Field axioms and properties Reflexive property:
$$x = x$$
Commutative property of addition:
$$a + b = b + a$$
Definition of subtraction:
$$a - b = a + \left( -b \right)$$
Associative property of addition:
$$a + b + c = \left(a + b\right) + c = a + \left(b + c\right)$$
Multiplication notation:
$$a \cdot b = ab$$
Commutative property of multiplication:
$$a \cdot b = b \cdot a$$
Distributive property:
$$a\left(b + c\right) = ab + ac$$
Division notation:
$$a \div b = \frac{a}{b}$$
Definition of division (multiplicative inverse):
$$\frac{a}{b} = a \cdot \frac{1}{b}$$
Multiplication of fractions:
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$
Signed zero property:
$$0 = -0$$
Additive identity:
$$x + 0 = x$$
Subtraction identity:
$$x - 0 = x$$
Multiplicative identity:
$$1x = x$$
Zero property of multiplication:
$$0x = 0$$
Division by identity:
$$\frac{x}{1} = x$$
Property of negation:
$$-1x = -x$$
Powers of one:
$$1^{x} = 1$$
Negative exponent / Reciprocal:
$$x^{-1} = \frac{1}{x}$$
PEMDAS/BODMAS (order of operations):
$$3 + 3 \cdot 3 = 3 + 9 = 12$$
$$3 + 3 \cdot 3 \neq 18$$
$$\left(3 + 3\right) \cdot 3 = 6 \cdot 3 = 18$$
Leibniz formula for $\pi$ :
$$\pi = 4 \sum_{k=0}^{\infty} \frac{\left(-1\right)^{k}}{2k + 1}$$
Chudnovsky algorithm for $\pi$ :
$$\pi = 1 \div \left(12 \sum_{k=0}^{\infty} \frac{\left(-1\right)^{k} \left(6k\right)! \left(545140134k + 13591409\right)}{\left(3k\right)! \left(k!\right)^{3} \left(640320\right)^{3k + 3/2}}\right)$$
Summation series for $e$ :
$$e = \sum_{k=0}^{\infty} \frac{1}{k!}$$
$e$ limit:
$$e = \lim_{k \to \infty} \left(1 + \frac{1}{k}\right)^{k}$$
Golden ratio:
$$\phi = \frac{1 + \sqrt{5}}{2}$$
$$\frac{1}{\phi} = \phi - 1$$
$$\sin\left(x\right) = \frac{e^{ix} - e^{-ix}}{2i}$$
$$\cos\left(x\right) = \frac{e^{ix} + e^{-ix}}{2}$$
$$\cos\left(x\right) = \sin\left(x + \frac{\pi}{2}\right)$$
$$\tan\left(x\right) = \frac{\sin\left(x\right)}{\cos\left(x\right)}$$
$$\cot\left(x\right) = \frac{\cos\left(x\right)}{\sin\left(x\right)}$$
$$\sec\left(x\right) = \frac{1}{\cos\left(x\right)}$$
$$\csc\left(x\right) = \frac{1}{\sin\left(x\right)}$$
$$\arcsin\left(x\right) = \sin^{-1}\left(x\right)$$
$$\arccos\left(x\right) = \cos^{-1}\left(x\right)$$
$$\arctan\left(x\right) = \tan^{-1}\left(x\right)$$
$$\text{arccot}\left(x\right) = \cot^{-1}\left(x\right)$$
$$\text{arcsec}\left(x\right) = \sec^{-1}\left(x\right)$$
$$\text{arccsc}\left(x\right) = \csc^{-1}\left(x\right)$$
$$\sinh\left(x\right) = \frac{e^{x} - e^{-x}}{2}$$
$$\cosh\left(x\right) = \frac{e^{x} + e^{-x}}{2}$$
$$\tanh\left(x\right) = \frac{\sinh\left(x\right)}{\cosh\left(x\right)}$$
$$\coth\left(x\right) = \frac{\cosh\left(x\right)}{\sinh\left(x\right)}$$
$$\text{sech}\left(x\right) = \frac{1}{\cosh\left(x\right)}$$
$$\text{csch}\left(x\right) = \frac{1}{\sinh\left(x\right)}$$
Factorial and gamma function Factorial definition:
$$z! = \Gamma\left(z + 1\right)$$
Gamma function:
$$\Gamma\left(z + 1\right) = z \Gamma\left(z\right)$$
$$\Gamma\left(z\right) = \int_{0}^{\infty} t^{z - 1} e^{-t}, dt$$
$$e^{ix} = \cos\left(x\right) + i\sin\left(x\right)$$
$$e^{i \pi} = -1$$
$$e^{i \pi} + 1 = 0$$
$$\zeta\left(s\right) = 2^{s} \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma\left(1 - s\right) \zeta\left(1 - s\right)$$
Speed of light in a vacuum:
$$c = 299,792,458\ \text{m} \cdot \text{s}^{-1}$$
Gravitational constant:
$$G \approx 6.6743 \cdot 10^{-11}\ \text{m}^{3} \cdot \text{kg}^{-1} \cdot \text{s}^{-2}$$
Planck's constant:
$$h = 6.62607015 \cdot 10^{-34}\ \text{J} \cdot \text{Hz}^{-1}$$
Reduced Planck's constant:
$$\hbar = \frac{h}{2\pi}$$
Boltzmann constant:
$$k_{\text{B}} = 1.380649 \cdot 10^{-23}\ \text{J} \cdot \text{K}^{-1}$$
Planck length:
$$l_{\text{P}} = \sqrt{\frac{\hbar G}{c^{3}}}$$
Planck mass:
$$m_{\text{P}} = \sqrt{\frac{\hbar c}{G}}$$
Planck time:
$$t_{\text{P}} = \sqrt{\frac{\hbar G}{c^{5}}}$$
Planck temperature:
$$T_{\text{P}} = \sqrt{\frac{\hbar c^{5}}{G k_{\text{B}}^{2}}}$$
Planck area:
$$l_{\text{P}}^{2} = \frac{\hbar G}{c^{3}}$$
Planck volume:
$$l_{\text{P}}^{3} = \left(\frac{\hbar G}{c^{3}}\right)^{\frac{3}{2}} = \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}}$$
Planck momentum:
$$m_{\text{P}}c = \frac{\hbar}{l_{\text{P}}} = \sqrt{\frac{\hbar c^{3}}{G}}$$
Planck energy:
$$E_{\text{P}} = m_{\text{P}}c^{2} = \frac{\hbar}{t_{\text{P}}} = \sqrt{\frac{\hbar c^{5}}{G}}$$
Planck force:
$$F_{\text{P}} = \frac{E_{\text{P}}}{l_{\text{P}}} = \frac{\hbar}{l_{\text{P}}t_{\text{P}}} = \frac{c^{4}}{G}$$
Planck density:
$$\rho_{\text{P}} = \frac{m_{\text{P}}}{l_{\text{P}}^{3}} = \frac{\hbar t_{\text{P}}}{l_{\text{P}}^{5}} = \frac{c^{5}}{\hbar G^{2}}$$
Planck acceleration:
$$a_{\text{P}} = \frac{c}{t_{\text{P}}} = \sqrt{\frac{c^{7}}{\hbar G}}$$
Glaisher-Kinkelin constant:
$$A = e^{\frac{1}{12} - \zeta'\left(-1\right)}$$
Hyperfactorial function:
$$\text{H}\left(x\right) = \prod_{k=1}^{x} k^{k}$$