aarjaneiro/planetmath huggingface dataset:
conjugation (mnemonic)

conjugation (mnemonic)

In pre-college mathematics, students typically learn how to rationalize the

denominator

(or, in some cases, numerator) of expressions such as

3

11

+

2

and

x

+

h

-

x

-

h

2

⁢

h

. In to do this, they multiply the numerator and denominator of the fraction by an

algebraic conjugate

(or, in some cases, its

negative

) to eliminate the

square root

(s) (

http://planetmath.org/SquareRoot

) in the appropriate part of the fraction. Typically, the only algebraic conjugates that pre-college students encounter are those in some

quadratic extension

.

Most students who have advanced far enough in mathematics to encounter rationalizing denominators or numerators have also encountered some (usually Indo-European) foreign . Such students are familiar with the concept of of verbs, in which the ending of the verb changes to make agreement with the person and number of the subject. A helpful mnemonic for students to the algebraic conjugates that they need to use is pointing out to them that the procedure in mathematics is (and actually easier) than in foreign . The algebraic conjugates (or their negatives) that they need are nothing more than changing the ending of the number. For example, the way that a pre-college student is taught to rationalize the denominator of an expression such as

3

11

+

2

is:

3

11

+

2

=

3

11

+

2

⋅

11

-

2

11

-

2

=

3

⁢

11

-

6

11

-

4

=

3

⁢

11

-

6

7

Title

conjugation

(mnemonic)

Canonical name

Conjugationmnemonic

Date of creation

2013-03-22 16:00:59

Last modified on

2013-03-22 16:00:59

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Classification

msc 11R04

Related topic

AlgebraicConjugates

Related topic

Division

Related topic

DifferenceOfSquares

Defines

rationalize the denominator

Defines

rationalize the numerator

Generated on Sat Feb 10 13:45:09 2018 by

LaTeXML
even-even-odd rule

even-even-odd rule

The is a mnemonic that is helpful for students for simplifying

radical expressions

. The phrase even-even-odd stands for the rule: If a

real variable

to an even exponent (

http://planetmath.org/Exponent2

) is under a with an even index (

http://planetmath.org/Radical6

) and, when the is eliminated, the resulting on the

variable

is odd, then

absolute value

signs must be placed around the variable. (All numbers to which ”” and ”” refer are natural numbers.) This rule is justified by the following:

Recall that, for any

positive integer

n

,

b

is the

n

th root (

http://planetmath.org/NthRoot

) of

a

if and only if

b

n

=

a

and

sign

⁡

(

b

)

=

sign

⁡

(

a

)

. Thus, for any positive integer

n

and

x

∈

ℝ

,

x

n

n

=

{

|

x

|

if

⁢

n

⁢

is even

x

if

⁢

n

⁢

is odd.

The following are some examples of how to use the even-even-odd rule.

Problem

.

Let

x

, and

y

be real variables. Simplify the

expression

x

12

⁢

y

8

4

.

Solution:

The on the

x

is even (12), the of the is even (4), and the that will occur on the

x

once the is eliminated will be odd (3). Thus, absolute values are necessary on the

x

.

The on the

y

is even (8), the of the is even (4), and the that will occur on the

y

once the is eliminated will be even (2). Thus, according to the rule, absolute values are not necessary on the

y

. (Note, though, that it would not be incorrect to have them.) The reason that the absolute values are not necessary is that

y

2

is nonnegative regardless of the value of

y

.

Thus, we have

x

12

⁢

y

8

4

=

|

x

|

3

⁢

y

2

. (The answer

|

x

3

|

⁢

y

2

is also acceptable.)

Some care is needed in applying the even-even-odd rule, as the next problem shows.

Problem

.

Let

x

be a real variable. Simplify the expression

x

2

4

.

Note that, as stated, the even-even-odd rule does not apply here, since, if the were eliminated, the resulting on the

x

will be

1

2

. On the other hand, it can still be used to provide a correct answer for this particular problem.

Solution:

x

2

4

=

x

2

=

|

x

|

The good news is that, for

square roots

, this issue discussed above does not arise: If the even-even-odd rule does not apply, then absolute values are not necessary. That is because, if

n

∈

ℕ

is odd, the expression

x

n

only makes sense in the

real numbers

when

x

is nonnegative.

I would like to thank Mrs. Sue Millikin, who taught me how to simplify expressions in this manner.

Title

even-even-odd rule

Canonical

name

EvenevenoddRule

Date of creation

2013-03-22 16:01:02

Last modified on

2013-03-22 16:01:02

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Related topic

NthRoot

Related topic

SquareRoot

Related topic

Radical6

Generated on Sat Feb 10 13:45:11 2018 by

LaTeXML
factoring a sum or difference of two cubes

factoring a sum or difference of two cubes

The formula for factoring a sum of two cubes is:

x

3

+

y

3

=

(

x

+

y

)

⁢

(

x

2

-

x

⁢

y

+

y

2

)

The formula for factoring a difference of two cubes is:

x

3

-

y

3

=

(

x

-

y

)

⁢

(

x

2

+

x

⁢

y

+

y

2

)

When teaching these

factorization

methods, it may be a good idea to encourage students to know one method for these factorizations rather than have them memorize two separate formulas.

First of all, the factorization is the

product

of a

binomial

and a

trinomial

. There is a

mnemonic device

for remembering the signs that works when the binomial is put in front of the trinomial as above.

Ignoring signs for the time being, let us determine how to remember the terms in the factorization. The two terms of the binomial are just the two terms of the original with the cubes dropped. The first term of the trinomial is the square of the first term of the binomial, and the last term of the trinomial is the square of the last term of the binomial. Finally, the middle term of the trinomial is the product of the two terms of the binomial.

Filling these in yields:

x

3

+

y

3

=

(

x

y

)

⁢

(

x

2

x

⁢

y

y

2

)

x

3

-

y

3

=

(

x

y

)

⁢

(

x

2

x

⁢

y

y

2

)

Now for the signs. A mnemonic for the signs is SDP, which stands for “same, different, plus”. This means that the first sign, the one in the binomial, is the

same

as that of the original problem; the second sign, the first one in the trinomial, is

different

from that of the original problem; and the third sign, the last one in the trinomial, is always

plus

.

Filling the signs in as indicated yields the correct formulas:

x

3

+

y

3

=

(

x

+

y

)

⁢

(

x

2

-

x

⁢

y

+

y

2

)

x

3

-

y

3

=

(

x

-

y

)

⁢

(

x

2

+

x

⁢

y

+

y

2

)

For students who have problems remembering SDP, the team name

San Diego Padres

comes in handy. (Who would have thought that a baseball team would be useful for remembering mathematical formulas?)

Title

factoring a sum or difference of two cubes

Canonical name

FactoringASumOrDifferenceOfTwoCubes

Date of creation

2013-03-22 15:59:17

Last modified on

2013-03-22 15:59:17

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 97D40

Related topic

IrreducibilityOfBinomialsWithUnityCoefficients

Related topic

DifferenceOfSquares

Defines

SDP

Defines

S.D.P.

Defines

same different plus

Defines

San Diego Padres

Generated on Sat Feb 10 13:45:13 2018 by

LaTeXML
factors with minus sign

factors with minus sign

The sign (cf. plus sign, opposite number) rule

(

+

a

)

⁢

(

-

b

)

=

-

(

a

⁢

b

)

,

(1)

derived in the
parent entry (

http://planetmath.org/productofnegativenumbers

)
and concerning numbers and elements

a

,

b

of an arbitrary
ring, may be generalised to the following

Theorem.

If the sign of one factor (

http://planetmath.org/Product

) in a

ring product

is changed, the sign of the product changes.

Corollary 1.

The product of

real numbers

is equal to the product of their

absolute values

equipped with the “

-

” sign if the number of

negative

factors is odd and with “

+

” sign if it is even. Especially, any

odd power

of a negative real number is negative and any even power of it is positive.

Corollary 2.

Let us consider natural powers of a ring element. If one changes the sign of the base, then an odd power changes its sign but an even power remains unchanged:

(

-

a

)

2

⁢

n

+

1

=

-

a

2

⁢

n

+

1

,

(

-

a

)

2

⁢

n

=

a

2

⁢

n

(

n

∈

ℕ

)

Title

factors with minus sign

Canonical name

FactorsWithMinusSign

Date of creation

2015-02-04 12:30:11

Last modified on

2015-02-04 12:30:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 97D40

Classification

msc 13A99

Synonym

sign rules for products

Related topic

GeneralAssociativity

Related topic

Multiplication

Related topic

DoublyEvenNumber

Generated on Sat Feb 10 13:45:15 2018 by

LaTeXML
inequality with absolute values

inequality with absolute values

Recalling that the

absolute value

(

http://planetmath.org/AbsoluteValue

) of a

real number

means on the

number line

(

real axis

) the distance of the point from the origin, we have the following three rules which remove the absolute value signs from an

inequality

(we use the logical symbol “

∨

” for alternativeness ‘or’).  Note that the symbols “

≥

” and “

≤

” may also be without the

equality

bar.

1.

|

a

|

≥

b

⇔

a

≤

-

b

∨

a

≥

b

2.

|

a

|

≤

b

⇔

-

b

≤

a

≤

b

3.

|

a

|

≥

|

b

|

⇔

a

2

≥

b

2

These rules are valid for all real values of

a

and

b

.  For example, if one has a case

|

x

|

<

-

5

corresponding the rule 2, this inequality seems to be impossible since no absolute value is

negative

; but now also the result

-

(

-

5

)

<

x

<

-

5

given by the rule 2 is impossible — no real number is simultaneously greater than

+

5

and less than

-

5

.

Examples.

We solve some inequalities with absolute values.

a)

|

2

⁢

x

+

1

|

>

5

⁢

x

2

⁢

x

+

1

<

-

5

⁢

x

or

2

⁢

x

+

1

>

5

⁢

x

(rule 1)

7

⁢

x

<

-

1

or

-

3

⁢

x

>

-

1

x

<

-

1

/

7

or

x

<

1

/

3

x

<

1

/

3

(combined)

b)

8

⁢

|

x

|

+

|

x

-

2

|

>

6

|

8

⁢

x

|

>

6

-

|

x

-

2

|

8

⁢

x

<

-

6

+

|

x

-

2

|

or

8

⁢

x

>

6

-

|

x

-

2

|

(rule 1)

|

x

-

2

|

>

8

⁢

x

+

6

or

|

x

-

2

|

>

6

-

8

⁢

x

x

-

2

<

-

8

⁢

x

-

6

or

x

-

2

>

8

⁢

x

+

6

or

x

-

2

<

-

6

+

8

⁢

x

or

x

-

2

>

6

-

8

⁢

x

(rule 1 twice)

9

⁢

x

<

-

4

or

-

7

⁢

x

>

8

or

-

7

⁢

x

<

-

4

or

9

⁢

x

>

8

x

<

-

4

/

9

or

x

<

-

8

/

7

or

x

>

4

/

7

or

x

>

8

/

9

x

<

-

4

/

9

or

x

>

4

/

7

(from the number line)

c)

|

1

-

5

⁢

x

|

≤

3

-

3

≤

1

-

5

⁢

x

≤

3

(rule 2)

-

4

≤

-

5

⁢

x

≤

2

(subtracted 1 from all parts)

4

/

5

≥

x

≥

-

2

/

5

(divided by

-

5

)

-

2

/

5

≤

x

≤

4

/

5

(rewritten from end to begin)

Title

inequality with absolute values

Canonical name

InequalityWithAbsoluteValues

Date of creation

2013-03-22 16:57:20

Last modified on

2013-03-22 16:57:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 97D40

Related topic

AbsoluteValue

Related topic

AbsoluteValueInequalities

Related topic

OrderOfSixMeans

Generated on Sat Feb 10 13:45:17 2018 by

LaTeXML
keep-flip-change

keep-flip-change

Let

a

,

b

,

c

,

d

∈

ℤ

-

{

0

}

. A

mnemonic device

for calculating

a

b

÷

c

d

is

KFC

. This stands for ”

keep-flip-change

”. This means that you keep the first

fraction

as it is, flip the second fraction, and change the problem to a

multiplication

problem. Thus, the

expression

above becomes

a

b

⋅

d

c

.

This yields that

a

b

÷

c

d

=

a

b

⋅

d

c

=

a

⁢

d

b

⁢

c

.

Title

keep-flip-change

Canonical

name

Keepflipchange

Date of creation

2013-03-22 15:59:52

Last modified on

2013-03-22 15:59:52

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Synonym

KFC

Synonym

K.F.C.

Generated on Sat Feb 10 13:45:19 2018 by

LaTeXML
mnemonic device

mnemonic device

A

mnemonic device

is a or phrase that can help people remember something. Mnemonic devices are especially important in teaching mathematics, as those who are not familiar with math can be overwhelmed by the and methods of solving problems that they need to memorize.

Title

mnemonic device

Canonical

name

MnemonicDevice

Date of creation

2013-03-22 15:59:15

Last modified on

2013-03-22 15:59:15

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Synonym

mnemonic

Generated on Sat Feb 10 13:45:20 2018 by

LaTeXML
sohcahtoa

sohcahtoa

Let

θ

be an

acute angle

and fix a

right triangle

having

θ

as one of its angles. The

trigonometric functions

sin

⁡

θ

,

cos

⁡

θ

, and

tan

⁡

θ

can be defined as:

sin

⁡

θ

=

opposite

hypotenuse

cos

⁡

θ

=

adjacent

hypotenuse

tan

⁡

θ

=

opposite

adjacent

A mnemonic device for remembering the three given above is

sohcahtoa

.

The sohcahtoa can be divided up into three portions of three letters each. The first portion, soh, should recall to memory that

sin

⁡

θ

is opposite over

hypotenuse

; the second portion, cah, should recall to memory that

cos

⁡

θ

is

adjacent

over hypotenuse; and the third portion, toa, should recall to memory that

tan

⁡

θ

is opposite over adjacent.

I would like to thank Mrs. Sue Millikin for teaching me trigonometric functions using this mnemonic device.

Title

sohcahtoa

Canonical name

Sohcahtoa

Date of creation

2013-03-22 15:59:23

Last modified on

2013-03-22 15:59:23

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Classification

msc 51-01

Related topic

Adjacent2

Related topic

Opposite2

Related topic

Hypotenuse

Related topic

Trigonometry

Generated on Sat Feb 10 13:45:29 2018 by

LaTeXML
example of integration by parts involving algebraic manipulation

example of integration by parts involving algebraic manipulation

For some

integrals

which require

integration by parts

, it may be to treat the integral like a variable and solve for it. For example, consider the integral

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

.

Using integration by parts with the substitutions

u

=

cos

⁡

x

d

⁢

v

=

e

x

⁢

d

⁢

x

d

⁢

u

=

-

sin

⁡

x

⁢

d

⁢

x

v

=

e

x

,

we obtain

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

=

e

x

⁢

cos

⁡

x

+

∫

e

x

⁢

sin

⁡

x

⁢

d

⁢

x

.

Using integration by parts on the integral on the

right hand side

with the substitutions

u

=

sin

⁡

x

d

⁢

v

=

e

x

⁢

d

⁢

x

d

⁢

u

=

cos

⁡

x

⁢

d

⁢

x

v

=

e

x

,

we obtain

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

=

e

x

⁢

cos

⁡

x

+

e

x

⁢

sin

⁡

x

-

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

.

The “trick” is to add

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

to both sides of the equation. Some people find this concept surprising at first sight, especially since most people who are taking

calculus

for the first time do not use equations when showing their work for integration. For integrals such as

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

, writing out an equation is essential.

After adding

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

to both sides of the above equation, we will need a

+

C

on the right hand side. Thus, we obtain

2

⁢

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

=

e

x

⁢

cos

⁡

x

+

e

x

⁢

sin

⁡

x

+

C

.

Therefore, we can figure out what

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

is by dividing both sides by

2

, which yields

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

=

1

2

⁢

(

e

x

⁢

cos

⁡

x

+

e

x

⁢

sin

⁡

x

+

C

)

.

On the other hand, since

C

is an arbitrary constant, we generally write

∫

e

x

⁢

cos

⁡

x

⁢

d

⁢

x

=

1

2

⁢

(

e

x

⁢

cos

⁡

x

+

e

x

⁢

sin

⁡

x

)

+

C

with the understanding that the constant

C

in the final equation may not have the same value as

C

appearing in equations in previous steps.

Title

example of integration by parts involving algebraic manipulation

Canonical name

ExampleOfIntegrationByPartsInvolvingAlgebraicManipulation

Date of creation

2013-03-22 17:39:52

Last modified on

2013-03-22 17:39:52

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 97D70

Classification

msc 26A36

Related topic

ALectureOnIntegrationByParts

Generated on Sat Feb 10 13:45:32 2018 by

LaTeXML
freshman’s dream error

freshman’s dream error

The name “freshman’s dream

theorem

” comes from the fact that people who are unfamiliar with mathematics commonly make the error of distributing exponents over

addition

and/or

subtraction

, typically when working in fields of characteristic zero. An example is the equation

(

x

+

y

)

2

=

x

2

+

y

2

for

x

,

y

∈

ℝ

. The equation is incorrect unless

x

=

0

or

y

=

0

. By no means does the exponent need to be a

natural number

or an integer for this error to occur. An example of this is the equation

x

+

y

=

x

+

y

for

x

,

y

∈

ℝ

with

x

≥

0

and

y

≥

0

. This equation can be rewritten using the exponent

1

2

, and again, the equation is incorrect unless

x

=

0

or

y

=

0

.

An easy way to explain to someone who is under the impression that exponents

distribute over

addition and/or subtraction is to provide a

counterexample

. For

instance

, when

x

=

3

and

y

=

4

, we have:

(

x

+

y

)

2

=

(

3

+

4

)

2

=

7

2

=

49

x

2

+

y

2

=

3

2

+

4

2

=

9

+

16

=

25

On the other hand, the freshman’s dream theorem yields some instances in which exponents can be distributed over addition and/or subtraction.

Title

freshman’s dream error

Canonical

name

FreshmansDreamError

Date of creation

2013-03-22 16:07:23

Last modified on

2013-03-22 16:07:23

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 97D70

Generated on Sat Feb 10 13:45:34 2018 by

LaTeXML
illustration of why SSA may not prove congruence

illustration of why SSA may not prove congruence

SSA cannot be used to prove that two

triangles

are

congruent

when the angles that are known to be congruent are acute. Below is an illustration of how this can happen.

In the picture below,

△

⁢

A

⁢

B

⁢

C

and

△

⁢

A

⁢

B

⁢

D

share the angle

∠

⁢

A

and the side

A

⁢

B

¯

, and the

line segments

B

⁢

C

¯

and

B

⁢

D

¯

are congruent; however,

△

⁢

A

⁢

B

⁢

C

and

△

⁢

A

⁢

B

⁢

D

are clearly not congruent.

Title

illustration of why SSA may not prove congruence

Canonical name

IllustrationOfWhySSAMayNotProveCongruence

Date of creation

2013-03-22 17:04:11

Last modified on

2013-03-22 17:04:11

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 97D70

Classification

msc 51M99

Classification

msc 51-01

Generated on Sat Feb 10 13:45:35 2018 by

LaTeXML
difference of squares

difference of squares

One of the most known and used formulas (

http://planetmath.org/

Equation

) of mathematics is the one concerning the product of sum and

difference

:

(

a

+

b

)

⁢

(

a

-

b

)

=

a

2

-

b

2

(1)

This form may be used for multiplying any sum of two numbers (terms) by the difference of the same numbers (terms).

In the form

a

2

-

b

2

=

(

a

+

b

)

⁢

(

a

-

b

)

(2)

the formula is used for factoring

binomials

which are the difference of two squares.

(1) is sometimes called the

conjugate rule

, especially in articles written in Sweden (in Swedish:

konjugatregel

).

(1) is an

identic equation

for all numbers

a

,

b

and, more generally, for arbitrary elements

a

,

b

of any

commutative ring

.  Conversely, it is easy to justify that if (1) is true for all elements

a

,

b

of a ring, then the ring is

commutative

.  By the way,

a

+

b

and

a

-

b

also commute with each other in a non-commutative ring.

Title

difference of squares

Canonical name

DifferenceOfSquares

Date of creation

2013-03-22 17:45:11

Last modified on

2013-03-22 17:45:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Topic

Classification

msc 97D99

Classification

msc 26C99

Classification

msc 13A99

Synonym

conjugate rule

Related topic

ConjugationMnemonic

Related topic

ExampleOnSolvingAFunctionalEquation

Related topic

SquareOfSum

Related topic

GroupingMethodForFactorizingPolynomials

Related topic

IncircleRadiusDeterminedByPythagoreanTriple

Related topic

FactoringASumOrDifferenceOfTwoCubes

Related topic

Polynomial

Related topic

SineOfAngleOfTriangle

Related topic

RepresentantsOfQuadraticRe

Generated on Sat Feb 10 13:45:38 2018 by

LaTeXML
empty sum

empty sum

The

empty sum

is such a borderline case of sum where the number of the addends is zero, i.e. the set of the addends is an empty set.

•

One may think that the zeroth

multiple

0

⁢

a

of a ring element

a

is the empty sum; it can spring up by adding in the ring two multiples whose

integer

coefficients are opposite numbers:

(

-

n

)

⁢

a

+

n

⁢

a

=

(

-

n

+

n

)

⁢

a

=

0

⁢

a

This empty sum equals the

additive identity

0 of the ring, since the multiple

(

-

n

)

⁢

a

is defined to be

(

-

a

)

+

(

-

a

)

+

…

+

(

-

a

)

⏟

n

⁢

copies

•

In using the

sigma notation

(

http://planetmath.org/Summing

)

∑

i

=

m

n

f

⁢

(

i

)

(1)

one sometimes sees a case

∑

i

=

m

m

-

1

f

⁢

(

i

)

.

(2)

It must be an empty sum, because in

∑

i

=

m

m

f

⁢

(

i

)

(3)

the number of addends is clearly one and therefore in (2) the number is zero.  Thus the value of (2) may be defined to be 0.

Note.

The sum (1) is not defined when

n

is less than

m

-

1

, but if one would want that the usual rule

∑

i

=

m

n

f

⁢

(

i

)

+

∑

i

=

n

+

1

k

f

⁢

(

i

)

=

∑

i

=

m

k

f

⁢

(

i

)

(4)

would be true also in such a cases, then one has to define

∑

i

=

m

n

f

(

i

)

=

-

∑

i

=

n

+

1

m

-

1

f

(

i

)

(

n

<

m

-

1

)

,

because by (4) one could calculate

0

=

-

∑

i

=

n

+

1

m

-

1

f

⁢

(

i

)

+

∑

i

=

n

+

1

m

-

1

f

⁢

(

i

)

=

∑

i

=

m

n

f

⁢

(

i

)

+

∑

i

=

n

+

1

m

-

1

f

⁢

(

i

)

=

∑

i

=

m

m

-

1

f

⁢

(

i

)

.

Title

empty sum

Canonical name

EmptySum

Date of creation

2013-03-22 18:40:57

Last modified on

2013-03-22 18:40:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Topic

Classification

msc 97D99

Classification

msc 05A19

Classification

msc 00A05

Related topic

EmptyProduct

Related topic

EmptySet

Related topic

AddingAndRemovingParenthesesInSeries

Generated on Sat Feb 10 13:45:40 2018 by

LaTeXML