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This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

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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$\left( a+\mathrm{b}\right)+ c= a+\left(\mathrm{b}+ c\right)$ |

The associative of multiplication | Show source$\left( a\cdot\mathrm{b}\right)\cdot c= a\cdot\left(\mathrm{b}\cdot c\right)$ |

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$a\cdot1= a$ |

The multiplication by zero | Show source$a\cdot0=0$ |

# Names of arguments (operands) and result

Operation name | Name of the first argument | Colloquial operation name | Name of second argument | 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**:

$w = a / b = a : b = \dfrac{a}{b}$

- addition, marked with a symbol
- 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**($a : b$).

- the result of the
- 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**

$\text{quotient} = \text{dividend} : \text{divisor}$

- numbers, which we
- 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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# Links to external sites (leaving Calculla?)

- mathsisfun.com: commutative, associative and distributive laws
- stackexchange.com: what are the formal names of operands and results for basic operations
- mathsteacher.com.au: basic math operations
- youtube.com: distributive property of multiplication over addition
- harvard.edu: abstract algebra open learning course