Number systems
Integers and Reals
Two integers $a$ and $b$ are congruent modulo $n>1$ if $a-b$ is an integer multiple of $n$ ( $a=b+kn$ for some $k \in \mathbb{Z}$). This can be written as $a \equiv b$ mod $n$ or $a \overset{n}{\equiv} b$.
Given any integers $a,n \in \mathbb{Z}$ with $n \neq 0$, there are unique integers $q,r \in \mathbb{Z}$ such that $a=qn+r$ and $0 \leq r < |n|$. $q$ is the quotient and $r$ is the remainder. The remainder is the smallest non-negative integer $b$ such that $a \equiv b$ mod $n$. The notation $a$ mod $n$ is used to denote the remainder.
Congruences with the same modulus can be added, subtracted and multiplied. If $x \equiv 3$ mod $n$ and $y \equiv 5$ mod $n$ then $(x+y) \equiv 8$ mod $n$ and $(x \times y) \equiv 15$ mod $n$.
Rational numbers
A rational number has the for $\frac{m}{n}$ where $m,n \in \mathbb{Z}$
and $n \neq 0$.
There is always a $m$ and $n$ such that $n \geq 1$ and
$\operatorname{gcd}(m,n)=1$.
Every non-zero rational number $q$ has an inverse $q^{-1}$ with the
product $q \times q^{-1}=1$.
Real numbers
Irrational numbers
An algebraic number is a real number (like $\sqrt{2}$) that is the solution of a polynomial equation with rational coefficients.
Transcendental numbers are real numbers which can't be the solutions of polynomial equations with rational coefficients such as $\pi$ and $\mathrm{e}$.
All the properties of the real number system can be derived from 13 axioms. These axioms hold $\forall x,y,z \in \mathbb{R}$. The first 6 axioms also hold for $\mathbb{N}$. The 7th holds for $\mathbb{Z}$. 8 to 12 hold for $\mathbb{Q}$. 13 only holds for $\mathbb{R}$.
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Commutativity ($x+y=y+x$ and $x \times y = y \times x$)
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Associativity ($x+(y+z)=(x+y)+z$ and $x(yz) = (xy) \times z$)
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Distributivity of multiplication over addition ($x(y+z) = xy + xz$)
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Additive identity (There exists 0 such that $x+0-x$)
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Multiplicative identity (There exists 1 such that $x \times 1= x$)
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The multiplicative and additive identities are distinct ($1 \neq 0$)
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Every element has an additive inverse. There exists $(-x)$ such that $x+(-x)=0$.
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Every non-zero number has a multiplicative inverse. If $x \neq 0$ then there exists $x^{-1}$ such that $x \times x^{-1}=1$.
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Transitivity of ordering ($x<y$ and $y<z$ then $x<z$)
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Trichotomy law. Exactly one of $x < y$, $y <x$ or $x=y$ is true.
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Preservation of ordering under addition (If $x<y$ then $x+z<y+z$)
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Preservation of ordering under multiplication (If $0<z$ and $x<y$ then $xz<yz$)
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The axiom of completeness. Every non-empty subset that is bounded above has a least upper bound.
Upper and lower bounds, supremum and infimum
Let $S$ be a set of real numbers.
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A real number $u$ is an upper bound of $S$ if $u \geq x$ for all $x \in S$.
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A real number $U$ is the least upper bound (supremum) of $S$ if $U$ is an upper bound of $S$ and $U \leq u$ for every upper bound $u$ of $S$.
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A real number $l$ is a lower bound of $S$ if $l \leq x$ for all $x \in S$.
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A real number $L$ is the greatest upper bound (infimum) of $S$ if $L$ is a lower bound of $S$ and $L \geq l$ for every lower bound $l$ of $S$.
The completeness axiom says that every non-empty set of real numbers which has an upper bound has a least upper bound. From this we can derive that every non-empty set of real numbers which has a lower bound has a greatest lower bound.
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From presentation slides
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The real numbers are completely characterised by 12 basic algebraic and order axioms and the completeness axiom. Any theorem about real numbers can be derived from these. Any structure satisfying these properties can be shown to be essentially identical to $\mathbb{R}$. The completeness axiom implies that $\{x\in \mathbb{R}\:|\:x^2 < 2\}$ has a least upper bound. It follows that there exists $x \in \mathbb{R}$ such that $x^2 = 2$.
Complex numbers
A complex number is of the form $a+ib$ where $a,b \in \mathbb{R}$ and $i = \sqrt{-1}$. The conjugate of a complex number $z$ is $\overline{z}$. For any $z,w \in \mathbb{C}$:
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$\overline{z+w} = \overline{z} + \overline{w}$
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$\overline{zw} = \overline{z} \times \overline{w}$
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$\overline{z \div w} = \overline{z} \div \overline{w}$
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$\operatorname{Re}(z) = \frac{z+\overline{z}}{2}$
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$\operatorname{Im}(z) = \frac{z-\overline{z}}{2}$
The modulus of a complex number $a+bi$, denoted $|a+bi|$, is $\sqrt{a^2+b^2}$. For any $z \in \mathbb{C}$:
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$|z| = |\overline{z}|$
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$|z| = \sqrt{z\overline{z}}$
For any $z,w \in \mathbb{C}$:
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$|zw|=|z||w|$
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$|z+w|\leq |z| + |w|$
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$||z|-|w|| = \leq |z-w|$
Any complex number $x+iy \neq 0$ can be expressed in polar coordinates. It uses the modulus $r = \sqrt{x^2+y^2}$ and argument $\theta = \arctan{(\frac{y}{x})}$, $-\pi < \theta \leq \pi$. This means $x+iy$ can be expressed as $r(\cos{\theta}+i\sin{\theta})$. $x=r\cos{\theta}$ and $y = r\sin{\theta}$.
De Moivre's theorem
$(\cos{\theta}+i\sin{\theta})^n = \cos{n\theta}+i\sin{n\theta}$.