User:Moro~enwiki/Maths

Addition
( a + b )+ c = a +( b + c )

a + b = b + a

a + 0 = a

a + -a = 0

Multiplication
λ( a + b )=λ a +λ b

(λ+μ) a =λ a +μ a

0 a = 0

Arithmetic
$$ (a \pm ib)+(c \pm id)=(a \pm c)+i(b\pm d)$$

$$ \begin{align}

(a + ib)(c + id)&=ac+ibd+ida+(ib)(id) \\ &= (ac-bd)+i(bc+ad)\\ \end{align} $$

$$ (a+ib)^{-1}=\frac{a}{a^2+b^2}-\frac{ib}{a^2+b^2}$$

Conjugate
If $$ z=a+ib $$

then the complex conjugaqte $$\overline{z}$$is defined as

$$\overline{z}=a-ib$$

and has the following properties

$$\overline{z_1 \pm z_2} = \overline{z_1}+\overline{z_2}$$

$$\overline{z_1z_2}=\overline{z_1} \overline{z_2}$$

$$ \overline{z^{-1}} =\overline{z}^{-1}$$

Sums...
Every integer can be expressed as a product of prime factors, where all the primes are of one of the following forms

$$ 2$$

$$ 4k+1$$

$$ 4k+3$$

...of two squares
An integer $$ n $$ can be expressed as the sum of two squares if and only if all its prime factors of the form

$$ 4k+3$$

occur an even number of times.

...of three squares
An integer $$ n $$ can be expressed as the sum of three squares if and only if $$ n $$ is not of the form

$$4^m(8k+7)$$

where $$ m $$ and $$ k $$ are integers and can be equal to 0.

Expressions with radicals
$$ \sqrt{a+\sqrt{b}} = \sqrt {\frac{a+\sqrt{a^2-b}}{2}}+\sqrt {\frac{a-\sqrt{a^2-b}}{2}}$$

$$ \sqrt{a-\sqrt{b}} = \sqrt {\frac{a+\sqrt{a^2-b}}{2}}-\sqrt {\frac{a-\sqrt{a^2-b}}{2}}$$

Maximum
If $$ \frac{dy}{dx} =0 \quad \text{and} \quad \frac{d^2y}{dx^2} < 0$$

then the point is a maximum.

Minimum
If $$ \frac{dy}{dx} =0 \quad \text{and} \quad \frac{d^2y}{dx^2} > 0$$

then the point is a minimum.

Point of Inflexion
If $$ \frac{dy}{dx} =0 \quad \text{and} \quad \frac{d^2y}{dx^2} = 0  \quad \text{but} \quad \frac{d^3y}{dx^3} \not = 0$$

then it is a point of inflexion.

All three
If $$ \frac{dy}{dx} =0 \quad \text{and} \quad \frac{d^2y}{dx^2} = 0$$

then the point could be a minimum, maximum or point of inflexion.