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.

N is the set of natural numbers

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

N* is the set of natural numbers except 0 (zero)

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).

Z* (Z star) is the set of integers except 0 (zero).

D is the set of decimal numbers, i.e. represented by a x 10^n, where a and n are elements of Z.

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.

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

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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- 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 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

Source : https://www.dcode.fr/number-sets

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