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Number Sets

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.

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Tag(s) : Arithmetics, Notation System

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# Number Sets

## Number sets calculator

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.

### What are common number sets?

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.

### What is the N number set?

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

### What is the Z number set?

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).

### What is the D number set?

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).

### What is the Q number set?

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).

### What is the R number set?

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.

### What is the I number set?

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

### What is the C number set?

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.

### What are E and O number sets?

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

### What are inclusions of sets?

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.

### Why the letter Q for Rationals?

Q was chosen for the word Quotient.

### What does R^2 means (or other power) of a set?

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 $\mathbb{Z} ^ 3$.

### How to write a number set in LaTeX?

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

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