Volume 03 Issue 11-2023
51
International Journal of Pedagogics
(ISSN
–
2771-2281)
VOLUME
03
ISSUE
11
P
AGES
:
51-53
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
6.
676
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
This article shows an unusual method for multiplying on the set of all positive real numbers.
KEYWORDS
Commutativity, associativity, distributivity.
INTRODUCTION
In the set of all composite numbers, a new
multiplication can be specified by the relations
,
b
a b
a
=
.
Find all positive rational numbers for which the
multiplication is as follows:
1)
commutatively:
,
,
a b
b a
=
;
2)
associatively:
, ,
,
,
a b c
a b c
=
;
3)
distribution right and left:
(
),
,
,
,
,(
)
,
,
a
b c
a c
b c
c a
b
c a
c b
+
=
+
+
=
+
.
Research Article
AN UNUSUAL METHOD OF REPRODUCTION
Submission Date:
November 01, 2023,
Accepted Date:
November 05, 2023,
Published Date:
November 10, 2023
Crossref doi:
https://doi.org/10.37547/ijp/Volume03Issue11-10
Gafforov Raxmatjon Abdukaxxorovich
Fergana State University, Fergana, Murabbiylar Street, 19, Uzbekistan
Toshbuvaev Boburmirzo Mashrab O‘G‘Li
Fergana State University, Fergana, Murabbiylar Street, 19, Uzbekistan
Gafforova Nafisa Xanifovna
Academic Lyceum No. 2 At Fergana State University, Uzbekistan
Sobirjonova Moxinur, Muxtarova Umida
Students Of The Faculty Of Mathematics And Informatics Of Fergana State University, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ijp
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 03 Issue 11-2023
52
International Journal of Pedagogics
(ISSN
–
2771-2281)
VOLUME
03
ISSUE
11
P
AGES
:
51-53
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
6.
676
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
I.
If
b
a
=
, then the solution is obvious.
Otherwise, we agree to consider
b
the greater of the
numbers
(
)
b
a
. Then
b
ra
=
, where
1
r
and
1
r
a
r
−
=
, or, what is the same,
1/(
1)
r
a
r
−
=
. Let
1/ (
1) /
r
u v
−
, where
u
and
v
are mutually prime
numbers. Since
a
and
r
are rational, the number
(
) /
r
v
u
u
= +
must be the
v
th power of a rational
number. But then the numbers
u
and
v
u
+
, as
mutually prime, are also equal to the
v
rd powers of
integers. But the difference of two
v
powers with
different bases is greater than
v
, unless it is equal to 1.
Therefore,
1
v
=
and
1 / (
1)
r
u
− =
are integers.
Thus,
[(
1) / ]
u
a
u
u
=
+
and
1
[(
1) / ]
u
b
u
u
+
=
+
.
II.
Note that otrelation
(
)
(
)
bc
b c
a
a
=
holds if
1
a
=
,
b
and
c
are arbitrary numbers or
1
c
=
,
a
and
b
are arbitrary numbers. If
1
a
,
1
c
, then the relation
c
b
bc
=
or
1
c
b
c
−
=
must be
satisfied. As arguments similar to those above show,
(
1) /
c
u
u
= +
,
[(
1) / ]
u
c
u
u
=
+
where
u
is any
integer, and the number
a
remains arbitrary.
III.
Without limiting loss of generality,
assume that
a
b
. Then
a
rb
=
, where
1
r
, and
the ratio
(
)
c
c
c
a
b
a
b
+
=
+
reduces to the relations
(1
)
1
c
c
r
r
+
= +
. Suppose for the time that the
parameter
r
changes
continuously.
Let
1
( )
(1
)
c
f r
r
= +
,
2
( ) 1
c
f r
r
= +
. Then at any
0
r
, derivatives
'
1
( )
f r
and
'
2
( )
f r
are related by one of
three
relations:
'
'
1
2
( )
( )
f r
f r
at
1
c
,
'
'
1
2
( )
( )
f r
f r
=
for
1
c
=
, and
'
'
1
2
( )
( )
f r
f r
for
1
c
.
But
1
2
(0)
(0)
f
f
=
.
Therefore,
1
2
( )
( )
f r
f r
for
1
c
,
1
2
( )
( )
f r
f r
=
for
1
c
=
and
1
2
( )
( )
f r
f r
for
1
c
. Thus, the complete
solution corresponds to
1
c
=
and arbitrary
a
and
b
.
Let
us
now
consider
the
relation
a b
a
b
c
c
c
+
=
+
. Without loss of generality, assume
that
a
b
.Then
(
1)
a
br r
=
. Let
/
b
c
v u
=
,
/
r
p q
=
, where
p
,
q
,
u
,
v
are positive integers,
and each of the fractions
/
v u
and
/
p q
is
irreducible and (because the
1
r
)
p
q
. Relation
a b
a
b
c
c
c
+
=
+
in the accepted notation is reduced to
the form
(
)
p q
p
v
v
u
u
−
−
=
. If
1
v
=
, then
(1
)
q
p
u
u
−
=
. Since
1
p
,
1
q
, then every prime
divisor of
u
must be a divisor of
1
u
−
and therefore a
divisor of unity. From this we conclude that
1
u
=
but
1
u
=
does not satisfy relation
(1
)
q
p
u
u
−
=
. The
resulting contradiction means that
1
v
. Since
v
, is
not a divisor of
p
u
, then
0
p
q
− =
. But then,
b
a
=
and
v
u
u
− =
, or
1
u
=
,
2
v
=
. Thus, in the case
under consideration, the complete solution is
determined by the relations
1 /
a
b
n
= =
,
2
n
c
=
,
where
n
is any integer.
Let us now consider how the sets of elements are
arranged, on which one of the properties of the new
multiplication I-III is satisfied for any choice of its
elements.
Let us first consider the set of elements on which the
new multiplication is commutative. Such a set contains
either one element, which is arbitrary, or two of the
above elements.
Volume 03 Issue 11-2023
53
International Journal of Pedagogics
(ISSN
–
2771-2281)
VOLUME
03
ISSUE
11
P
AGES
:
51-53
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
6.
676
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
We move on to the set of elements, on which the new
multiplication is associative. First of all, let us turn to
the case when the set contains the identity element
and
element
1
a
.
From
relation
[[ ,1],
[ ,[ ,1]]
a
a
a a
=
, it follows that
a
a
a
=
, where
1
a
=
. Thus, if a set bunch of contains unit, then it does
not contain other elements. Let us assume that no
element of the set is equal to unit. Then from relation
[ ,[ , ]] [[ , ], ]
a a a
a a a
=
it follows that
2
a
a
a
=
,
where
2
a
=
. Thus, the set of elements on which the
new multiplication is associative contains only one
element equal to either 1 or 2.
Finally, consider the set of elements on which the new
multiplication is distributive. In the case of
distributivity, the right is
(
)
2
a
a
a
a
a
+
=
or
2
2
a
=
.
Therefore,
1
a
=
and unit-single element belonging to
the set. In the case of distributivity on the left
(
)
2
a a
a
a
a
+
=
, or
2
a
a
=
. Since this relation does not
hold for rational
a
, the set under consideration is
empty.
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Yusupova A.K., Gafforov R.A. The role of student
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November 2021
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Gafforov
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Texas Journal of Multidisciplinary Studies. Vol. 11,
2022.
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