# Category:Real Numbers

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This category contains results about **Real Numbers**.

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

A **real number** is defined as a **number** which is identified with a point on the **real number line**.

#### Real Number Line

From the Cantor-Dedekind Hypothesis, the set of real numbers is isomorphic to any infinite straight line.

The **real number line** is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real numbers is consistent with the length of the line between those two points.

## Subcategories

This category has the following 39 subcategories, out of 39 total.

### B

### C

### E

- Equality of Real Numbers (2 P)

### I

### N

### R

- Real Division (13 P)

### S

### T

### U

- Unbounded Above Sets of Real Numbers (empty)
- Unbounded Below Sets of Real Numbers (empty)
- Unbounded Sets of Real Numbers (empty)

## Pages in category "Real Numbers"

The following 90 pages are in this category, out of 90 total.

### A

- User:Abcxyz/Sandbox/Real Numbers
- User:Abcxyz/Sandbox/Real Numbers/Identity for Real Addition
- User:Abcxyz/Sandbox/Real Numbers/Identity for Real Multiplication
- User:Abcxyz/Sandbox/Real Numbers/Inverses for Real Addition
- User:Abcxyz/Sandbox/Real Numbers/Inverses for Real Multiplication
- User:Abcxyz/Sandbox/Real Numbers/Ordering on Real Numbers is Compatible with Addition
- User:Abcxyz/Sandbox/Real Numbers/Ordering on Real Numbers is Total Ordering
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Associative
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Closed
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Commutative
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication Distributes over Addition
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Associative
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Closed
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Commutative
- User:Abcxyz/Sandbox/Real Numbers/Real Numbers are Dedekind Complete

### C

- Canonical Injection of Real Number Line into Complex Plane
- Characterizing Property of Infimum of Subset of Real Numbers
- Characterizing Property of Supremum of Subset of Real Numbers
- Choice Function/Examples/Doubletons of Real Numbers
- Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order
- Continuum Property implies Well-Ordering Principle
- Convergent Real Sequence has Unique Limit
- Cross-Relation on Real Numbers is Equivalence Relation

### E

### F

### I

### O

### P

### Q

### R

- Rational Numbers form Subfield of Real Numbers
- Rational Numbers form Subset of Real Numbers
- Real Number between Zero and One is Greater than Power/Natural Number
- Real Number Inequalities can be Added
- Real Number is Greater than Zero iff its Negative is Less than Zero
- Real Number is not necessarily Rational Number
- Real Number Line is Metric Space
- Real Number Line is not Topological Continuum
- Real Number Ordering is Transitive
- Real Numbers are Densely Ordered
- Real Numbers are not Well-Ordered under Conventional Ordering
- Real Numbers are Uncountably Infinite
- Real Numbers form Algebra
- Real Numbers form Field
- Real Numbers form Integral Domain
- Real Numbers form only Ordered Field which is Complete
- Real Numbers form Ordered Field
- Real Numbers form Ordered Integral Domain
- Real Numbers form Perfect Set
- Real Numbers form Ring
- Real Numbers form Subfield of Complex Numbers
- Real Numbers form Valued Field
- Real Numbers form Vector Space
- Real Vector Space is Vector Space
- Real Zero is Less than Real One
- Real Zero is Zero Element
- Reals are Isomorphic to Dedekind Cuts
- Reciprocal of Real Number is Non-Zero
- Reciprocal of Strictly Negative Real Number is Strictly Negative
- Reciprocal of Strictly Positive Real Number is Strictly Positive

### S

- Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering
- Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum
- Square of Non-Zero Real Number is Strictly Positive
- Square Root is Strictly Increasing
- Sub-Basis for Real Number Line
- Subgroup of Real Numbers is Discrete or Dense
- Sum of Indices of Real Number