# Category:Rational Numbers

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This category contains results about Rational Numbers.

Definitions specific to this category can be found in Definitions/Rational Numbers.

A number in the form $\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a **rational number**.

The set of all **rational numbers** is usually denoted $\Q$.

Thus:

- $\Q = \set {\dfrac p q: p \in \Z, q \in \Z_{\ne 0} }$

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### F

### I

### R

## Pages in category "Rational Numbers"

The following 81 pages are in this category, out of 81 total.

### A

### C

### E

### I

### M

### N

### O

### P

### R

- Rational Addition Identity is Zero
- Rational Addition is Associative
- Rational Addition is Closed
- Rational Addition is Commutative
- Rational Division is Closed
- Rational Multiplication Distributes over Addition
- Rational Multiplication Identity is One
- Rational Multiplication is Associative
- Rational Multiplication is Closed
- Rational Multiplication is Commutative
- Rational Number is Algebraic
- Rational Number is Algebraic of Degree 1
- Rational Number plus Irrational Number is Irrational
- Rational Numbers are Countably Infinite
- Rational Numbers are not Connected
- Rational Numbers are not Discrete Space
- Rational Numbers are not Extremally Disconnected
- Rational Numbers are Totally Disconnected
- Rational Numbers form F-Sigma Set in Reals
- Rational Numbers form Field
- Rational Numbers form Integral Domain
- Rational Numbers form Metric Space
- Rational Numbers form Ordered Integral Domain
- Rational Numbers form Prime Field
- Rational Numbers form Ring
- Rational Numbers form Subfield of Complex Numbers
- Rational Numbers form Subfield of Real Numbers
- Rational Numbers form Subset of Real Numbers
- Rational Numbers form Totally Ordered Field
- Rational Numbers form Vector Space
- Rational Numbers under Addition form Abelian Group
- Rational Numbers under Addition form Monoid
- Rational Numbers under Multiplication do not form Group
- Rational Numbers under Multiplication form Commutative Monoid
- Rational Numbers under Multiplication form Monoid
- Rational Numbers with Denominator Power of Two form Integral Domain
- Rational Numbers with Denominators Coprime to Prime under Addition form Group
- Rational Square Root of Integer is Integer
- Rational Subtraction is Closed
- Real Number is not necessarily Rational Number
- Ring Monomorphism from Integers to Rationals

### S

- Set of Rational Numbers is not Closed in Reals
- Set of Rational Numbers is not G-Delta Set in Reals
- Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element
- Set of Rational Numbers whose Numerator Divisible by p is Closed under Addition
- Set of Rational Numbers whose Numerator Divisible by p is Closed under Multiplication
- Set of Rationals Greater than Root 2 has no Smallest Element
- Set of Rationals Less than Root 2 has no Greatest Element
- Smallest Strictly Positive Rational Number does not Exist
- Strictly Positive Rational Numbers are Closed under Addition
- Strictly Positive Rational Numbers are Closed under Multiplication
- Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group
- Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers