- 2 + 3 = 3 + 2 (because both are 5);
- 3 + 5 = 5 + 3 (because both are 8);
- 4 + (−9) = (−9) + 4 (because both are −5).

Name of the identity | Statement | For what numbers? |
---|---|---|

Commutative law of addition; | a + b = b + a; |
any real numbers a and b. |

Associative law of addition; | (a + b) + c = a + (b + c); |
any real numbers a, b, and c. |

Commutative law of multiplication; | a · b = b · a; |
any real numbers a and b. |

Associative law of multiplication; | (a · b) · c
= a · (b · c); |
any real numbers a, b, and c. |

Additive identity (right); | a + 0 = a; |
any real number a. |

Additive identity (left); | 0 + a = a; |
any real number a. |

Multiplicative identity (right); | a · 1 = a; |
any real number a. |

Multiplicative identity (left); | 1 · a = a; |
any real number a. |

Zero law (right); | a · 0 = 0; |
any real number a. |

Zero law (left); | 0 · a = 0; |
any real number a. |

Distributive law (left); | a · (b + c)
= (a · b) + (a · c); |
any real numbers a, b, and c. |

Distributive law (right); | (a + b) · c
= (a · c) + (b · c); |
any real numbers a, b, and c. |

Additive inverse (right); | a + (−a) = 0; |
any real number a. |

Additive inverse (left); | (−a) + a = 0; |
any real number a. |

Multiplicative inverse (right); | a · (1/a) = 1; |
any nonzero real number a. |

Multiplicative inverse (left); | (1/a) · a = 1; |
any nonzero real number a. |

You don't really want to try to *memorise* this list.
In any case, these are not all of the identities!
(They are just in a way the most fundamental.)
Instead, you should try to make these laws
pare of your *intuition* for mathematics.
Then the only thing worth memorising are the names
—but I only included the names for reference,
and I will not test you on them in any way.

So, the point of the commutative laws
is that you can add or multiply two numbers in either order.
In the associative laws,
I included parentheses
to indicate which addition or multiplication is to be performed first;
but really the whole point of the associative laws
is that it doesn't matter which you do first!
So if the associative laws seem silly to you,
because you don't need the parentheses in these expressions in the first place,
then that's good
—when a rule seems obvious, that's a sign that you understand it!
The distributive law is probably the most important rules for algebra;
it's complicated and may not seem very obvious at all,
and yet it's used a great deal.
You should think about the distributive law until it too is second nature!
Notice that inverse law
is true only for a *nonzero* real number *a*;
if *a* were 0, then the statement would not make sense,
because 1/0 is undefined.

Statement | For what numbers? |
---|---|

a − b = a + (−b); |
any real numbers a and b. |

a/b = (1/b) · a; |
any real number a and any nonzero real number b. |

−(−a) = a; |
any real number a. |

a − b = −(b − a); |
any real numbers a and b. |

−a − b = −(a + b); |
any real numbers a and b. |

−a + b = b − a; |
any real numbers a and b. |

a − (−b) = a + b; |
any real numbers a and b. |

−a = (−1) · a; |
any real number a. |

a · (−b)
= −(a · b); |
any real numbers a and b. |

(−a) · b
= −(a · b); |
any real numbers a and b. |

(−a) · (−b)
= a · b; |
any real numbers a and b. |

a/(−b) = −(a/b); |
any real number a and any nonzero real number b. |

(−a)/b = −(a/b); |
any real number a and any nonzero real number b. |

(−a)/(−b) = a/b; |
any real number a and any nonzero real number b. |

1/(1/a) = a; |
any nonzero real number a. |

(a · b)/b = a; |
any real number a and any nonzero real number b. |

(a · c)/(b · c)
= a/b; |
any real number a
and any nonzero real numbers b and c. |

(a/c) + (b/c)
= (a + b)/c; |
any real numbers a and b
and any nonzero real number c. |

(a/c) − (b/c)
= (a − b)/c; |
any real numbers a and b
and any nonzero real number c. |

(a/c) · (b/d)
= (a · b)/(c · d); |
any real numbers a and b
and any nonzero real numbers c and d. |

1/(a/b) = b/a; |
any nonzero real numbers a and b. |

(a/c)/(b/d)
= (a · d)/(c · b); |
any real number a
and any nonzero real numbers b, c and d. |

- round parentheses: ‘(’ and ‘)’;
- square brackets: ‘[’ and ‘]’;
- curly braces: ‘{’ and ‘}’.

- inside absolute values;
- above or below a horizontal division bar;
- inside an exponent.

In principle, one should always use some grouping symbols,
which I've done in the tables above.
But in practice, we don't need them all the time.
For example, the associative laws
mean that we don't need to use grouping symbols
between addition and multiplication,
because it simply doesn't matter what we do first.
Between addition and mutliplication, the order does matter,
but we can always use the distributive law
to turn an addition-first expression into a multiplication-first one.
By default we *always* do multiplication first;
then we only need grouping symbols if we (unusually) want addition first.

In general, here is the standard order of operations:

- Exponentiation;
- Multiplication;
- Division;
- Addition, subtraction, and taking opposites.

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