Tool to understand sets of numbers N, Z, Q, R, I, C. Number sets are groups of numbers constructed by mathematicians in order to define them and classify them.

Number Sets - dCode

Tag(s) : Arithmetics, Notation System

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Tool to understand sets of numbers N, Z, Q, R, I, C. Number sets are groups of numbers constructed by mathematicians in order to define them and classify them.

In mathematics, there are multiple **sets: the natural numbers** N, the set of integers Z, all decimal numbers D, the **set of rational numbers** Q, the **set of real numbers** R and the **set of complex numbers** C.

Other sets such as quaternions, or hyper-complex numbers exist but are reserved for advanced mathematical theories, NZQRC are the most common sets.

N is the **set of natural numbers**

__Example:__ 0, 1, 2, 3, 4, 5, ... 10, 11, ..., 100, ...

$ \mathbb{N}^* $ (N asterisk) is the **set of natural numbers** except 0 (zero), it is also referred as $ \mathbb{N}^{+} $

Some (old) textbooks indicate the letter W instead of N for this set, W stands for *Whole numbers*

Z is the set of integers, ie. positive, negative or zero.

__Example:__ ..., -100, ..., -12, -11, -10, ..., -5, -4, -3, -2, - 1, 0, 1, 2, 3, 4, 5, ... 10, 11, 12, ..., 100, ...

The set N is included in the set Z (because all natural numbers are part of the relative integers).

$ \mathbb{Z}^* $ (Z asterisk) is the set of integers except 0 (zero).

D is the set of decimal numbers (its use is rare and mainly limited to Europe)

$$ \mathbb {D} = \left\{ \frac{a}{10^{p}} , a \in \mathbb{Z}, p \in \mathbb {N} \right\} $$

All decimals in D are numbers that can be written with a finite number of digits.

__Example:__ -123.45, -2.1, -1, 0, 5, 6.7, 8.987654

The sets N and Z are included in the set D (because all integers are decimal numbers that have no decimal places).

Q is the **set of rational numbers**, ie. represented by a fraction a/b with a belonging to Z and b belonging to Z * (excluding division by 0).

__Example:__ 1/3, -4/1, 17/34, 1/123456789

Sets N, Z and D are included in the set Q (because all these numbers can be written in fraction).

R is the **set of real numbers**, ie. all numbers that can actually exist, it contains in addition to rational numbers, non-rational numbers or irrational as $ \pi $ or $ \sqrt{2} $.

__Example:__ $ \Pi $ $ \sqrt{2} $ $ \sqrt{3} $, ...

Sets N, Z, D and Q are included in the set R.

I is the **set of imaginary numbers**, ie. the numbers that can not actually exist, these numbers have been created by mathematicians to solve certain equations.

__Example:__ i, i^2=-1

C is the **set of complex numbers**, ie. the **set of real numbers** R and all imaginary numbers I.

__Example:__ A + ib

Sets N, Z, D, Q, R and I are included in the set C.

Some books define the sets E for *even* numbers and O for *odd* numbers. This is not a standard notation.

The links between the different sets are represented by inclusions: $$ N \subset Z \subset D \subset Q \subset R \subset C $$

The subset symbol ⊆ is that of inclusion (broad sense), A ⊆ B if every element of A is an element of B.

The subset symbol ⊂ or ⊊ is that of proper inclusion (strict sense), A ⊂ B if every element of A is an element of B and A ≠ B.

Q was chosen for the word Quotient.

If an element belongs to $ \mathbb{X}^d $ where $ X $ is a set and $ d $ an integer, then it is a tuple of numbers (containing $ d $ numbers).

__Example:__ The point P (a, b) of the 2D plane belongs to $ \mathbb{R}^2 $.

__Example:__ The point P (a, b, c) has integer coordinates, it belongs to the 3D grid $ \mathbb{Z} ^ 3 $.

A set of numbers is written with the mathbb tag: \mathbb{Z} for $ \mathbb{Z} $

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Please, check our community Discord for help requests!

- Number sets calculator
- What are common number sets?
- What is the N number set?
- What is the Z number set?
- What is the D number set?
- What is the Q number set?
- What is the R number set?
- What is the I number set?
- What is the C number set?
- What are E and O number sets?
- What are inclusions of sets?
- Why the letter Q for Rationals?
- What does R^2 means (or other power) of a set?
- How to write a number set in LaTeX?

set,theory,number,integer,natural,rational,real,complex,imaginary,nzqrc

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