 + 3 = 5$	-	$3 \cdot 2 = 6$ $2 \cdot 3 = 6$	-
Associative example	$3 + \left(4 + 5\right) = 12$ $\left(3 + 4\right) + 5 = 12$	-	$3 \cdot \left(4 \cdot 5\right) = 60$ $\left(3 \cdot 4\right) \cdot 5 = 60$	-
Commmutativity counter-example (why this operation has NO commutativity property)	-	$3 - 2 = 1$ $2 - 3 = -1$	-	$6 : 3 = 2$ $3 : 6 = \dfrac{1}{2}$
Associative counter-example (why this operation has NO associative property)	-	$5 - \left(4 + 3\right) = 4$ $\left(5 - 4\right) - 3 = -2$	-	$24 : \left(6 : 2\right) = 8$ $\left(24 : 6\right) : 2 = 2$

Some facts#

Basic matemathematics operation, that we can do on numbers are:
- addition, marked with a symbol $+$ :
  
  $w = a + b$
- subtraction, marked with a symbol $-$ :
  
  $w = a - b$
- multiplication, marked with a symbol $\cdot$ or $\times$ :
  
  $w = a \cdot b = a \times b$
- division, marked with a symbol $/$ , $:$ or by using fraction bar:
  
  $w = a / b = a : b = \dfrac{a}{b}$
Depending on the type of operation, we will name the obtained result in a different way:
- the result of the addition is called sum ( $a + b$ ),
- the result of the subtraction is called difference ( $a - b$ ),
- the result of the multiplication is called product ( $a \cdot b$ ),
- the result of the division is called quotient ( $a : b$ ).
Depending on the type of operation, we also call differently the numbers on which we perform this operation (so-called arguments or operands):
- numbers, which we add to each other, we call summands or addends:
  
  $\text{sum} = \text{the first summand} + \text{second summand}$
- numbers that we subtract from each other, we call minuend and subtrahend:
  
  $\text{difference} = \text{minuend} - \text{subtrahend}$
- numbers, which we multiply, we call factors:
  
  $\text{product} = \text{the first factor} \cdot \text{second factor}$
- numbers that we divide, we call dividend and divisor
  
  $\text{quotient} = \text{dividend} : \text{divisor}$
If you want to learn more about names of operands and results of various math operations check out our another calculator: Operands and results names.

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Tags:

math_operation_properties · real_numbers_properties · properties_of_algebra_operations · commutativity_of_multiplication · commutativity_of_addition · distributive_property_of_multiplication_over_addition · associative_of_addition · associative_of_multiplication

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wlasnosci_dzialan · wlasnosci_operacji_algebraicznych · wlasnosci_liczb_rzeczywistych · przemiennosc_mnozenia · przemiennosc_dodawanie · rozdzielnosc_mnozenia_wzgledem_dodawania · lacznosc_mnozenia · lacznosc_dodawania

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