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

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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$6 : 3 = 2$ |

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$a\cdot0=0$ |

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 |

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) | - | $3 - 2 = 1$ $2 - 3 = -1$ | - | $6 : 3 = 2$ $3 : 6 = \frac{1}{2}$ |

Associative counter-example (why this operation has NO associative property) | - | $5 - \left(4 + 3\right) = 4$ $\left(5 - 4\right) - 3 = -2$ | - | $24 : \left(6 : 2\right) = 8$ $\left(24 : 6\right) : 2 = 2$ |

- 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**:

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

- addition, marked with a symbol
- Depending on the kind 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 kind 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

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- 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

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