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

NameFormula
The commutativity of additionShow sourcea+b=b+a a+\mathrm{b}=\mathrm{b}+ a
The commutativity of multiplicationShow sourceab=ba a\cdot\mathrm{b}=\mathrm{b}\cdot a
The associative of additionShow source(a+b)+c=a+(b+c)\left( a+\mathrm{b}\right)+ c= a+\left(\mathrm{b}+ c\right)
The associative of multiplicationShow source(ab)c=a(bc)\left( a\cdot\mathrm{b}\right)\cdot c= a\cdot\left(\mathrm{b}\cdot c\right)
The distributive property of multiplication over additionShow sourcea(b+c)=ab+ac a\cdot\left(\mathrm{b}+ c\right)= a\cdot\mathrm{b}+ a\cdot c
The addition of zeroShow sourcea+0=a a+0= a
The multiplication by oneShow sourcea1=a a\cdot1= a
The multiplication by zeroShow sourcea0=0 a\cdot0=0

Names of arguments (operands) and result

Operation nameName of the first argumentColloquial operation nameName of second argumentName of result
Additionfirst summandplussecond summand=sum
Subtractionminuendminussubtrahend=difference
Multiplicationfirst factortimessecond factor=product
Divisiondividendperdivisor=product

Commutativity and associative laws

PropertyAdditionSubtractionMultiplicationDivision
Operation has commmutativity property
(the order of terms does not matter)
yesnoyesno
Operation has associative property
(it does not matter where the bracket stands, i.e. how terms are grouped)
yesnoyesno
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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