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 nameOperation symbolExample
AdditionShow source++Show source6+3=96 + 3 = 9
SubtractionShow source-Show source63=36 - 3 = 3
MultiplicationShow source\cdotShow source63=186 \cdot 3 = 18
DivisionShow source::Show source6:3=26 : 3 = 2

Properties of algebra operations#

The commutativity of additionShow sourcea+b=b+aa+\mathrm{b}=\mathrm{b}+a
The commutativity of multiplicationShow sourceab=baa \cdot \mathrm{b}=\mathrm{b} \cdot a
The associative of additionShow sourcea+b+c=a+b+ca+\mathrm{b}+c=a+\mathrm{b}+c
The associative of multiplicationShow sourceabc=abca \cdot \mathrm{b} \cdot c=a \cdot \mathrm{b} \cdot c
The distributive property of multiplication over additionShow sourcea(b+c)=ab+aca \cdot \left(\mathrm{b}+c\right)=a \cdot \mathrm{b}+a \cdot c
The addition of zeroShow sourcea+0=aa+0=a
The multiplication by oneShow sourcea1=aa \cdot 1=a
The multiplication by zeroShow sourcea0=0a \cdot 0=0

Names of arguments (operands) and result#

Operation nameName of the first argumentColloquial operation nameName of second argumenteqSymbolName of result
Additionfirst summandplussecond summand=sum
Multiplicationfirst factortimessecond factor=product

Commutativity and associative laws#

Operation has commmutativity property
(the order of terms does not matter)
Operation has associative property
(it does not matter where the bracket stands, i.e. how terms are grouped)
Commmutativity example3+2=53 + 2 = 5
2+3=52 + 3 = 5
-32=63 \cdot 2 = 6
23=62 \cdot 3 = 6
Associative example3+(4+5)=123 + \left(4 + 5\right) = 12
(3+4)+5=12\left(3 + 4\right) + 5 = 12
-3(45)=603 \cdot \left(4 \cdot 5\right) = 60
(34)5=60\left(3 \cdot 4\right) \cdot 5 = 60
Commmutativity counter-example
(why this operation has NO commutativity property)
-32=13 - 2 = 1
23=12 - 3 = -1
-6:3=26 : 3 = 2
3:6=123 : 6 = \dfrac{1}{2}
Associative counter-example
(why this operation has NO associative property)
-5(4+3)=45 - \left(4 + 3\right) = 4
(54)3=2\left(5 - 4\right) - 3 = -2
-24:(6:2)=824 : \left(6 : 2\right) = 8
(24:6):2=2\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+bw = a + b
    • subtraction, marked with a symbol -:
      w=abw = a - b
    • multiplication, marked with a symbol \cdot or ×\times:
      w=ab=a×bw = a \cdot b = a \times b
    • division, marked with a symbol //, :: or by using fraction bar:
      w=a/b=a:b=abw = 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+ba + b),
    • the result of the subtraction is called difference (aba - b),
    • the result of the multiplication is called product (aba \cdot b),
    • the result of the division is called quotient (a:ba : 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:
      sum=the first summand+second summand\text{sum} = \text{the first summand} + \text{second summand}
    • numbers that we subtract from each other, we call minuend and subtrahend:
      difference=minuendsubtrahend\text{difference} = \text{minuend} - \text{subtrahend}
    • numbers, which we multiply, we call factors:
      product=the first factorsecond factor\text{product} = \text{the first factor} \cdot \text{second factor}
    • numbers that we divide, we call dividend and divisor
      quotient=dividend: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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