In mathematics, everything is based on a set of five *fundamental* definitions called **axioms**. Based on those, we have several more definitions (such as ‘nothing happens when you multiply with 1’), some requirements (such as ‘we can’t divide by zero’) and some laws and rules (such as the arithmetic laws, and rules for symbol rearrangement and manipulation).

The formulas shown use letters $a$, $b$, $c$, $d$ etc. which are numbers with any sign.

Axioms^{[1,2]} |

Reflexive | $$ab=ab$$ |

Symmetric | $$a=b\Leftrightarrow b=a$$ |

Transitive | If $a=b$ and $b=c$, then $a=c$ |

Additive | If $a=b$ and $c=d$, then $a+c=b+d$ |

Multiplicative | If $a=b$ and $c=d$, then $ac=bd$ |

Arithmetic rules^{[3,4,5]} |

Identities | $$1\cdot a=a$$ | Definitions |

$$0+a=a$$ |

Subtraction as addition | $$a-b=a+(-b)$$ |

Rule of substitution | If $a=b+c$, then $ad=(\underbrace{b+c}_a) d$ | Rule |

The commutative law | $$a+b=b+a$$ | Arithmetic laws |

$$ab=ba$$ |

$a\%$ of $b=b\%$ of $a$ |

The distributive law | $$a(b+c)=ab+ac$$ |

The associative law | $$(a+b)+c=a+(b+c)$$ |

$$(ab)c=a(bc)$$ |

The four groups below contain a few definitions and requirements for the different types of mathematical operations and notation, and then a set of algebraic rules that are very convenient to know. They are proven (or at least justified intuitively) in Proof 16, Proof 17, Proof 18 and Proof 19.

Fraction rules |

$$\frac1a \text{ requires }a\neq 0$$ | Requirement |

$$\frac aa = 1$$ | Definition |

$$a\cdot \frac bc=\frac{ab}c$$ | Rules |

$$\frac ab\cdot\frac 1c=\frac a{bc}$$ |

$$a/\frac bc=a\cdot \frac cb$$ |

$$\frac ac+\frac bc =\frac{a+b}c$$ |

$$\frac{-a}b=\frac a{-b}$$ |

Power rules |

$$a^0=1$$ | Definitions |

$$a^{-b}=\frac1{a^b}$$ |

$$a^{1/b}=\sqrt[b]a$$ |

$$a^b\cdot a^c=a^{b+c}$$ | Rules |

$$\left(a^b\right)^c=a^{b\cdot c}$$ |

$$(ab)^c=a^c\cdot b^c$$ |

$$\left(\frac ab\right)^c=\frac{a^c}{b^c}$$ |

$$a^{b-c}=\frac{a^b}{a^c}$$ |

Root rules |

$\sqrt a$, $\sqrt[4] a$, $\sqrt[6] a$... require $a\geq 0$ | Requirement |

$$\sqrt{ab}=\sqrt a\cdot \sqrt b$$ | Rules |

$$\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}$$ |

$$\sqrt[c]{\sqrt[b] a}=\sqrt[b\cdot c] a$$ |

Logarithm rules |

$\log(a)$ requires $a>0$ | Requirement |

$$\log_a(a)=1$$ | Definitions |

$$\log(a)=\log_{10}(a)$$ |

$$\ln(a)=\log_e(a)$$ |

$$\log(1)=0$$ | Rules |

$$\log(ab)=\log(a)+\log(b)$$ |

$$\log\left(\frac ab\right)=\log(a)-\log(b)$$ |

$$\log\left(a^b\right)=b\log(a)$$ |

$$\log\left(\sqrt[b] a\right)=\frac 1b \log(a)$$ |

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