aarjaneiro/planetmath huggingface dataset:
opposite number

opposite number

The

opposite number

of a number

a

is such a number

-

a

that

-

a

+

a

=

0

.

Some

properties

:

•

-

a

=

(

-

1

)

⋅

a

•

-

0

=

0

•

-

(

-

a

)

=

a

•

-

(

a

+

b

)

=

(

-

a

)

+

(

-

b

)

•

-

(

a

⋅

b

)

=

a

⋅

(

-

b

)

=

(

-

a

)

⋅

b

•

-

(

a

-

b

)

=

b

-

a

•

-

∑

j

=

1

n

a

j

=

∑

j

=

1

n

(

-

a

j

)

•

-

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

∫

b

a

f

⁢

(

x

)

⁢

𝑑

x

Exactly similar properties (except the last) are valid in every ring.  The fifth property implies the

Corollary

.

If one changes the sign of one factor of a ring product, then the sign of the whole product changes.

Title

opposite number

Canonical

name

OppositeNumber

Date of creation

2013-03-22 15:03:25

Last modified on

2013-03-22 15:03:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 97D99

Classification

msc 12D99

Synonym

negative

[as a noun]

Related topic

Ring

Related topic

OppositePolynomial

Related topic

ConditionOfOrthogonality

Related topic

Automorphism4

Related topic

ProductOfNegativeNumbers

Related topic

PlusSign

Generated on Sat Feb 10 13:45:42 2018 by

LaTeXML
per cent

per cent

The

per cent

may be in general interpreted to mean a ‘

hundredth

’. So e.g. 5 per cent is ‘5 hundredths’, i.e.

5

100

.

In practice, giving some number of per cents, one means so many hundredths of a quantity given in the same or being clear from the context; for example, we can say that the illiteracy in the world is about 20 per cent – meaning that 20/100 of the adults of the world cannot read. If we say that the

interest

(rate) of a loan is 8 per cent, it means that one must pay interest for the loan 8/100 of the amount of the loan in a year.

If a

percentage

of a quantity has changed e.g. from 12% to 15%, we must not say that it has grown 3% but that it has grown 3

percentage points

.

Determination of percentage

How many percent a number

a

is of a second number

b

? The answer, the

per cent number

p

, is obtained from

p

=

a

b

⋅

100

.

(1)

The number

b

here is called the

base value

and

a

the

per cent value

(?). Essentially, the procedure in (1) may be replaced by converting the ratio

a

b

to hundredths, which can be done formally by multiplying this ratio by

1

=

100

100

=

100

%

:

a

b

=

a

b

⋅

100

%

.

Title

per cent

Canonical

name

PerCent

Date of creation

2013-03-22 17:32:43

Last modified on

2013-03-22 17:32:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 97D99

Classification

msc 00A69

Synonym

percent

Defines

percentage point

Defines

per cent number

Defines

base value

Defines

per cent value

Generated on Sat Feb 10 13:45:44 2018 by

LaTeXML
strange root

strange root

In solving certain of equations, one may obtain besides the proper () roots (

http://planetmath.org/Equation

) also some

strange roots

which do not

satisfy

the original equation. Such a thing can happen especially when one has in some stage squared both sides of the treated equation; in this situation one must check all “roots” by substituting them to the original equation.

Example.

x

-

x

=

12

x

-

12

=

x

(

x

-

12

)

2

=

(

x

)

2

x

2

-

24

⁢

x

+

144

=

x

x

2

-

25

⁢

x

+

144

=

0

x

=

25

±

25

2

-

4

⋅

144

2

=

25

±

7

2

x

=

16

∨

x

=

9

Substituting these values of

x

into the left side of the original equation yields

16

-

4

=

12

,

9

-

3

=

6

.

Thus, only

x

=

16

is valid,

x

=

9

is a strange root. (How

x

=

9

is related to the solved equation, is explained by that it may be written

(

x

)

2

-

x

-

12

=

0

, from which one would obtain via the

quadratic formula

that

x

=

1

±

7

2

,  i.e.

x

=

4

or

x

=

-

3

.  The latter corresponds the value

x

=

9

,  but it were relevant to the original equation only if we would allow

negative

values for

square roots

of positive numbers; the practice excludes them.)

The general explanation of strange roots when squaring an equation is, that the two equations

a

=

b

,

a

2

=

b

2

are not

equivalent

(

http://planetmath.org/Equivalent3

) (but the equations

a

=

±

b

and

a

2

=

b

2

would be such ones).

Title

strange root

Canonical

name

StrangeRoot

Date of creation

2013-03-22 17:55:53

Last modified on

2013-03-22 17:55:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 97D99

Classification

msc 26A09

Synonym

wrong root

Synonym

extraneous root

Related topic

QuadraticFormula

Related topic

LogicalOr

Related topic

SquaringConditionForSquareRootInequality

Generated on Sat Feb 10 13:45:46 2018 by

LaTeXML
proportion equation

proportion equation

The

proportion equation

, or usually simply
, is an equation whose both are ratios (

http://planetmath.org/

Division

) of (non-zero) numbers:

a

b

=

c

d

or

a

:

b

=

c

:

d

(1)

The numbers

a

,

b

,

c

,

d

are the

members

of the ;

a

and

d

are the

extreme members

and

b

and

c

are the

middle members

.  The number

d

is called the

fourth proportional

of the numbers

a

,

b

and

c

.

.

•

The product of the extreme members of the is equal to the product of the middle members.

•

The

a

c

=

b

d

,

i.e., the middle members can be swapped.

•

The

a

+

b

a

-

b

=

c

+

d

c

-

d

if the do not vanish.

•

If any three members of a are known, then the fourth member may be determined (often by using the first property).

•

If the number

b

satisfies the proportion

a

b

=

b

c

(2)

then

b

is called the

central proportional

of

a

and

c

.  We have

b

=

a

⁢

c

,

i.e., the central proportional of two

real numbers

(of same sign) equals to their

geometric mean

.

•

In (2), the number

c

is called the

third proportional

of

a

and

b

.

Title

proportion equation

Canonical name

ProportionEquation

Date of creation

2014-02-23 21:37:10

Last modified on

2014-02-23 21:37:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Topic

Classification

msc 97U99

Classification

msc 12D99

Synonym

proportion

Related topic

Equation

Related topic

SimilarityInGeometry

Related topic

GoldenRatio

Related topic

ContraharmonicProportion

Related topic

ConstructionOfFourthProportional

Defines

proportion

Defines

extreme members

Defines

middle members

Defines

fourth proportional

Defines

central proportional

Defines

third proportional

Generated on Sat Feb 10 13:45:48 2018 by

LaTeXML
proportionality of numbers

proportionality of numbers

The nonzero numbers

a

1

,

a

2

,

…

,

a

n

are (

directly

)

proportional

to the nonzero numbers

b

1

,

b

2

,

…

,

b

n

if

a

1

:

a

2

:

…

:

a

n

=

b

1

:

b

2

:

…

:

b

n

,

(1)

which special notation means the simultaneous

validity

of the proportion equations

a

1

:

a

2

=

b

1

:

b

2

,

a

2

:

a

3

=

b

2

:

b

3

,

…

,

a

n

-

1

:

a

n

=

b

n

-

1

:

b

n

.

(2)

It follows however that

a

i

:

a

j

=

b

i

:

b

j

for all

⁢

i

,

j

.

(3)

In fact, if one multiplies the

left hand sides

of e.g. two first equations (2) and similarly their right hand sides, then one obtains

a

1

:

a

3

=

b

1

:

b

3

.

Swapping the middle members of the proportions (2), which by the parent entry (

http://planetmath.org/ProportionEquation

) is allowable, one gets the system of equations

a

1

b

1

=

a

2

b

2

=

…

=

a

n

b

n

(4)

which is

equivalent

(

http://planetmath.org/Equivalent3

) with (1) and (2).

The numbers

a

1

,

a

2

,

…

,

a

n

are

inversely proportional

to the numbers

b

1

,

b

2

,

…

,

b

n

if

a

1

:

a

2

:

…

:

a

n

=

1

b

1

:

1

b

2

:

…

:

1

b

n

.

Then we have

a

i

:

a

j

=

b

j

:

b

i

for all

⁢

i

,

j

.

Note.

The notation

a

1

:

a

2

:

…

:

a

n

expressing the “ratio of several numbers” is, of course, , but it behaves as the ratio (= the

quotient

) of two numbers in the sense that all of its members

a

i

may be multiplied by a nonzero number without injuring the validity of (1).

Example.

Let

a

:

b

=

2

:

3

and

b

:

c

=

4

:

5

.  Determine the least

positive integers

to which the numbers

a

,

b

,

c

are  a) directly,  b) inversely proportional.

a) The least common

multiple

of 3 and 4, the members corresponding the members

b

in the given proportions, is 12.  Thus we must multiply the right hand sides of these proportions respectively by

12

3

=

4

and

12

4

=

3

:

a

:

b

=

2

:

3

=

8

:

12

,

b

:

c

=

4

:

5

=

12

:

15

.

Accordingly,

a

:

b

:

c

=

8

:

12

:

15

.

b) We may write

a

:

b

=

1

3

:

1

2

=

1

15

:

1

10

,

b

:

c

=

1

5

:

1

4

=

1

10

:

1

8

,

where the

denominators

of the right hand sides have been multiplied by 5 and 2, respectively.  Consequently,

a

:

b

:

c

=

1

15

:

1

10

:

1

8

,

i.e. the required

integers

are 15, 10, 8.

References

1

K. Väisälä

:

Geometria

.  Reprint of the tenth edition.  Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).

Title

proportionality of numbers

Canonical name

ProportionalityOfNumbers

Date of creation

2014-02-23 21:26:01

Last modified on

2014-02-23 21:26:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 97U99

Classification

msc 12D99

Synonym

proportionality

Related topic

Variation

Related topic

KalleVaisala

Defines

proportional

Defines

directly proportional

Defines

inversely proportional

Generated on Sat Feb 10 13:45:51 2018 by

LaTeXML