aarjaneiro/planetmath huggingface dataset:
reciprocal polynomial

reciprocal polynomial

Definition

[

1

]

Let

p

:

ℂ

→

ℂ

be a

polynomial

of degree

n

with

complex

(or real)
coefficients. Then

p

is a

reciprocal polynomial

if

p

⁢

(

z

)

=

±

z

n

⁢

p

⁢

(

1

/

z

)

for all

z

∈

ℂ

.

Examples of reciprocal polynomials are

Gaussian polynomials

, as well as the

characteristic polynomials

of

orthogonal matrices

(including the

identity matrix

as a special case), symplectic matrices,

involution

matrices (

http://planetmath.org/LinearInvolution

), and the

Pascal matrices

[

2

]

.

It is clear that if

z

is a zero for a reciprocal polynomial, then

1

/

z

is also a zero. This

property

motivates the name. This
means that the spectra of matrices of above type is

symmetric

with respect to the

unit circle

in

ℂ

; if

λ

∈

ℂ

is an

eigenvalue

, so is

1

/

λ

.

The sum,

difference

, and product of two reciprocal polynomials is again a reciprocal polynomial. Hence, reciprocal polynomials form an algebra over the

complex numbers

.

References

1

H. Eves,

Elementary Matrix

Theory

,
Dover publications, 1980.

2

N.J. Higham,

Accuracy and Stability of Numerical Algorithms

,
2nd ed., SIAM, 2002.

Title

reciprocal polynomial

Canonical

name

ReciprocalPolynomial

Date of creation

2013-03-22 13:36:33

Last modified on

2013-03-22 13:36:33

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 12D10

Related topic

CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial

Generated on Fri Feb 9 16:38:23 2018 by

LaTeXML
root

root

Let

f

⁢

(

x

)

:

ℝ

→

ℝ

be a function in a

real variable

x

. Then, a

root

of

f

is a

real number

a

that

f

⁢

(

a

)

=

0

.

Example. If

f

⁢

(

x

)

=

x

2

-

4

, then

x

=

2

is a root, since

f

⁢

(

2

)

=

2

2

-

4

=

0

.

Graphically, a root of

f

is a value where the graph of the function intersects the

x

-axis.

Of course the definition can be generalized to functions on other sets. In particular, neither the domain nor the codomain need be the set of real numbers. All that is required is that the codomain have a well-defined

0

element. A root of the function will then be an element of the domain belonging to the preimage of the

0

.

Note that, for example, the function

f

:

ℝ

→

ℝ

given by

f

⁢

(

x

)

=

x

2

+

1

has no roots, but the function

f

:

ℂ

→

ℂ

given by

f

⁢

(

x

)

=

x

2

+

1

has

i

as a root.

In the special case of

polynomials

, there are general formulas for finding roots of polynomials with degree up to 4: the

quadratic formula

, the

cubic formula

and the

quartic formula

.

If we have a root

a

for a polynomial

f

⁢

(

x

)

, we divide

f

⁢

(

x

)

by

x

-

a

(either by

polynomial long division

or

synthetic division

) and we are left with a polynomial with smaller degrees whose roots are the other roots of

f

. We can use that result together with the

rational root theorem

to find a

rational root

if exists, and then get a polynomial with smaller degree which possibly we can find easily the other roots.

Considering the general case of functions

y

=

f

⁢

(

x

)

(not necessarily polynomials) there are several numerical methods (like Newton’s method) to approximate roots. This could be handy too for polynomials whose roots are not

rational numbers

.

Title

root

Canonical name

Root

Date of creation

2013-03-22 12:08:12

Last modified on

2013-03-22 12:08:12

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 12D10

Synonym

zero

Related topic

FactorTheorem

Related topic

APolynomialOfDegreeNOverAFieldHasAtMostNRoots

Generated on Fri Feb 9 16:38:25 2018 by

LaTeXML
trigonometric cubic formula

trigonometric cubic formula

Given a

cubic polynomial

of the form

f

⁢

(

X

)

=

X

3

+

a

⁢

X

2

+

b

⁢

X

+

c

=

0

, one may reduce

f

⁢

(

x

)

via the substitution

X

→

(

x

-

a

/

3

)

to obtain

f

~

⁢

(

x

)

=

f

⁢

(

x

-

a

/

3

)

where the

reduced

polynomial

may be represented

f

~

⁢

(

x

)

=

x

3

+

q

⁢

x

+

r

(1)

The roots to (1) are given by Viéte in the following cases:

Case I

The roots of

f

~

⁢

(

x

)

are real:

Define

t

≡

-

4

⁢

q

/

3

and

α

=

arccos

⁡

(

-

4

⁢

r

/

t

3

)

.
Then the roots of

f

~

⁢

(

x

)

are

t

⁢

cos

⁡

(

α

/

3

)

,

t

⁢

cos

⁡

(

α

/

3

+

2

⁢

π

/

3

)

,

t

⁢

cos

⁡

(

α

/

3

+

4

⁢

π

/

3

)

Case II

The roots of

f

~

⁢

(

x

)

are

complex

:

Keeping the definition of

t

from Case I, if

-

4

⁢

q

/

3

≥

0

, then the

real root

of

f

~

⁢

(

x

)

is

t

⁢

cosh

⁡

(

β

/

3

)

where

cosh

⁡

(

β

)

=

(

-

4

⁢

r

/

t

3

)

If

-

4

⁢

q

/

3

<

0

, then the real root of

f

~

⁢

(

x

)

is

t

⁢

sinh

⁡

(

γ

/

3

)

where

sinh

⁡

(

γ

)

=

(

-

4

⁢

r

/

t

3

)

One may then

inverse

transform the roots of

f

~

⁢

(

x

)

to obtain the roots of the desired cubic

f

⁢

(

x

)

We note there are no other cases for the possibilities of the roots of a cubic (i.e. there is no

instance

where one finds one complex and two real roots). This result is intuitively obvious after graphing cubic polynomials and taking into account that

imaginary

roots may only

occur in

conjugate pairs.

Title

trigonometric cubic formula

Canonical name

TrigonometricCubicFormula

Date of creation

2013-03-22 15:02:07

Last modified on

2013-03-22 15:02:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 12D10

Synonym

Alternate cubic formula

Related topic

CardanosFormulae

Generated on Fri Feb 9 16:38:26 2018 by

LaTeXML
variant of Cardano’s derivation

variant of Cardano’s derivation

By a linear change of variable, a

cubic polynomial

over

ℂ

can
be given the form

x

3

+

3

⁢

b

⁢

x

+

c

. To find the zeros of this cubic in the
form of surds in

b

and

c

, make the substitution

x

=

y

1

/

3

+

z

1

/

3

,

thus replacing one

unknown

with two, and then write down

identities

which are suggested by the resulting equation in two unknowns. Specifically, we get

y

+

3

⁢

(

y

1

/

3

+

z

1

/

3

)

⁢

y

1

/

3

⁢

z

1

/

3

+

z

+

3

⁢

b

⁢

(

y

1

/

3

+

z

1

/

3

)

+

c

=

0

.

(1)

This will be true if

y

+

z

+

c

=

0

(2)

3

⁢

y

1

/

3

⁢

z

1

/

3

+

3

⁢

b

=

0

,

(3)

which in turn requires

y

⁢

z

=

-

b

3

.

(4)

The pair of equations (2) and (4) is a quadratic system in

y

and

z

,
readily solved. But notice that (3) puts a

restriction

on a certain
choice of cube roots.

Title

variant of Cardano’s

derivation

Canonical name

VariantOfCardanosDerivation

Date of creation

2013-03-22 13:38:43

Last modified on

2013-03-22 13:38:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Proof

Classification

msc 12D10

Generated on Fri Feb 9 16:38:28 2018 by

LaTeXML
Euclidean field

Euclidean field

An

ordered field

F

is

Euclidean

if every non-negative element

a

(

a

≥

0

) is a square in

F

(there exists

b

∈

F

such that

b

2

=

a

).

1

Examples

•

ℝ

is Euclidean.

•

ℚ

is not Euclidean because

2

is not a square in

ℚ

(i.e. (

http://planetmath.org/Ie

),

±

2

∉

ℚ

).

•

ℂ

is not a Euclidean field because

ℂ

is not an ordered field (

http://planetmath.org/MathbbCIsNotAnOrderedField

).

•

The

field of real constructible numbers

(

http://planetmath.org/ConstructibleNumbers

) is Euclidean.

A Euclidean field is an ordered

Pythagorean field

.

There are ordered fields that are Pythagorean but not Euclidean.

Title

Euclidean field

Canonical name

EuclideanField

Date of creation

2013-03-22 14:22:39

Last modified on

2013-03-22 14:22:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

34

Author

CWoo (3771)

Entry type

Definition

Classification

msc 12D15

Related topic

ConstructibleNumbers

Related topic

EuclideanNumberField

Defines

Euclidean

Generated on Fri Feb 9 16:38:32 2018 by

LaTeXML
formally real field

formally real field

A field

F

is called

formally real

if

-

1

can not be expressed as a sum of squares (of elements of

F

).

Given a field

F

, let

S

F

be the set of all sums of squares in

F

. The following are

equivalent

conditions that

F

is formally real:

1.

-

1

∉

S

F

2.

S

F

≠

F

and

char

⁡

(

F

)

≠

2

3.

∑

a

i

2

=

0

implies each

a

i

=

0

, where

a

i

∈

F

4.

F

can be ordered (There is a

total order

<

which makes

F

into an

ordered field

)

Some Examples:

•

ℝ

and

ℚ

are both formally real fields.

•

If

F

is formally real, so is

F

⁢

(

α

)

, where

α

is a root of an

irreducible polynomial

of odd degree in

F

⁢

[

x

]

. As an example,

ℚ

⁢

(

2

3

⁢

ω

)

is formally real, where

ω

≠

1

is a third root of unity.

•

ℂ

is not formally real since

-

1

=

i

2

.

•

Any field of

characteristic

non-zero is not formally real; it is not orderable.

A formally real field is said to be

real closed

if any of its

algebraic extension

which is also formally real is itself. For any formally real field

k

, a formally real field

K

is said to be a

real closure

of

k

if

K

is an algebraic extension of

k

and is real closed.

In the example above,

ℝ

is real closed, and

ℚ

is not, whose real closure is

ℚ

~

. Furthermore, it can be shown that the real closure of a

countable

formally real field is countable, so that

ℚ

~

≠

ℝ

.

Title

formally real field

Canonical name

FormallyRealField

Date of creation

2013-03-22 14:22:22

Last modified on

2013-03-22 14:22:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 12D15

Related topic

PositiveCone

Related topic

RealRing

Defines

formally real

Defines

real closed

Defines

real closure

Generated on Fri Feb 9 16:38:34 2018 by

LaTeXML
positive semidefinite

positive semidefinite

A rational function

f

∈

ℝ

⁢

(

X

1

,

…

,

X

n

)

is

positive semidefinite

if

f

is nonnegative whenever it is defined.

Title

positive semidefinite

Canonical

name

PositiveSemidefinite1

Date of creation

2013-03-22 17:43:32

Last modified on

2013-03-22 17:43:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

4

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 12D15

Generated on Fri Feb 9 16:38:38 2018 by

LaTeXML
Pythagorean field

Pythagorean field

Let

F

be a field. A

field extension

K

of

F

is called a

Pythagorean extension

if

K

=

F

⁢

(

1

+

α

2

)

for some

α

in

F

, where

1

+

α

2

denotes a root of the

polynomial

x

2

-

(

1

+

α

2

)

in the

algebraic closure

F

¯

of

F

. A field

F

is

Pythagorean

if every Pythagorean extension of

F

is

F

itself.

The following are

equivalent

:

1.

F

is Pythagorean

2.

Every

sum of two squares

in

F

is a square

3.

Every sum of (finite number of) squares in

F

is a square

Examples:

•

ℝ

and

ℂ

are Pythagorean.

•

ℚ

is not Pythagorean.

Remark

. Every field is

contained

in some

Pythagorean field

. The smallest Pythagorean field over a field

F

is called the

Pythagorean closure

of

F

, and is written

F

p

⁢

y

. Given a field

F

, one way to construct its Pythagorean closure is as follows: let

K

be an

extension

over

F

such that there is a tower

F

=

K

1

⊆

K

2

⊆

⋯

⊆

K

n

=

K

of fields with

K

i

+

1

=

K

i

⁢

(

1

+

α

i

2

)

for some

α

i

∈

K

i

, where

i

=

1

,

…

,

n

-

1

. Take the

compositum

L

of the family

𝒦

of all such

K

’s. Then

L

=

F

p

⁢

y

.

Title

Pythagorean field

Canonical name

PythagoreanField

Date of creation

2013-03-22 14:22:36

Last modified on

2013-03-22 14:22:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 12D15

Defines

Pythagorean extension

Defines

Pythagorean closure

Generated on Fri Feb 9 16:38:40 2018 by

LaTeXML
real closed fields are o-miminal

real closed fields are o-miminal

It is clear that the axioms for a

structure

to be an

ordered field

can be written in

L

, the

first order language

of ordered rings. It is also true that the condition

for each odd degree

polynomial

p

∈

K

⁢

[

x

]

,

p

has a root

can be written in a schema of

first order

sentences

in this

language

.

Let

A

be all these sentences together with one that states that all

positive elements

have a

square root

.
Then one can show that the

consequences

of

A

are a

complete theory

T

.
It is clear that this theory is the theory of the

real numbers

. We call any

L

structure a

real closed field

(which can be defined purely algebraically also, see here (

http://planetmath.org/RealClosed

)).

The

semi

algebraic sets

on a real closed field are

Boolean

combinations

of solution sets of polynomial

equalities

and

inequalities

.
Tarski showed that

T

has

quantifier elimination

, which is

equivalent

to the class of semi algebraic sets being

closed under

projection

.

Let

K

be a real closed field. Consider the

definable

subsets of

K

. By quantifier elimination,
each is definable by a

quantifier free formula

, i.e. a boolean combination of atomic formulas.
An atomic formula in one variable has one of the following forms:

•

f

⁢

(

x

)

>

g

⁢

(

x

)

for some

f

,

g

∈

K

⁢

[

x

]

•

f

⁢

(

x

)

=

g

⁢

(

x

)

for some

f

,

g

∈

K

⁢

[

x

]

.

The first defines a finite union of

intervals

, the second defines a finite union of points. Every definable subset of

K

is a finite union of these kinds of sets, so is a finite union of intervals and points.
Thus any real closed field is

o-minimal

.

Title

real closed fields are o-miminal

Canonical name

RealClosedFieldsAreOmiminal

Date of creation

2013-03-22 13:23:04

Last modified on

2013-03-22 13:23:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Example

Classification

msc 12D15

Classification

msc 14P10

Classification

msc 03C64

Classification

msc 12D99

Related topic

Theory

Generated on Fri Feb 9 16:38:42 2018 by

LaTeXML
theorem on constructible numbers

theorem on constructible numbers

Theorem 1

.

Let

F

be the

field of constructible numbers

and

α

∈

F

. Then there exists a

nonnegative integer

k

such that

[

Q

(

α

)

:

Q

]

=

2

k

.

Before proving this

theorem

, some preliminaries must be addressed.

First of all, within this entry, the following nonconventional definition will be used:

Let

S

be a subset of

ℂ

that contains a nonzero

complex number

and

α

∈

ℂ

. Then

α

is

immediately constructible from

S

if any of the following hold:

•

α

=

a

+

b

for some

a

,

b

∈

S

;

•

α

=

a

-

b

for some

a

,

b

∈

S

;

•

α

=

a

⁢

b

for some

a

,

b

∈

S

;

•

α

=

a

/

b

for some

a

,

b

∈

S

with

b

≠

0

;

•

α

=

|

z

|

⁢

e

i

⁢

θ

2

for some

z

∈

S

with

z

≠

0

and

θ

=

arg

⁡

(

z

)

with

0

≤

θ

<

2

⁢

π

.

The following lemmas are clear from this definition:

Lemma 1

.

Let

S

be a subset of

C

that contains a nonzero complex number and

α

∈

C

. Then

α

is constructible from

S

if and only if there exists a

finite sequence

α

1

,

…

,

α

n

∈

C

such that

α

1

is immediately constructible from

S

,

α

2

is immediately constructible from

S

∪

{

α

1

}

,

…

, and

α

is immediately constructible from

S

∪

{

α

1

,

…

,

α

n

}

.

Lemma 2

.

Let

F

be a

subfield

of

C

and

α

∈

C

. If

α

is immediately constructible from

F

, then either

[

F

(

α

)

:

F

]

=

1

or

[

F

(

α

)

:

F

]

=

2

.

Now to prove the theorem.

Proof.

By the first lemma, there exists a finite sequence

α

1

,

…

,

α

n

∈

ℂ

such that

α

1

is immediately constructible from

ℚ

,

α

2

is immediately constructible from

ℚ

∪

{

α

1

}

,

…

, and

α

is immediately constructible from

ℚ

∪

{

α

1

,

…

,

α

n

}

. Thus,

α

2

is immediately constructible from

ℚ

⁢

(

α

1

)

,

…

, and

α

is immediately constructible from

ℚ

⁢

(

α

1

,

…

,

α

n

)

. By the second lemma,

[

ℚ

⁢

(

α

1

)

:

ℚ

]

is equal to either

1

or

2

,

[

ℚ

⁢

(

α

1

,

α

2

)

:

ℚ

⁢

(

α

1

)

]

is equal to either

1

or

2

,

…

, and

[

ℚ

⁢

(

α

1

,

…

,

α

n

,

α

)

:

ℚ

⁢

(

α

1

,

…

,

α

n

)

]

is equal to either

1

or

2

. Therefore, there exists a nonnegative integer

m

such that

[

ℚ

(

α

1

,

…

,

α

n

,

α

)

:

ℚ

]

=

2

m

. Since

ℚ

⊆

ℚ

⁢

(

α

)

⊆

ℚ

⁢

(

α

1

,

…

,

α

n

,

α

)

, it follows that there exists a nonnegative integer

k

such that

[

ℚ

(

α

)

:

ℚ

]

=

2

k

.
∎

Title

theorem on constructible numbers

Canonical name

TheoremOnConstructibleNumbers

Date of creation

2013-03-22 17:16:28

Last modified on

2013-03-22 17:16:28

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 12D15

Related topic

ConstructibleNumbers

Related topic

ClassicalProblemsOfConstructibility

Defines

immediately constructible from

Generated on Fri Feb 9 16:38:45 2018 by

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Archimedean property

Archimedean property

Let

x

be any

real number

. Then there exists a

natural number

n

such that

n

>

x

.

This

theorem

is known as the

Archimedean property

of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a

consequence

of the least

upper bound

property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).

Proof.

Let

x

be a real number, and let

S

=

{

a

∈

ℕ

:

a

≤

x

}

. If

S

is empty, let

n

=

1

; note that

x

<

n

(otherwise

1

∈

S

).

Assume

S

is nonempty. Since

S

has an upper bound,

S

must have a least upper bound; call it

b

. Now consider

b

-

1

. Since

b

is the least upper bound,

b

-

1

cannot be an upper bound of

S

; therefore, there exists some

y

∈

S

such that

y

>

b

-

1

. Let

n

=

y

+

1

; then

n

>

b

. But

y

is a natural, so

n

must also be a natural. Since

n

>

b

, we know

n

∉

S

; since

n

∉

S

, we know

n

>

x

. Thus we have a natural

greater

than

x

.
∎

Corollary 1

.

If

x

and

y

are real numbers with

x

>

0

, there exists a natural

n

such that

n

⁢

x

>

y

.

Proof.

Since

x

and

y

are reals, and

x

≠

0

,

y

/

x

is a real. By the Archimedean property, we can choose an

n

∈

ℕ

such that

n

>

y

/

x

. Then

n

⁢

x

>

y

.
∎

Corollary 2

.

If

w

is a real number greater than

0

, there exists a natural

n

such that

0

<

1

/

n

<

w

.

Proof.

Using Corollary 1, choose

n

∈

ℕ

satisfying

n

⁢

w

>

1

. Then

0

<

1

/

n

<

w

.
∎

Corollary 3

.

If

x

and

y

are real numbers with

x

<

y

, there exists a

rational number

a

such that

x

<

a

<

y

.

Proof.

First examine the case where

0

≤

x

. Using Corollary 2, find a natural

n

satisfying

0

<

1

/

n

<

(

y

-

x

)

. Let

S

=

{

m

∈

ℕ

:

m

/

n

≥

y

}

. By Corollary 1

S

is non-empty, so let

m

0

be the least element of

S

and let

a

=

(

m

0

-

1

)

/

n

. Then

a

<

y

. Furthermore, since

y

≤

m

0

/

n

, we have

y

-

1

/

n

<

a

; and

x

<

y

-

1

/

n

<

a

. Thus

a

satisfies

x

<

a

<

y

.

Now examine the case where

x

<

0

<

y

. Take

a

=

0

.

Finally consider the case where

x

<

y

≤

0

. Using the first case, let

b

be a rational satisfying

-

y

<

b

<

-

x

. Then let

a

=

-

b

.
∎

Title

Archimedean property

Canonical name

ArchimedeanProperty

Date of creation

2013-03-22 13:00:47

Last modified on

2013-03-22 13:00:47

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

9

Author

Daume (40)

Entry type

Theorem

Classification

msc 12D99

Synonym

axiom of Archimedes

Synonym

Archimedean principle

Related topic

ArchimedeanSemigroup

Related topic

ExistenceOfSquareRootsOfNonNegativeRealNumbers

Generated on Fri Feb 9 16:38:52 2018 by

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associativity of multiplication

associativity of multiplication

It’s important to know the following

interpretation

of the associative law

a

⋅

(

b

⋅

c

)

=

(

a

⋅

b

)

⋅

c

(1)

of

multiplication

in

arithmetics

and elementary algebra:

A

product

(

b

⋅

c

) is multiplied by a number (

a

) so that only one (

b

) of the product is multiplied by that number.

This rule is sometimes violated even in high school e.g. like

10

⋅

2.5

⋅

0.3

=

25

⋅

3

=

75

,

which is wrong.  Or when solving an equation like

x

⋅

2

⁢

x

-

1

3

=

1

one would like to multiply both sides by 3 for removing the

denominator

, getting perhaps

3

⁢

x

⁢

(

2

⁢

x

-

1

)

=

3

;

then the both of left side have incorrectly been multiplied by 3.

The reason of such mistakes is very likely that one confuses the associative law with the distributive law;  cf. (1) with this latter

a

⋅

(

b

+

c

)

=

a

⋅

b

+

a

⋅

c

,

(2)

which two different

operations

, multiplication and

addition

; both

addends must be multiplied separately

.

Title

associativity of multiplication

Canonical

name

AssociativityOfMultiplication

Date of creation

2013-03-22 15:09:22

Last modified on

2013-03-22 15:09:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Application

Classification

msc 12D99

Classification

msc 00A35

Generated on Fri Feb 9 16:38:54 2018 by

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commensurable numbers

commensurable numbers

Two positive real numbers

a

and

b

are

commensurable

, iff there exists a positive real number

u

such that

a

=

m

⁢

u

,

b

=

n

⁢

u

(1)

with some

positive integers

m

and

n

.  If the positive numbers

a

and

b

are not commensurable, they are

incommensurable

.

Theorem

.

The positive numbers

a

and

b

are commensurable if and only if their ratio is a

rational number

m

n

(

m

,

n

∈

ℤ

).

Proof.

The

equations

(1) imply the

proportion

(

http://planetmath.org/ProportionEquation

)

a

b

=

m

n

.

(2)

Conversely

, if (2) is valid with

m

,

n

∈

ℤ

,  then we can write

a

=

m

⋅

b

n

,

b

=

n

⋅

b

n

,

which means that

a

and

b

are

multiples

of

b

n

and thus commensurable.  Q.E.D.

Example.

The lengths of the side and the diagonal of

http://planetmath.org/node/1086

square are always incommensurable.

0.1

Commensurability as relation

•

The commensurability is an

equivalence relation

in the set

ℝ

+

of the

positive

reals:  the

reflexivity

and the

symmetry

are

trivial

;  if

a

:

b

=

r

and

b

:

c

=

s

,  then

a

:

c

=

(

a

:

b

)

(

b

:

c

)

=

r

s

,  whence one obtains the

transitivity

.

•

The

equivalence classes

of the commensurability are of the form

[

ϱ

]

:=

{

r

ϱ

⋮

r

∈

ℚ

+

}

.

•

One of the equivalence classes is the set

[

1

]

=

ℚ

+

of the positive rationals, all others consist of positive

irrational numbers

.

•

If one sets

[

ϱ

]

⋅

[

σ

]

:=

[

ϱ

⁢

σ

]

,  the equivalence classes form with respect to this

binary operation

an

Abelian group

.

Title

commensurable numbers

Canonical name

CommensurableNumbers

Date of creation

2013-03-22 18:11:14

Last modified on

2013-03-22 18:11:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 12D99

Classification

msc 03E02

Related topic

RationalAndIrrational

Related topic

CommensurableSubgroups

Defines

commensurable

Defines

incommensurable

Defines

commensurability

Generated on Fri Feb 9 16:39:01 2018 by

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