Mathematical tables: operations properties
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.