Numbers

Numbers are the basic objects used to count, measure, quantify and calculate things.

Sets of numbers

There are multiple sets of numbers:

• Natural numbers: $N = \{0, 1, 2, 3, 4, ...\}$
• Integers: $Z = \{..., -5, -4, -3, -2, -1, 0\} \cup N$
• Rational: $Q = \{\frac{5}{3}, 1.5, ...\} \cup Z$
• Real: $R = \{\pi,\sqrt{2},e, ...\} \cup Q$
• Complex: $C=\{1 + i, 2 + 3i, ...\} \cup R$ , where $\pi$, $e$ and $i$ are constants.

Natural numbers

Numbers from $1$ to $\infty$ with step of 1.

Integers

Numbers from $-\infty$ to 0 with step of 1 and all natural numbers.

Rational numbers

Numbers that are fractions in form of $\frac{m}{n}$, where $n \neq 0$ and all integers.

Real numbers

Numbers with infinitely long decimal expansion that does not repeat and all rational numbers. Example is $\pi$ which is $3.1415926...$ and never repeats.

Complex numbers

All real numbers and numbers that have two parts - real part plus real constant multiplied by $i$. Example is $\pi + ei$ or $3 + 2i$.

Number operations

There are a couple of operations we can do with numbers:

• Addition ($+$) and inverse operation Subtraction ($-$);
• Multiplication ($*$) and inverse operation Division ($/$);
• Exponentiation ($x^a$) and inverse operations Logarithm ($log_ab$) and Roots ($\sqrt[a]{b}$).

Number representations

There are a few ways to represent a number:

• Decimal (Denary) $a_n...a_4a_3a_2a_1a_0.a_{-1}a_{-2}...a_{-m}$, where $a_i \in \{0, 1, 2, 3, ..., 9\}$ and each $a_i$ represents $a \times 10^i$, for example $123.456$;
• Binary $a_n...a_4a_3a_2a_1a_0.a_{-1}a_{-2}...a_{-m}$, where $a_i \in \{0, 1\}$ and each $a_i$ represents $a \times 2^i$, for example $100101.110001$;
• Fraction $\frac{m}{n}$, where $m \in R$, $n \in R$ and $n \neq 0$, for example $\frac{56}{23}$;
• Scientific $a_0.a_{-1}a_{-2}...a_{-m} \times base^b$, where $a_i \in \{0, ..., base-1\}$, $b \in R$ and each $a_i$ represents $a \times base^i$, for example $1.2345 \times 10^4$.