aarjaneiro/planetmath huggingface dataset:
complex conjugate

complex conjugate

1

Definition

1.1

Scalar Complex Conjugate

Let

z

be a

complex number

with

real part

a

and

imaginary part

b

,

z

=

a

+

b

⁢

i

Then the

complex conjugate

of

z

is

z

¯

=

a

-

b

⁢

i

Complex conjugation represents a

reflection

about the

real axis

on the

Argand diagram

representing a complex number.

Sometimes a star (

*

) is used instead of an overline, e.g. in physics you might see

∫

-

∞

∞

Ψ

*

⁢

Ψ

⁢

𝑑

x

=

1

where

Ψ

*

is the complex conjugate of a wave .

1.2

Matrix Complex Conjugate

Let

A

=

(

a

i

⁢

j

)

be a

n

×

m

matrix with complex
entries. Then the

complex conjugate

of

A

is the matrix

A

¯

=

(

a

i

⁢

j

¯

)

. In particular, if

v

=

(

v

1

,

…

,

v

n

)

is a complex row/

column vector

, then

v

¯

=

(

v

1

¯

,

…

,

v

n

¯

)

.

Hence, the matrix complex conjugate is what we would expect: the same matrix with all of its scalar components conjugated.

2

Properties of the Complex Conjugate

2.1

Scalar Properties

If

u

,

v

are complex numbers, then

1.

u

⁢

v

¯

=

(

u

¯

)

⁢

(

v

¯

)

2.

u

+

v

¯

=

u

¯

+

v

¯

3.

(

u

¯

)

-

1

=

u

-

1

¯

4.

(

u

¯

)

¯

=

u

5.

If

v

≠

0

, then

(

u

v

)

¯

=

u

¯

/

v

¯

6.

Let

u

=

a

+

b

⁢

i

. Then

u

¯

⁢

u

=

u

⁢

u

¯

=

a

2

+

b

2

≥

0

(the

complex modulus

).

7.

If

z

is written in polar form as

z

=

r

⁢

e

i

⁢

ϕ

, then

z

¯

=

r

⁢

e

-

i

⁢

ϕ

.

2.2

Matrix and Vector Properties

Let

A

be a matrix with complex entries, and let

v

be a complex row/column vector.

Then

1.

A

T

¯

=

(

A

¯

)

T

2.

A

⁢

v

¯

=

A

¯

⁢

v

¯

, and

v

⁢

A

¯

=

v

¯

⁢

A

¯

. (Here we assume that

A

and

v

are

compatible

size.)

Now assume further that

A

is a complex

square matrix

, then

1.

trace

⁡

A

¯

=

(

trace

⁡

A

)

¯

2.

det

⁡

A

¯

=

(

det

⁡

A

)

¯

3.

(

A

¯

)

-

1

=

A

-

1

¯

Title

complex conjugate

Canonical name

ComplexConjugate

Date of creation

2013-03-22 12:12:03

Last modified on

2013-03-22 12:12:03

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 12D99

Classification

msc 30-00

Classification

msc 32-00

Related topic

Complex

Related topic

ModulusOfComplexNumber

Related topic

AlgebraicConjugates

Related topic

TriangleInequalityOfComplexNumbers

Related topic

Antiholomorphic2

Defines

complex conjugation

Defines

matrix complex conjugate

Generated on Fri Feb 9 16:39:07 2018 by

LaTeXML
complex number

complex number

The ring of

complex numbers

ℂ

is defined to be the

quotient ring

of the

polynomial ring

ℝ

⁢

[

X

]

in one

variable

over the reals by the

principal ideal

(

X

2

+

1

)

. For

a

,

b

∈

ℝ

, the

equivalence class

of

a

+

b

⁢

X

in

ℂ

is usually denoted

a

+

b

⁢

i

, and one has

i

2

=

-

1

.

The complex numbers form an algebraically closed field. There is a standard metric on the complex numbers, defined by

d

⁢

(

a

1

+

b

1

⁢

i

,

a

2

+

b

2

⁢

i

)

:=

(

a

2

-

a

1

)

2

+

(

b

2

-

b

1

)

2

.

Title

complex number

Canonical name

ComplexNumber

Date of creation

2013-03-22 11:52:35

Last modified on

2013-03-22 11:52:35

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 12D99

Classification

msc 30-00

Classification

msc 32-00

Classification

msc 46L05

Classification

msc 18B40

Classification

msc 46M20

Classification

msc 17B37

Classification

msc 22A22

Classification

msc 81R50

Classification

msc 22D25

Synonym

ℂ

Related topic

Complex

Generated on Fri Feb 9 16:39:09 2018 by

LaTeXML
continuous functions on the extended real numbers

continuous functions on the extended real numbers

Within this entry,

ℝ

¯

will be used to refer to the

extended real numbers

.

Theorem 1

.

Let

f

:

R

→

R

be a

function

. Then

f

¯

:

R

¯

→

R

¯

defined by

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

A

if

⁢

x

=

∞

B

if

⁢

x

=

-

∞

is

continuous

if and only if

f

is continuous such that

lim

x

→

∞

⁡

f

⁢

(

x

)

=

A

and

lim

x

→

-

∞

⁡

f

⁢

(

x

)

=

B

for some

A

,

B

∈

R

¯

.

Proof.

Note that

f

¯

is continuous if and only if

lim

x

→

c

⁡

f

¯

⁢

(

x

)

=

f

¯

⁢

(

c

)

for all

c

∈

ℝ

¯

. By defintion of

f

¯

and the

topology

of

ℝ

¯

,

lim

x

→

c

⁡

f

¯

⁢

(

x

)

=

lim

x

→

c

⁡

f

⁢

(

x

)

for all

c

∈

ℝ

¯

. Thus,

f

¯

is continuous if and only if

lim

x

→

c

⁡

f

⁢

(

x

)

=

f

¯

⁢

(

c

)

for all

c

∈

ℝ

¯

. The latter condition is

equivalent

(

http://planetmath.org/Equivalent3

) to the hypotheses that

f

is continuous on

ℝ

,

lim

x

→

∞

⁡

f

⁢

(

x

)

=

A

, and

lim

x

→

-

∞

⁡

f

⁢

(

x

)

=

B

.
∎

Note that, without the

universal assumption

that

f

is a function from

ℝ

to

ℝ

,

necessity

holds, but sufficiency does not. As a

counterexample

to sufficiency, consider the function

f

¯

:

ℝ

→

ℝ

defined by

f

¯

⁢

(

x

)

=

{

1

x

2

if

⁢

x

∈

ℝ

∖

{

0

}

∞

if

⁢

x

=

0

0

if

⁢

x

=

±

∞

.

Title

continuous functions

on the extended real numbers

Canonical name

ContinuousFunctionsOnTheExtendedRealNumbers

Date of creation

2013-03-22 16:59:31

Last modified on

2013-03-22 16:59:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 12D99

Classification

msc 28-00

Generated on Fri Feb 9 16:39:11 2018 by

LaTeXML
every ordered field with the least upper bound property is isomorphic to the real numbers

every ordered field with the least upper bound property is isomorphic to the real numbers

Let

F

be an

ordered field

. If

F

satisfies the least

upper bound

property
then

F

is

isomorphic

as an ordered field to the

real numbers

ℝ

.

Title

every ordered field with the least upper bound property is isomorphic to the real numbers

Canonical name

EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicToTheRealNumbers

Date of creation

2013-03-22 14:10:33

Last modified on

2013-03-22 14:10:33

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Theorem

Classification

msc 12D99

Classification

msc 26-00

Classification

msc 54C30

Generated on Fri Feb 9 16:39:26 2018 by

LaTeXML
examples of continuous functions on the extended real numbers

examples of continuous functions on the extended real numbers

Within this entry,

ℝ

¯

will be used to refer to the extended real numbers.

Examples of

continuous functions

on

ℝ

¯

include:

•

Polynomial functions: Let

f

∈

ℝ

⁢

[

x

]

with

f

⁢

(

x

)

=

∑

j

=

0

n

a

n

⁢

x

n

for some

n

∈

ℕ

and

a

0

,

…

,

a

n

∈

ℝ

with

a

n

≠

0

if

n

≠

0

. Then

f

¯

is defined in the following manner:

(a)

If

n

=

0

, then

f

¯

⁢

(

x

)

=

a

0

for all

x

∈

ℝ

¯

.

(b)

If

n

is odd and

a

n

>

0

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

x

if

⁢

x

∉

ℝ

.

(c)

If

n

is odd and

a

n

<

0

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

-

x

if

⁢

x

∉

ℝ

.

(d)

If

n

≠

0

is even and

a

n

>

0

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

∞

if

⁢

x

∉

ℝ

.

(e)

If

n

≠

0

is even and

a

n

<

0

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

-

∞

if

⁢

x

∉

ℝ

.

•

Exponential functions

: Let

f

⁢

(

x

)

=

a

x

for some

a

∈

ℝ

with

a

>

0

and

a

≠

1

. Then

f

¯

is defined in the following manner:

(a)

If

a

<

1

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

0

if

⁢

x

=

∞

∞

if

⁢

x

=

-

∞

.

(b)

If

a

>

1

, then

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

∞

if

⁢

x

=

∞

0

if

⁢

x

=

-

∞

.

•

Miscellaneous

(a)

Let

f

⁢

(

x

)

=

arctan

⁡

x

. Then

f

¯

is defined by

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

π

2

if

⁢

x

=

∞

-

π

2

if

⁢

x

=

-

∞

.

(b)

Let

f

⁢

(

x

)

=

tanh

⁡

x

. Then

f

¯

is defined by

f

¯

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

ℝ

1

if

⁢

x

=

∞

-

1

if

⁢

x

=

-

∞

.

Of course, not every

function

f

that is

continuous

on

ℝ

extends to a continuous function on

ℝ

¯

. Common examples of these include the

real functions

x

↦

sin

⁡

x

and

x

↦

cos

⁡

x

. (It is proven that these are continuous on

ℝ

in the entry

continuity of sine and cosine

.)

On the other hand, there are some continuous functions

f

¯

:

ℝ

¯

→

ℝ

¯

that have no analogous function

f

:

ℝ

→

ℝ

. For example, consider

f

¯

⁢

(

x

)

=

{

1

x

2

if

⁢

x

∈

ℝ

∖

{

0

}

∞

if

⁢

x

=

0

0

if

⁢

x

=

±

∞

.

Title

examples of continuous functions on the extended real numbers

Canonical name

ExamplesOfContinuousFunctionsOnTheExtendedRealNumbers

Date of creation

2013-03-22 16:59:34

Last modified on

2013-03-22 16:59:34

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 12D99

Classification

msc 28-00

Generated on Fri Feb 9 16:39:30 2018 by

LaTeXML
examples of totally real fields

examples of totally real fields

Here we present

examples of totally real fields

,

totally imaginary
fields

and CM-fields.

Examples

:

1.

Let

K

=

ℚ

⁢

(

d

)

with

d

a

square-free

positive

integer

. Then

Σ

K

=

{

Id

K

,

σ

}

where

Id

K

:

K

↪

ℂ

is
the identity map (

Id

K

⁡

(

k

)

=

k

, for all

k

∈

K

),
whereas

σ

:

K

↪

ℂ

,

σ

⁢

(

a

+

b

⁢

d

)

=

a

-

b

⁢

d

Since

d

∈

ℝ

it
follows that

K

is a totally real field.

2.

Similarly, let

K

=

ℚ

⁢

(

d

)

with

d

a square-free

negative

integer. Then

Σ

K

=

{

Id

K

,

σ

}

where

Id

K

:

K

↪

ℂ

is
the identity map (

Id

K

⁡

(

k

)

=

k

, for all

k

∈

K

),
whereas

σ

:

K

↪

ℂ

,

σ

⁢

(

a

+

b

⁢

d

)

=

a

-

b

⁢

d

Since

d

∈

ℂ

and
it is not in

ℝ

, it follows that

K

is a totally imaginary
field.

3.

Let

ζ

n

,

n

≥

3

, be a primitive

n

t

⁢

h

root of
unity

and let

L

=

ℚ

⁢

(

ζ

n

)

, a

cyclotomic extension

. Note
that the only roots of unity that are real are

±

1

. If

ψ

:

L

↪

ℂ

is an

embedding

, then

ψ

⁢

(

ζ

n

)

must be a

conjugate

of

ζ

n

, i.e. one of

{

ζ

n

a

∣

a

∈

(

ℤ

/

n

⁢

ℤ

)

×

}

but those are all

imaginary

. Thus

ψ

⁢

(

L

)

⊈

ℝ

. Hence

L

is a totally
imaginary field.

4.

In fact,

L

as in

(

3

)

is a CM-field. Indeed, the maximal
real

subfield

of

L

is

F

=

ℚ

⁢

(

ζ

n

+

ζ

n

-

1

)

Notice
that the

minimal polynomial

of

ζ

n

over

F

is

X

2

-

(

ζ

n

+

ζ

n

-

1

)

⁢

X

+

1

so we obtain

L

from

F

by adjoining the

square root

of the

discriminant

of this

polynomial

which is

ζ

n

2

+

ζ

n

-

2

-

2

=

2

⁢

cos

⁡

(

4

⁢

π

n

)

-

2

<

0

and any other conjugate is

ζ

n

2

⁢

a

+

ζ

n

-

2

⁢

a

-

2

=

2

⁢

cos

⁡

(

4

⁢

a

⁢

π

n

)

-

2

<

0

,

a

∈

(

ℤ

/

n

⁢

ℤ

)

×

Hence,

L

is a CM-field.

5.

Notice that any

quadratic imaginary number field

is
obviously a CM-field.

Title

examples of totally real fields

Canonical name

ExamplesOfTotallyRealFields

Date of creation

2013-03-22 13:55:05

Last modified on

2013-03-22 13:55:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Example

Classification

msc 12D99

Related topic

TotallyRealAndImaginaryFields

Related topic

NumberField

Defines

examples of totally imaginary fields

Defines

examples of CM-fields

Generated on Fri Feb 9 16:39:32 2018 by

LaTeXML
factorization of primitive polynomial

factorization of primitive polynomial

As an application of the parent entry (

http://planetmath.org/EliminationOfUnknown

) we take the

factorization

of a

primitive polynomial

of

ℤ

⁢

[

x

]

into

primitive

(

http://planetmath.org/PrimitivePolynomial

)

prime factors

.  We shall see that the procedure may be done by performing a finite number of tests.

Let

a

(

x

)

=

:

a

n

x

n

+

a

n

-

1

x

n

-

1

+

…

+

a

0

be a primitive polynomial in

ℤ

⁢

[

x

]

.

By the

rational root theorem

and the

factor theorem

, one finds all first-degree prime factors

x

-

a

and thus all

primitive prime factors

of the

polynomial

a

⁢

(

x

)

.

If

a

⁢

(

x

)

has a primitive quadratic factor, then it has also a factor

x

2

+

p

⁢

x

+

q

(1)

where

p

and

q

are

rationals

(and conversely).  For settling the existence of such a factor we treat

p

and

q

as

unknowns

and perform the

long division

a

⁢

(

x

)

:

(

x

2

+

p

⁢

x

+

q

)

.

It gives finally the remainder

b

⁢

(

p

,

q

)

⁢

x

+

c

⁢

(

p

,

q

)

where

b

⁢

(

p

,

q

)

and

c

⁢

(

p

,

q

)

belong to

ℤ

⁢

[

p

,

q

]

.  According to the parent entry (

http://planetmath.org/EliminationOfUnknown

) we bring the system

{

b

⁢

(

p

,

q

)

=

0

c

⁢

(

p

,

q

)

=

0

to the form

{

b

¯

⁢

(

q

)

=

0

c

¯

⁢

(

p

,

q

)

=

0

and then can determine the possible rational solutions

(

p

,

q

)

of the system via a finite number of tests.  Hence
we find the possible quadratic factors (1) having rational

coefficients

.  Such a factor is converted into a primitive one when it is multiplied by the gcd of the

denominators

of

p

and

q

.

Determining a possible cubic factor

x

3

+

p

⁢

x

2

+

q

⁢

x

+

r

with rational coefficients requires examination of a remainder of the form

b

⁢

(

p

,

q

,

r

)

⁢

x

2

+

c

⁢

(

p

,

q

,

r

)

⁢

x

+

d

⁢

(

p

,

q

,

r

)

.

In the needed system

{

b

⁢

(

p

,

q

,

r

)

=

0

c

⁢

(

p

,

q

,

r

)

=

0

d

⁢

(

p

,

q

,

r

)

=

0

we have to perform two eliminations.  Then we can act as above and find a primitive cubic factor of

a

⁢

(

x

)

.  Similarly also the primitive factors of higher degree.  All in all, one needs only look for factors of degree

≤

n

2

.

References

1

K. Väisälä:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

factorization of primitive polynomial

Canonical name

FactorizationOfPrimitivePolynomial

Date of creation

2013-03-22 19:20:30

Last modified on

2013-03-22 19:20:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 12D99

Classification

msc 26C05

Synonym

primitive factors of primitive polynomial

Related topic

EliminationOfUnknown

Generated on Fri Feb 9 16:39:34 2018 by

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fundamental theorem of algebra

fundamental theorem of algebra

Theorem

.

Let

f

∈

C

⁢

[

Z

]

be a non-constant

polynomial

. Then there is a

z

∈

C

with

f

⁢

(

z

)

=

0

.

In other ,

ℂ

is

algebraically closed

.

As a corollary, a non-constant polynomial in

ℂ

⁢

[

Z

]

factors completely
into linear factors.

Title

fundamental theorem of algebra

Canonical name

FundamentalTheoremOfAlgebra

Date of creation

2013-03-22 12:18:56

Last modified on

2013-03-22 12:18:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

14

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 12D99

Classification

msc 30A99

Related topic

ComplexNumber

Related topic

Complex

Related topic

TopicEntryOnComplexAnalysis

Related topic

ZeroesOfDerivativeOfComplexPolynomial

Generated on Fri Feb 9 16:39:36 2018 by

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fundamental theorem of algebra result

fundamental theorem of algebra result

This leads to the following theorem:

Given a

polynomial

p

⁢

(

x

)

=

a

n

⁢

x

n

+

a

n

-

1

⁢

x

n

-

1

+

…

+

a

1

⁢

x

+

a

0

of degree

n

≥

1

where

a

i

∈

ℂ

, there are exactly

n

roots in

ℂ

to the equation

p

⁢

(

x

)

=

0

if we count

multiple roots

.

Proof

The non-constant polynomial

a

1

⁢

x

-

a

0

has one root,

x

=

a

0

/

a

1

.
Next, assume that a polynomial of degree

n

-

1

has

n

-

1

roots.

The polynomial of degree

n

has then by the

fundamental theorem of algebra

a root

z

n

. With polynomial

division

we find the unique polynomial

q

⁢

(

x

)

such that

p

⁢

(

x

)

=

(

x

-

z

n

)

⁢

q

⁢

(

x

)

. The original equation has then

1

+

(

n

-

1

)

=

n

roots.
By induction, every non-constant polynomial of degree

n

has exactly

n

roots.

For example,

x

4

=

0

has four roots,

x

1

=

x

2

=

x

3

=

x

4

=

0

.

Title

fundamental theorem of algebra result

Canonical name

FundamentalTheoremOfAlgebraResult

Date of creation

2013-03-22 14:22:01

Last modified on

2013-03-22 14:22:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 12D99

Classification

msc 30A99

Generated on Fri Feb 9 16:39:38 2018 by

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imaginary

imaginary

An

imaginary number

is the product of a nonzero

real number

multiplied by an

imaginary unit

(such as

i

) but having having

real part

0.

Any

complex number

c

∈

ℂ

may be written in the form

c

=

a

+

b

⁢

i

where

i

is the imaginary unit

i

=

-

1

and

a

and

b

are real numbers (

a

,

b

∈

ℝ

). So an imaginary number is a complex number

c

such that

a

=

0

in the above formulation, i.e. such that

c

can be written as

c

=

b

⁢

i

. Such a complex number is then sometimes called a

purely imaginary number

. For a purely imaginary number

c

, then it is the case that

ℜ

⁡

(

c

)

=

0

and

ℑ

⁡

(

c

)

≠

0

and

b

∈

ℝ

*

. A few examples of purely imaginary numbers:

0

+

47

⁢

i

,

0

-

π

⁢

i

,

3

4

⁢

i

.

Note that the imaginary numbers are closed under

addition

but not under

multiplication

, since

i

×

i

=

-

1

is not imaginary.

Whether

0

+

0

⁢

i

is an imaginary number is a matter of debate. It is in the middle of the number line of the real numbers but it is also in the middle of the number line of purely imaginary numbers.

The imaginary has had its

definition

expanded

by analogy to several areas of mathematics. For example, if

V

is

vector space

with a

linear involution

(denoted

v

→

v

*

), then an imaginary element in

V

is one such that

v

*

=

-

v

.

Much in the same way that the Impressionist painters came to be known by a term initially intended to be derogatory, the term “imaginary number” comes from a sneer by Descartes as to the validity of the concept: “For any equation one can imagine as many roots [as its degree would suggest], but in many cases no quantity exists which corresponds to what one imagines.”

References

1

Titu Andreescu & Dorin Andrica,

Complex Numbers from A to… Z

.

2

Bryan E. Blank, Book Review of

An Imaginary Tale: The Story of

-

1

,

Notices of the AMS

46

10 (1999): 1236

3

Paul Nahin,

An Imaginary Tale: The Story of

-

1

. Princeton: Princton University Press (1998)

Title

imaginary

Canonical

name

Imaginary

Date of creation

2013-03-22 12:21:11

Last modified on

2013-03-22 12:21:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 12D99

Synonym

pure complex

Related topic

Complex

Related topic

ComplexNumber

Related topic

ImaginaryUnit

Related topic

ImaginaryQuadraticField

Defines

imaginary

Defines

imaginary number

Defines

imaginary element

Defines

purely imaginary number

Defines

purely imaginary

Defines

pure imaginary number

Defines

pure imaginary

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

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imaginary unit

imaginary unit

The

imaginary unit

i

:=

-

1

. Any

imaginary number

m

may be written as

m

=

b

⁢

i

,

b

∈

ℝ

. Any

complex number

c

∈

ℂ

may be written as

c

=

a

+

b

⁢

i

,

a

,

b

∈

ℝ

.

Note that there are two

complex

square roots of

-

1

(i.e. the two solutions to the equation

x

2

+

1

=

0

in

ℂ

), so there is always some ambiguity in which of these we choose to call “

i

” and which we call “

-

i

”, though this has little bearing on any applications of complex numbers.

In physics and some engineering fields, the symbol

j

is used for the imaginary unit.

Title

imaginary unit

Canonical name

ImaginaryUnit

Date of creation

2013-03-22 12:21:14

Last modified on

2013-03-22 12:21:14

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 12D99

Synonym

i

Related topic

Imaginary

Related topic

Complex

Generated on Fri Feb 9 16:39:41 2018 by

LaTeXML
indeterminate form

indeterminate form

The

expression

0

0

is known as the

indeterminate form

. The motivation for this name is that there are no rules for comparing the value of

0

0

to the other

real numbers

. Note that, for example,

1

0

is

not

indeterminate

, since we can justifiably

associate

it with

+

∞

, which

does

compare with the rest of the real numbers (in particular, it is defined to be greater than all of them.)

1

Other Indeterminate Forms

Although

0

0

is often called “the” indeterminate form, there are many others. Some of these are:

1.

∞

∞

, for the same motivating reasons as

0

0

.

2.

0

0

; which is the result of much impassioned debate (especially since

0

!

is defined to be 1, counter-intuitively, but not unreasonably).

3.

1

∞

; notably because of the

derivation

of

e

:

lim

n

→

∞

⁡

(

1

+

1

n

)

n

=

e

A direct substitution would yield

1

∞

.

Title

indeterminate form

Canonical

name

IndeterminateForm

Date of creation

2013-03-22 12:28:19

Last modified on

2013-03-22 12:28:19

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 12D99

Synonym

indeterminate value

Related topic

LHpitalsRule

Related topic

ImproperLimits

Related topic

EmptyProduct

Generated on Fri Feb 9 16:39:43 2018 by

LaTeXML
limit of nth root of n

limit of nth root of n

The

n

th root (

http://planetmath.org/NthRoot

) of

n

tends to 1 as

n

tends to

infinity

, i.e. the

real number

sequence

1

1

,

2

2

,

3

3

,

…

,

n

n

,

…

converges to the limit

lim

n

→

∞

⁡

n

n

=

1

.

(1)

Proof.

If we denote

n

n

:=

1

+

δ

n

, we may write by the

binomial theorem

that

n

=

(

1

+

δ

n

)

n

=

1

+

(

n

1

)

⁢

δ

n

+

(

n

2

)

⁢

δ

n

2

+

…

+

(

n

n

)

⁢

δ

n

n

.

This implies, since all hand side are

positive

(when

n

>

1

), that

n

>

(

n

2

)

⁢

δ

n

2

=

n

⁢

(

n

-

1

)

2

!

⁢

δ

n

2

,

δ

n

2

<

2

n

-

1

,

0

<

δ

n

<

2

n

-

1

,

whence

lim

n

→

∞

⁡

δ

n

=

0

.  Accordingly,

lim

n

→

∞

⁡

n

n

=

lim

n

→

∞

⁡

(

1

+

δ

n

)

=

1

,

Q.E.D.

Note.

(1) follows also from the

corollary

3 in the entry

growth of exponential function

.

Title

limit of nth root of n

Canonical

name

LimitOfNthRootOfN

Date of creation

2014-09-28 13:20:59

Last modified on

2014-09-28 13:20:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 12D99

Classification

msc 30-00

Synonym

sequence of nth roots of n

Generated on Fri Feb 9 16:39:48 2018 by

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