# 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 26 subcategories, out of 26 total.

### A

### C

### E

### M

### N

### O

### R

### T

## Pages in category "Real Numbers"

The following 138 pages are in this category, out of 138 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
- Absolute Value Equals Square Root of Square

### C

- Characterizing Property of Infimum of Subset of Real Numbers
- Characterizing Property of Supremum of Subset 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

### I

### M

- Mean of Unequal Real Numbers is Between them
- Minus One is Less than Zero
- Multiplication by Negative Real Number
- Multiplication by Negative Real Number/Corollary
- Multiplication of Positive Number by Real Number Greater than One
- Multiplication of Real Numbers Distributes over Subtraction
- Multiplication of Real Numbers is Left Distributive over Subtraction
- Multiplication of Real Numbers is Right Distributive over Subtraction
- Multiplicative Group of Rationals is Subgroup of Reals
- Multiplicative Group of Reals is Subgroup of Complex

### N

- Negative of Negative Real Number
- Negative of Quotient of Real Numbers
- Negative of Real Zero equals Zero
- Negative of Sum of Real Numbers
- Negative of Sum of Real Numbers/Corollary
- Non-Zero Integers under Multiplication are not Subgroup of Reals
- Non-Zero Real Numbers Closed under Multiplication
- Non-Zero Real Numbers under Multiplication form Abelian Group
- Non-Zero Real Numbers under Multiplication form Group
- Numbers of Type Integer a plus b root 3 form Integral Domain

### O

- Odd Power Function is Strictly Increasing/Real Numbers
- Order of Real Numbers is Dual of Order Multiplied by Negative Number
- Order of Real Numbers is Dual of Order of their Negatives
- Order of Strictly Positive Real Numbers is Dual of Order of their Reciprocals
- Ordering of Reciprocals
- Ordering on Real Numbers from Decimal Expansion
- Ordering Properties of Real Numbers

### P

- Pointwise Addition on Real-Valued Functions is Associative
- Pointwise Addition on Real-Valued Functions is Commutative
- Pointwise Multiplication on Real-Valued Functions is Associative
- Pointwise Multiplication on Real-Valued Functions is Commutative
- Positive Real has Real Square Root
- Positive Real Numbers Closed under Division
- Positive Real Numbers Closed under Multiplication
- Positive Real Numbers under Max Operation form Monoid
- Product of Negative Real Numbers is Positive
- Product of Quotients of Real Numbers
- Product of Real Number with Quotient
- Product of Real Numbers is Positive iff Numbers have Same Sign
- Product of Reciprocals of Real Numbers
- Product of Strictly Positive Real Numbers is Strictly Positive
- Properties of Real Numbers

### Q

### R

- Rational Numbers form Subfield of Real Numbers
- Rational Numbers form Subset of Real Numbers
- Real Addition Identity is Zero/Corollary
- Real Addition is Associative
- Real Addition is Closed
- Real Addition is Commutative
- Real Addition is Well-Defined
- Real Division by One
- Real Multiplication Distributes over Addition
- Real Multiplication Identity is One/Corollary
- Real Multiplication is Associative
- Real Multiplication is Closed
- Real Multiplication is Commutative
- Real Multiplication is Well-Defined
- Real Number between Zero and One is Greater than Power/Natural Number
- Real Number Divided by Itself
- 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 Ordering is Compatible with Addition
- Real Number Ordering is Compatible with Multiplication
- Real Number Ordering is Transitive
- Real Numbers are not Well-Ordered under Conventional Ordering
- Real Numbers are Uncountable
- Real Numbers form Algebra
- Real Numbers form Field
- Real Numbers form Integral Domain
- 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 Totally Ordered Field
- Real Numbers form Vector Space
- Real Numbers under Addition form Monoid
- Real Numbers under Multiplication do not form Group
- Real Numbers under Multiplication form Monoid
- Real Numbers under Subtraction do not form Semigroup
- Real Subtraction is Closed
- Real Vector Space is Vector Space
- Real Zero is Less than Real One
- Real Zero is Zero Element
- Reciprocal of Quotient of Real Numbers
- 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
- Square of Non-Zero Real Number is Strictly Positive
- Square Root is Strictly Increasing
- Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers
- Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group
- Sub-Basis for Real Number Line
- Subgroup of Real Numbers is Discrete or Dense
- Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup/Examples/Non-Zero Integers in Non-Zero Reals under Multiplication
- Sum of Quotients of Real Numbers
- Sum of Strictly Positive Real Numbers is Strictly Positive