##### Numbers Systems

A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

There are some ways to categorise numbers, like:

- ###### Natural Numbers

  Common counting numbers, starting by 1 e.g.

  N= 1, 2, 3, 4, ...

- ###### Prime Numbers

  A natural number greater than 1 which has only 1 and itself as factors e.g.

  P= 2, 3, 5, 7, 11, 13, 17, ...

- ###### Composite Numbers

  A natural number greater than 1 which has more factors than 1 and itself e.g.

  4, 6, 8, 9, 10, 12, ...

- ###### Integers

  Whole Numbers with their opposites (negative numbers) adjoined e.g.

  Z= …, −3, −2, −1, 0, 1, 2, 3, ...

- ###### Rational Numbers

  All numbers which can be written as fractions e.g.

  Q=−12, 0.33333…, 52, 1110, …

- ###### Irrational Numbers

  All numbers which cannot be written as fractions e.g.

  F=..., π, 2–√, 0.121221222..., ...

- ###### Real Numbers

  The set of Rational Numbers with the set of Irrational Numbers adjoined e.g.

  R=..., −3, −1, 0, 15, 1.1, 2–√2, 3, π, …

- ###### Complex Numbers

  A number which can be written in the form a + bi where a and b are real numbers and i is the square root of -1 e.g.

  C=..., −3+2i, 0, 1+ 3i, …

- ###### Positive Numbers

  All real numbers greater than 0 e.g.

  ..., 1, 1.1, 1.2, √2, π, ...

- ###### Negative Numbers

  All real numbers smaller than 0(opposite to a positive numbers, excluding 0) e.g.

  ..., -1, -1.1, -1.2, -√2, -π, ...

##### Numeral Systems

During Centuries many cultures used to use various numeral systems. 2 most important for us is:

###### Roman Numerals

Invented around year 1000 BCE in ancient Rome. Remained asa usual way of writting numbers throughout Europe untill the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, each letter with a fixed integer value. The 7 of them are most popular:

Roman symbol:
Value:

- I
  1
- V
  5
- X
  10
- L
  50
- C
  100
- D
  500
- M
  1000

To write down bigger values we can use "additive notation", what means combaning with each other roman symbols

Hovewer, the numerals for 4 (IV) and 9 (IX) are written using "subtractive notation", where the first symbol (I) is subtracted from the larger one (V, or X), thus avoiding the clumsier IIII and VIIII. Subtractive notation is also used for bigger values like 40 (XL), 90 (XC), 400 (CD) and 900 (CM).

###### Individual decimal places

| Thousands | Hundreds | Tens | Units
--- | --- | --- | --- | ---
1. | M | C | X | I
2. | MM | CC | XX | II
3. | MMM | CCC | XXX | III
4. |  | CD | XL | IV
5. |  | D | L | V
6. |  | DC | LX | VI
7. |  | DCC | LXX | VII
8. |  | DCCC | LXXX | VIII
9. |  | CM | XC | IX

###### Arabic Numerals

Invented around 9th Century. Most common numeral system nowdays. Is based on ten symbols used to write decimal numbers:

###### 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0