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cc
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Yo
rù
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á
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m
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b
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m
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n
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m
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r
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l
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.
In
th
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a
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t
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rd
Yo
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b
á
f
o
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m
.
T
h
e
c
o
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p
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tatio
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p
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ss
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s
u
n
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m
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th
e
m
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th
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.
Un
if
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M
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(
UML
)
a
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.
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e
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s
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ted
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it
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e
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f
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a
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Nu
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In
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S
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.
Al
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rig
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rv
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C
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p
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ing
A
uth
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:
Ag
b
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a
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A
.
O
.
,
Dep
ar
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C
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E
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in
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A
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ic,
A
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.
E
m
ail:
ab
a
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m
ia
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b
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y
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n
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i
@
g
m
ail.
co
m
1.
I
NT
RO
D
UCT
I
O
N
A
cc
o
r
d
in
g
to
[
1
]
,
n
u
m
b
er
i
s
an
ar
it
h
m
e
tical
v
a
lu
e,
e
x
p
r
ess
ed
b
y
a
w
o
r
d
,
s
y
m
b
o
l,
o
r
f
ig
u
r
e,
r
ep
r
esen
tin
g
a
p
ar
ticu
lar
q
u
a
n
tit
y
an
d
u
s
ed
i
n
co
u
n
ti
n
g
a
n
d
m
a
k
i
n
g
ca
lc
u
latio
n
s
.
Nat
u
r
al
n
u
m
b
er
s
ar
e
th
e
f
a
m
iliar
p
o
s
itiv
e
w
h
o
le
n
u
m
b
er
s
1
,
2
,
3
,
etc.
,
an
d
th
e
y
clea
r
l
y
p
la
y
an
es
s
e
n
tial
p
ar
t
in
m
an
y
m
at
h
e
m
a
tical
ac
tiv
itie
s
;
f
o
r
ex
a
m
p
le,
co
u
n
t
in
g
a
n
d
a
r
ith
m
et
ic.
I
n
ad
d
itio
n
to
th
eir
p
r
ac
tical
r
o
le,
n
u
m
b
er
s
also
h
a
v
e
a
ce
n
tr
al
p
lace
in
m
at
h
e
m
atica
l
th
eo
r
y
[
2
]
.
A
ls
o
in
[
3
]
,
n
u
m
b
e
r
is
ex
p
lain
ed
as
an
ab
s
tr
ac
t
id
ea
o
f
co
llectio
n
o
f
th
i
n
g
s
b
u
t
n
u
m
er
als
ar
e
m
an
-
m
ad
e
s
y
m
b
o
ls
th
a
t
r
ep
r
esen
t
t
h
e
n
u
m
b
er
s
.
N
u
m
b
er
s
ar
e
al
wa
y
s
t
h
e
s
a
m
e
v
al
u
e,
n
o
m
atter
w
h
at
s
y
m
b
o
l
o
r
w
o
r
d
is
u
s
ed
to
r
ep
r
esen
t
t
h
e
m
.
F
o
r
ex
a
m
p
le,
1
,
I
,
i
ar
e
n
u
m
er
al
s
th
at
r
ep
r
ese
n
t
th
e
s
a
m
e
n
u
m
b
er
as
o
n
e.
I
n
[
4
]
,
th
e
n
u
m
er
al
s
y
s
te
m
o
r
s
y
s
te
m
o
f
n
u
m
er
atio
n
i
s
s
ee
n
as
a
m
a
th
e
m
atica
l
n
o
tatio
n
f
o
r
r
ep
r
esen
tin
g
n
u
m
b
er
s
o
f
a
g
iv
e
n
s
e
t,
u
s
i
n
g
d
ig
it
s
o
r
o
th
er
s
y
m
b
o
l
s
in
a
co
n
s
is
ten
t
m
a
n
n
er
.
Nu
m
er
al
in
w
h
ate
v
er
lan
g
u
ag
e
an
d
in
w
h
ic
h
ev
er
f
o
r
m
s
/p
att
er
n
s
it
ta
k
es
i
s
a
clea
r
m
a
n
i
f
e
s
tatio
n
o
f
th
e
ab
ilit
y
o
f
m
an
to
m
an
ip
u
late
h
is
l
in
g
u
is
tic
r
eso
u
r
ce
s
t
o
ca
ter
f
o
r
h
is
co
m
m
u
n
icati
v
e
n
ee
d
s
.
Nu
m
er
als
ap
p
ea
r
an
asp
ec
t
o
f
la
n
g
u
a
g
e
s
tu
d
y
w
h
ich
r
esear
ch
er
s
,
s
ch
o
lar
s
an
d
li
n
g
u
is
t
s
p
a
y
le
s
s
atten
tio
n
to
.
T
h
is
n
eg
lec
t
co
u
ld
b
e
co
n
n
ec
ted
to
ce
r
tain
i
m
p
r
es
s
io
n
t
h
at
th
er
e
is
lit
tle
o
r
n
o
t
h
i
n
g
else
to
s
a
y
ab
o
u
t
it.
P
ar
t
o
f
o
u
r
lin
g
u
i
s
tic
k
n
o
w
led
g
e
i
s
co
n
ta
in
ed
in
th
e
ab
il
it
y
to
ex
p
r
ess
o
u
r
id
ea
s
i
n
o
u
r
m
o
th
er
to
n
g
u
e
s
.
O
u
r
lin
g
u
is
tic
k
n
o
w
led
g
e
r
e
m
ain
s
i
n
s
u
f
f
icie
n
t
i
f
w
e
ar
e
u
n
ab
le
to
co
u
n
t
f
l
u
en
t
l
y
,
a
n
al
y
ze
h
o
w
t
h
e
co
u
n
t
in
g
i
s
d
o
n
e,
a
n
d
u
n
d
er
s
ta
n
d
t
h
e
m
a
n
ip
u
latio
n
s
th
er
ein
[
5
]
.
A
s
n
o
te
d
in
[
6
]
,
co
u
n
ti
n
g
an
d
o
r
n
u
m
b
er
i
n
g
is
an
i
n
te
g
r
al
a
n
d
in
s
ep
ar
ab
le
p
ar
t
o
f
t
h
e
g
r
a
m
m
ar
o
f
a
n
y
la
n
g
u
a
g
e
b
ec
au
s
e
t
h
er
e
is
h
ar
d
l
y
an
y
m
ea
n
i
n
g
f
u
l
l
in
g
u
i
s
tic
d
i
s
co
u
r
s
e
in
a
la
n
g
u
a
g
e
th
a
t
d
o
es
n
o
t
m
ak
e
r
ef
er
e
n
ce
to
q
u
an
tit
y
,
s
ize
,
ti
m
e,
d
is
ta
n
ce
a
n
d
w
e
ig
h
t
i
n
d
e
f
in
i
te
n
u
m
b
er
s
.
A
l
s
o
,
th
e
u
s
e
o
f
n
u
m
b
er
s
a
n
d
th
eir
p
o
w
er
i
n
ca
p
tu
r
i
n
g
co
n
ce
p
ts
h
a
v
e
m
a
k
es
t
h
e
m
i
n
d
is
p
en
s
ab
le
i
n
ef
f
ec
ti
v
e
co
m
m
u
n
icatio
n
[
7
]
.
I
n
t
h
e
s
o
ciet
y
to
d
a
y
,
n
u
m
b
er
s
ar
e
s
ee
n
a
s
a
n
i
n
d
is
p
e
n
s
ab
le
to
o
l
in
o
u
r
d
a
y
-
to
-
d
a
y
ac
tiv
itie
s
.
[
8
]
s
tated
th
at,
k
e
y
ad
v
a
n
ce
m
e
n
t
in
ci
v
il
izatio
n
ca
n
b
e
tr
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to
th
e
co
n
ce
p
tio
n
,
in
v
en
t
io
n
,
r
ep
r
esen
tatio
n
,
an
d
m
a
n
ip
u
lati
o
n
o
f
n
u
m
b
er
s
to
f
ac
i
litate
ac
cu
r
ate
r
en
d
er
in
g
o
f
m
ea
s
u
r
ab
l
e
o
b
j
ec
ts
.
T
h
is
h
as
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I
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2
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8938
IJ
-
AI
Vo
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5
,
No
.
4
,
Dec
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b
er
2
0
1
6
:
1
2
7
–
1
3
4
128
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izat
io
n
to
r
ep
r
esen
t
p
r
ac
ticall
y
a
n
y
n
u
m
b
er
[
9
]
.
I
n
a
n
ea
r
l
y
s
t
u
d
y
o
f
t
h
e
Yo
r
ù
b
á
n
u
m
er
al
s
y
s
te
m
,
[
1
0
]
s
h
o
w
s
h
o
w
lar
g
e
n
u
m
b
er
s
co
u
ld
b
e
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ted
as
a
n
ar
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etic
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m
b
i
n
atio
n
o
f
th
e
b
asic
n
u
m
b
er
u
n
it
s
a
n
d
r
ev
ea
ls
t
h
at
th
e
s
u
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tr
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o
p
er
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n
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y
s
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n
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m
p
o
r
tan
t
r
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le
in
its
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u
m
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er
n
a
m
in
g
.
T
h
e
p
ec
u
liar
it
y
i
n
th
e
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r
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b
á
n
u
m
er
als
w
as
t
h
at:
v
er
y
d
i
f
f
er
e
n
t
i
s
th
e
f
r
a
m
e
w
o
r
k
o
f
t
h
e
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ù
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á,
it
ca
n
b
o
ast
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g
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ter
n
u
m
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ad
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m
e
s
o
f
n
u
m
er
als
,
an
d
to
a
lar
g
e
ex
te
n
t
m
a
k
es
u
s
e
o
f
s
u
b
tr
ac
tio
n
[
1
0
]
.
A
r
g
u
ab
l
y
,
E
n
g
li
s
h
n
u
m
e
r
al
u
s
es
t
h
e
d
ec
i
m
al
(
b
ase
1
0
)
f
o
r
m
o
f
n
u
m
er
al
n
o
tat
io
n
w
h
i
le
Yo
r
ù
b
á
n
u
m
er
al
us
e
s
v
i
g
esi
m
al
(
b
ased
2
0
)
ex
ten
s
i
v
el
y
.
Fig
u
r
e
1
.
Hin
d
u
-
A
r
ab
ic
Nu
m
e
r
als (
s
o
u
r
ce
: (
[
1
5
]
)
)
Yo
r
ù
b
á
is
a
m
aj
o
r
eth
n
ic
g
r
o
u
p
in
Nig
er
ia.
P
r
esen
tl
y
,
th
e
y
a
r
e
f
o
u
n
d
i
n
lar
g
e
co
n
ce
n
tr
atio
n
alo
n
g
th
e
W
est
Af
r
ican
co
as
t
as
w
ell
as
o
th
er
m
aj
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r
cities
o
f
t
h
e
wo
r
ld
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h
eir
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m
u
n
itie
s
i
n
t
h
e
Dia
s
p
o
r
a
ca
n
b
e
f
o
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n
d
in
B
r
az
il,
th
e
C
ar
ib
b
ea
n
: T
r
in
id
ad
an
d
T
o
b
ag
o
,
J
am
aic
a,
E
u
r
o
p
e
an
d
th
e
Un
ited
State
s
o
f
Am
er
ica
[
1
1
]
.
T
h
e
r
est
o
f
t
h
e
p
ap
er
is
o
r
g
an
ized
as
f
o
llo
w
s
:
s
ec
t
io
n
2
e
x
a
m
i
n
es
r
elate
d
w
o
r
k
s
;
s
ec
tio
n
3
g
iv
e
s
t
h
e
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ese
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ch
m
et
h
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;
s
ec
tio
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4
d
is
cu
s
s
e
s
th
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lt
s
,
w
h
ile
s
ec
tio
n
5
co
n
cl
u
d
es th
e
p
ap
er
.
2.
RE
L
AT
E
D
WO
RK
S
I
n
[
1
2
]
,
Yo
r
ù
b
á
n
u
m
er
al
w
as
ex
p
lain
a
s
a
f
o
r
m
o
f
a
v
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g
esi
m
al
n
u
m
er
al
s
y
s
te
m
i.e
.
it
s
n
u
m
b
er
s
ar
e
m
o
s
tl
y
r
ep
r
ese
n
ted
in
b
ase
2
0
.
B
u
t
in
[
1
3
]
,
it
w
as
ar
g
u
ed
th
at
w
h
er
e
th
e
E
u
r
o
p
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ec
i
m
al
s
y
s
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m
is
b
ased
u
p
o
n
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its
o
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n
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th
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f
u
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m
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ltip
licatio
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as
it
is
u
s
ed
in
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n
g
li
s
h
n
u
m
er
al,
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r
ù
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á
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m
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s
b
ased
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p
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o
th
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y
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m
u
ltip
li
ca
tio
n
a
n
d
s
u
b
tr
ac
tio
n
.
[
1
4
]
F
u
r
th
er
ar
g
u
ed
th
at,
Yo
r
ù
b
á
n
u
m
er
al
s
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s
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m
i
s
n
o
t
f
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ll
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v
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g
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al,
an
d
th
at
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m
e
n
ts
o
f
d
ec
i
m
al
an
d
q
u
i
n
ar
y
(
b
ase
5
)
ar
e
also
u
s
ed
in
it
s
n
u
m
er
al
r
ep
r
esen
tatio
n
.
I
n
th
e
s
t
u
d
y
b
y
[
1
2
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n
v
ig
e
s
i
m
al
n
u
m
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d
er
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atio
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a
l
m
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p
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y
,
i
t
w
as
s
tated
t
h
at
n
u
m
er
als
ar
e
in
n
u
m
er
ab
l
y
i
n
f
i
n
ite
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th
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is
n
o
lo
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g
est
n
u
m
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d
th
at
n
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m
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als
ca
n
b
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cr
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ted
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esp
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with
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f
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er
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ated
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l
s
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,
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ltip
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e
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v
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b
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ted
o
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au
s
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f
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.
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th
e
n
class
if
ied
Yo
r
ù
b
á
n
u
m
er
als
i
n
to
s
ix
tee
n
(
1
6
)
b
asic
f
o
r
m
s
w
h
ic
h
ar
e:
On
e
(
1
)
to
ten
(
1
0
)
,
T
w
e
n
t
y
(
2
0
)
,
th
ir
t
y
(
3
0
)
,
tw
o
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n
d
r
ed
(
2
0
0
)
,
th
r
ee
h
u
n
d
r
ed
(
3
0
0
)
,
f
o
u
r
h
u
n
d
r
ed
(
4
0
0
)
an
d
tw
e
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t
y
t
h
o
u
s
a
n
d
(
2
0
0
0
0
)
.
I
t
w
a
s
f
u
r
th
er
id
e
n
ti
f
ied
t
h
at
s
o
m
e
v
ar
io
u
s
s
u
b
clas
s
es
o
f
n
u
m
er
als
ex
is
t
f
o
r
Yo
r
ù
b
á
n
u
m
b
er
s
a
n
d
t
h
e
y
ar
e
g
iv
e
n
as: th
e
n
o
u
n
f
o
r
m
(
t
h
o
s
e
w
h
ic
h
ca
n
o
p
er
ate
li
k
e
n
o
u
n
s
)
,
ca
r
d
in
al
f
o
r
m
(
e.
g
m
éta,
m
ér
in
,
et
c.
)
an
d
o
r
d
in
al
f
o
r
m
(
e.
g
.
èk
ín
í,
èk
ej
ì,
èk
eta
etc.
)
.
I
n
th
e
s
tu
d
y
co
n
d
u
cted
b
y
[
1
4
]
,
it
w
as
e
x
p
lai
n
th
a
t,
ap
ar
t
f
r
o
m
Yo
r
ù
b
á,
o
th
er
Af
r
ican
la
n
g
u
ag
e
s
w
it
h
v
ig
e
s
i
m
al
n
u
m
er
al
s
y
s
te
m
ar
e;
Ma
d
in
g
o
,
Mu
n
d
o
,
L
o
g
o
n
e,
N
u
p
e,
Ne
m
b
e,
B
in
g
o
,
E
f
i
k
,
Vie,
I
g
b
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A
f
a
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I
t
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m
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asize
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at
Yo
r
ù
b
á
co
u
n
ti
n
g
s
y
s
te
m
h
as
lex
e
m
es
f
o
r
b
asic
n
u
m
b
er
s
f
r
o
m
o
n
e
(
1
)
to
ten
(
1
0
)
an
d
s
ix
h
i
g
h
er
n
u
m
er
als
(
t
w
e
n
t
y
(
2
0
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,
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ir
t
y
(
3
0
)
,
t
w
o
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n
d
r
ed
(
2
0
0
)
,
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r
ee
h
u
n
d
r
ed
(
3
0
0
)
,
f
o
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r
h
u
n
d
r
ed
(
4
0
0
)
,
an
d
t
w
e
n
t
y
t
h
o
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s
an
d
(
2
0
0
0
0
)
.
I
t
w
as
also
s
tated
t
h
at
an
o
t
h
e
r
s
et
o
f
b
asic
n
u
m
er
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u
s
ed
i
n
Yo
r
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b
á
n
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m
er
al
ar
e
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kò
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(
2
0
)
,
ò
jì
(
4
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a
(
6
0
)
an
d
(
8
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as
th
e
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u
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in
m
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ltip
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f
1
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d
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2
0
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d
2
0
0
0
0
.
Ok
ò
ó
is
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s
ed
w
h
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n
ó
l
é
(
ad
d
itio
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o
r
ó
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í
n
(
s
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ac
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is
ad
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to
th
e
v
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m
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o
g
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n
an
d
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w
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th
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m
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g
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2
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.
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Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
-
AI
I
SS
N:
2252
-
8938
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s
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129
T
h
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s
tu
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b
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[
1
6
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,
s
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h
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4
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o
p
h
o
n
es,
t
h
er
ef
o
r
e,
w
h
ile
'n
'
g
o
es
w
i
th
n
a
s
alis
ed
v
o
w
el,
'l'
g
o
es
w
i
th
o
r
al
v
o
wels.
Su
c
h
th
at
i
n
d
ín
-
ní
-
o
g
ú
n
(
less
th
a
n
t
w
en
t
y
)
,
th
e
co
m
b
in
atio
n
o
f
ní
-
o
g
ú
n
b
ec
o
m
e
s
ló
g
ú
n
b
ec
au
s
e
th
e
n
a
s
al
v
o
w
el
'l'
af
ter
'n
'
is
d
elete
d
at
th
e
j
u
n
ctio
n
.
T
h
e
p
atter
n
f
o
r
t
h
e
g
e
n
er
atio
n
o
f
Yo
r
u
b
a
n
u
m
er
al
s
w
as
h
ig
h
l
ig
h
ted
in
[
1
7
]
as f
o
llo
w
s
: O
n
e
to
ten
ar
e
b
asic
w
o
r
d
s
a
n
d
elev
e
n
to
f
o
u
r
teen
is
e
x
p
r
ess
ed
a
s
1
+
1
0
,
2
+
1
0
,
3
+
1
0
an
d
4
+
1
0
r
esp
ec
tiv
el
y
.
Fi
f
tee
n
to
n
in
e
teen
ar
e
ex
p
r
es
s
ed
as
2
0
-
5
,
2
0
-
4
,
2
0
-
3
,
2
0
-
2
,
2
0
-
1
an
d
t
w
e
n
t
y
"
o
g
u
n
"
is
a
b
asic
w
o
r
d
.
T
w
en
t
y
-
o
n
e
to
t
w
e
n
t
y
-
f
o
u
r
ar
e
ex
p
r
es
s
ed
as 2
0
+
1
,
2
0
+
2
,
2
0
+
3
an
d
2
0
+
4
.
T
w
e
n
t
y
-
f
v
e
to
t
w
en
t
y
-
n
in
e
ar
e
ex
p
r
ess
ed
as
30
-
5
,
3
0
-
4
,
3
0
-
-
-
ab
o
v
e
is
f
o
llo
w
ed
f
o
r
th
ir
t
y
-
o
n
e
to
th
ir
t
y
-
f
o
u
r
a
n
d
th
ir
t
y
-
f
i
v
e
to
th
ir
t
y
-
n
i
n
e.
Fo
r
t
y
"
o
g
ó
j
ì"
f
r
o
m
o
g
ú
n
+
èj
ì
is
ex
p
r
ess
ed
as
2
0
x
2
.
T
h
e
p
at
ter
n
o
f
t
h
e
ad
d
itio
n
o
f
-
1
0
.
T
h
is
p
atter
n
is
f
o
l
lo
w
ed
f
o
r
àá
d
ó
ò
r
in
7
0
(
8
0
-
1
0
)
,
àá
d
ó
ò
r
ù
n
-
ú
n
9
0
(
1
0
0
-
1
0
)
,
àá
d
ó
ò
f
à
1
1
0
(
1
2
0
-
1
0
)
,
àá
d
ó
j
e
1
3
0
(
1
4
0
-
10)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Kan
)
[
1
2
]
.
A
ll
o
th
er
n
u
m
er
al
s
ap
ar
t
f
r
o
m
t
h
e
b
asic
n
u
m
er
al
s
ar
e
d
er
iv
ed
u
s
i
n
g
r
u
les.
T
h
ese
r
u
l
es
w
ill t
h
e
d
is
cu
s
s
ed
ex
te
n
s
iv
e
l
y
u
n
d
er
m
et
h
o
d
o
lo
g
y
.
3.
M
E
T
H
O
DO
L
O
G
Y
T
h
e
s
y
s
te
m
is
d
es
ig
n
ed
as
a
n
o
n
lin
e
w
eb
ap
p
licatio
n
f
o
r
ea
s
y
ac
ce
s
s
to
u
s
er
s
.
T
h
e
p
atter
n
f
o
r
th
e
Yo
r
ù
b
á
n
u
m
er
al
s
y
s
te
m
w
a
s
u
s
ed
to
d
ev
elo
p
th
e
r
e
-
w
r
i
tes r
u
le
f
o
r
th
e
T
r
an
s
latio
n
p
r
o
ce
s
s
.
T
h
e
r
e
-
w
r
ite
r
u
les ar
e
g
i
v
e
n
b
elo
w
:
S
NUM
NUM
NP
|
N
UM
SN
SN
VV
NP
NP
DI
GI
T
|
MP
|
VP
NP
NP
R
E
DUCE
MP
|
NP
M
MP
M
|
MP
NP
VP
DI
GI
T
V
T
h
e
p
r
o
d
u
ctio
n
r
u
le
s
g
iv
e
n
a
b
o
v
e
ca
p
tu
r
e
th
e
v
ar
io
u
s
s
e
g
m
en
ts
i
n
t
h
e
tr
an
s
latio
n
p
r
o
ce
s
s
.
I
n
t
h
e
p
r
o
d
u
ctio
n
r
u
le,
NP
,
r
ep
r
esen
t
n
o
u
n
p
h
r
ase
o
r
s
i
m
p
l
y
n
u
m
b
er
p
h
r
ase
,
MP
r
e
p
r
esen
ts
m
u
l
t
ip
licativ
e
b
ase,
an
d
NUM
is
t
h
e
g
e
n
er
ated
n
u
m
b
er
.
A
s
r
ep
o
r
ted
in
[
1
4
]
,
Yo
r
ù
b
á
n
u
m
er
al
s
y
s
te
m
is
v
er
y
m
eth
o
d
ical,
th
er
ef
o
r
e,
an
ef
f
icien
t
co
m
p
u
ta
tio
n
al
s
y
s
te
m
is
n
ec
es
s
ar
y
to
ac
cu
r
atel
y
d
er
iv
ed
it
n
u
m
er
al
r
ep
r
esen
ta
ti
o
n
.
I
t
is
as
s
ee
n
a
s
a
n
ea
r
d
if
f
ic
u
lt
f
o
r
m
o
f
n
u
m
er
al
s
y
s
te
m
.
T
h
er
e
ar
e
f
iv
e
i
m
p
o
r
tan
t
p
r
o
ce
s
s
es
in
t
h
e
m
o
d
el.
First,
is
th
e
p
ar
t
th
at
co
n
v
er
ts
t
h
e
tex
t
u
al
n
u
m
b
er
to
n
u
m
b
er
s
i
n
f
i
g
u
r
e
(
th
e
alg
o
r
ith
m
u
s
ed
f
o
r
th
i
s
is
s
h
o
w
n
i
n
Fig
u
r
e
2
)
,
s
ec
o
n
d
s
tag
e
is
t
h
e
n
u
m
b
er
d
ec
o
m
p
o
s
itio
n
p
r
o
ce
s
s
,
w
h
er
e
t
h
e
n
u
m
b
er
s
ar
e
ex
p
r
ess
ed
as
a
s
u
m
o
f
s
m
al
ler
n
u
m
b
er
s
in
h
ar
m
o
n
y
w
i
th
t
h
eir
s
u
b
-
g
r
o
u
p
in
g
.
T
h
e
o
u
tp
u
t
o
f
t
h
is
p
r
o
ce
s
s
is
t
h
e
m
a
g
n
it
u
d
e
s
tac
k
.
Nex
t
,
th
er
e
is
a
p
r
o
ce
s
s
th
at
g
e
n
er
ates
t
h
e
p
o
s
s
ib
le
f
o
r
m
s
o
f
a
s
i
n
g
le
n
u
m
b
er
.
T
h
is
is
d
o
n
e
b
y
ca
r
e
f
u
l
co
m
b
in
at
io
n
s
o
f
t
h
e
n
eig
h
b
o
u
r
i
n
g
ele
m
en
t
s
o
f
t
h
e
m
a
g
n
itu
d
e
s
tac
k
a
n
d
p
ar
s
in
g
th
e
m
w
i
th
t
h
e
d
esi
g
n
ed
n
u
m
er
al
g
r
a
m
m
ar
.
T
h
e
th
ir
d
p
r
o
ce
s
s
is
w
h
er
e
to
k
e
n
s
o
f
t
h
e
n
u
m
b
er
f
o
r
m
ar
e
co
n
v
er
ted
to
th
eir
eq
u
iv
ale
n
t
lex
ical
f
o
r
m
s
,
an
d
th
e
f
i
n
al
p
r
o
ce
s
s
is
w
h
er
e
th
e
m
o
r
p
h
o
p
h
o
n
o
lo
g
ical
r
u
les e
m
p
lo
y
ed
in
n
u
m
b
er
s
n
a
m
i
n
g
.
T
h
e
g
en
er
ated
p
r
o
d
u
ctio
n
r
u
l
e
w
as
s
i
m
u
lated
u
s
in
g
Au
to
m
ata.
I
ts
b
eh
av
io
u
r
w
a
s
ex
p
r
ess
ed
as
an
ab
s
tr
ac
t
m
ac
h
in
e.
J
F
L
A
P
,
a
J
a
v
a
-
b
a
s
ed
Au
to
m
ata
s
i
m
u
latio
n
to
o
l
w
as
u
s
ed
.
T
h
e
C
FG
f
o
r
th
e
Yo
r
ù
b
á
n
u
m
er
a
l
tr
a
n
s
latio
n
s
y
s
te
m
i
s
as
s
h
o
w
n
in
Fi
g
u
r
e
3
.
T
h
e
FS
A
h
as
s
ix
s
tates
(
q
0
to
q
5
)
.
T
h
e
in
itial
s
t
ate
is
q
0
w
h
ile
t
h
e
f
i
n
al
s
tate
w
h
er
e
th
e
s
ta
te
tr
an
s
itio
n
s
ar
e
ac
ce
p
ted
o
r
r
e
j
ec
te
d
is
q
3
.
I
n
Fig
u
r
e
4
,
a
r
u
le
is
s
i
m
u
lated
to
s
h
o
w
i
f
th
e
au
to
m
ato
n
ac
ce
p
t t
h
e
in
p
u
t
o
r
n
o
t.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8938
IJ
-
AI
Vo
l.
5
,
No
.
4
,
Dec
em
b
er
2
0
1
6
:
1
2
7
–
1
3
4
130
Fig
u
r
e
2
.
A
l
g
o
r
ith
m
to
C
o
n
v
er
t T
ex
tu
al
Nu
m
b
er
to
Nu
m
b
er
Fig
u
r
e
3
.
R
e
-
W
r
ite
R
u
le
Au
to
m
ato
n
Fig
u
r
e
4
.
R
e
-
w
r
ite
R
u
le
A
u
to
m
ato
n
S
i
m
u
lat
io
n
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
-
AI
I
SS
N:
2252
-
8938
W
eb
-
B
a
s
ed
Yo
r
ù
b
á
N
u
mera
l Tr
a
n
s
la
tio
n
S
ystem
(
A
g
b
ey
a
n
g
i A
)
131
Fig
u
r
e
5
also
s
h
o
w
th
e
p
r
o
d
u
ctio
n
r
u
le
s
i
m
u
latio
n
w
it
h
m
u
l
tip
le
in
p
u
ts
w
h
er
e
ac
ce
p
ted
a
n
d
r
ej
ec
te
d
in
p
u
t c
o
m
b
i
n
atio
n
ar
e
ex
p
r
ess
ed
.
Fig
u
r
e
5
.
R
e
-
w
r
ite
R
u
le
A
u
to
m
ato
n
S
i
m
u
lat
io
n
w
it
h
A
cc
ep
t
ed
an
d
R
ej
ec
ted
I
n
p
u
ts
.
3
.
1
.
Da
t
a
Co
llect
io
n
T
h
e
v
ar
io
u
s
d
ata
u
s
ed
f
o
r
th
e
r
esear
ch
ar
e
as f
o
llo
w
:
1.
T
h
e
s
et
o
f
le
x
e
m
es
al
s
o
k
n
o
w
n
as
th
e
b
asic
n
u
m
b
er
i
n
Y
o
r
ù
b
á
n
u
m
er
al
s
y
s
te
m
,
w
h
ich
ar
e:
DI
GI
T
=
{o
k
an
(
1
)
,
ej
i
(
2
)
,
eta
(
3
)
,
er
in
(
4
)
,
ar
u
n
-
u
n
(
5
)
,
ef
a
(
6
)
,
ej
e
(
7
)
,
ejo
(
8
)
,
esan
-
a
n
(
9
)
,
e
w
a
(
1
0
)
,
o
g
b
o
n
(
3
0
)
,
o
d
u
n
r
u
n
(
3
0
0
)
,
ir
in
w
o
(
4
0
0
)
}
2.
T
h
e
s
ets o
f
m
u
ltip
lica
tiv
e
b
ase
; M
=
{o
g
u
n
(
2
0
)
,
ig
b
a
(
2
0
0
)
,
o
k
e
(
2
0
0
0
0
)
}
3.
T
h
e
s
e
t o
f
lex
ical
a
f
f
ix
e
s
s
h
o
win
g
ar
it
h
m
etic
o
p
er
atio
n
; V
=
{le
n
i (
+)
,
d
in
n
i (
-
)}
4.
T
h
e
s
et
o
f
o
p
er
ato
r
b
et
w
ee
n
p
h
r
ases
,
VV
=
{o
le
(
++
)
,
o
d
in
(
--
)}
5.
T
h
e
s
et
o
f
i
m
p
lied
s
u
b
tr
ac
tio
n
o
p
er
ato
r
s
;
R
E
DU
C
E
=
{a
a
d
in
(
r
ed
u
ctio
n
b
y
1
0
)
,
ee
d
in
(
r
ed
u
ctio
n
b
y
5
,
1
0
0
an
d
1
0
0
0
)
.
E
v
er
y
o
th
er
e
n
tr
y
i
n
th
e
d
atab
ase
(
s
tack
)
is
g
e
n
er
ated
b
y
t
h
e
r
u
les.
3
.
2
.
Dev
elo
p
m
e
nt
T
o
o
ls
T
h
e
s
y
s
te
m
i
m
p
le
m
e
n
tatio
n
was d
o
n
e
u
s
i
n
g
th
e
f
o
llo
w
i
n
g
s
o
f
t
w
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2
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8938
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4
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4.
RE
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AND
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Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
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2252
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W
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-
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133
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o
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b
á
lan
g
u
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is
b
ec
o
m
in
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s
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its
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ith
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o
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t
to
an
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e
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n
ter
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t
ed
in
Yo
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ù
b
á
n
u
m
er
al
s
y
s
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m
.
Ot
h
er
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ea
f
o
r
f
u
r
t
h
er
r
esear
ch
i
n
clu
d
es
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e
d
ev
elo
p
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en
t
o
f
tr
an
s
latio
n
m
o
d
el
f
o
r
Yo
r
u
b
a
d
ates,
o
r
d
in
al
n
u
m
b
er
s
a
n
d
th
e
tr
a
n
s
latio
n
f
r
o
m
Yo
r
u
b
a
to
E
n
g
l
is
h
n
u
m
er
al.
RE
F
E
R
E
NC
E
S
[
1
]
Hu
r
f
o
r
d
,
J
R.
T
h
e
lin
g
u
is
tic
t
h
e
o
r
y
o
f
n
u
m
er
al
s
,
v
o
l
u
m
e
1
6
.
C
a
m
b
r
id
g
e
Un
iv
er
s
it
y
P
r
ess
.
1975.
[
2
]
R
ip
s
,
L
J
,
B
lo
o
m
f
ield
,
A
,
As
m
u
th
,
J
.
Fro
m
n
u
m
er
ical
co
n
ce
p
ts
to
co
n
ce
p
ts
o
f
n
u
m
b
er
.
B
eh
av
io
r
al
an
d
B
r
ain
Scien
ce
s
,
2
0
0
8
; 3
1
(
6
)
: 6
2
3
-
642.
[
3
]
C
h
o
w
d
h
u
r
y
,
K,
B
ais
h
y
a,
A
A
C
o
m
p
ar
ati
v
e
St
u
d
y
o
f
E
ar
l
y
E
g
y
p
tian
,
B
ab
y
lo
n
ian
a
n
d
Ma
y
an
N
u
m
b
er
S
y
s
te
m
.
I
n
tern
a
tio
n
a
l J
o
u
r
n
a
l
o
f Ma
th
ema
tica
l A
r
ch
ive
(
I
JMA
)
.
2011.
[
4
]
I
f
r
ah
,
G.
T
h
e
u
n
i
v
er
s
al
h
is
to
r
y
o
f
n
u
m
b
er
s
:
Fro
m
p
r
eh
i
s
to
r
y
to
t
h
e
i
n
v
en
tio
n
o
f
t
h
e
co
m
p
u
ter
(
D.
B
ello
s
,
I
.
Mo
n
k
,
E
F Ha
r
d
in
g
,
&
S.
W
o
o
d
,
T
r
an
s
.
)
.
1
9
9
9
.
[
5
]
B
ab
ar
in
d
e,
O
.
T
h
e
L
i
n
g
u
i
s
tic
An
al
y
s
i
s
o
f
t
h
e
Str
u
ct
u
r
e
o
f
t
h
e
Yo
r
u
b
a
N
u
m
er
als.
Jo
u
r
n
a
l
o
f
Lit
e
r
a
tu
r
e,
La
n
g
u
a
g
es a
n
d
Lin
g
u
is
tics
,
2
0
1
3
;
1
:7
8
-
88.
[
6
]
O
m
ac
h
o
n
u
,
GS.
Der
i
v
atio
n
a
l
P
r
o
ce
s
s
es
in
I
g
a
la
N
u
m
er
al
S
y
s
te
m
.
Jo
u
r
n
a
l
o
f
u
n
ivers
a
l
la
n
g
u
a
g
e
.
2
0
1
1
;
1
2
(
2
)
:
8
1
-
101.
[
7
]
Go
y
v
ae
r
t
s
,
D.
C
o
u
n
ti
n
g
i
n
L
o
g
o
.
A
n
th
r
o
p
o
lo
g
ica
l Lin
g
u
is
tics
,
1
9
8
0
:
317
-
3
2
8
.
[
8
]
Z
asla
v
s
k
y
,
C
.
Af
r
ica
C
o
u
n
t
s
a
n
d
eth
n
o
m
at
h
e
m
atic
s
.
Fo
r
th
e
L
ea
r
n
i
n
g
o
f
Ma
t
h
e
m
atics,
1994:
3
-
8.
[
9
]
B
aile
y
,
DH,
B
o
r
w
ei
n
,
J
M.
H
ig
h
-
p
r
ec
is
io
n
n
u
m
er
ical
i
n
te
g
r
atio
n
:
P
r
o
g
r
ess
a
n
d
ch
alle
n
g
es.
Jo
u
r
n
a
l
o
f
S
ymb
o
lic
C
o
mp
u
ta
tio
n
.
2
0
1
1
;
4
6
(
7
)
:
741
-
7
5
4
.
[
1
0
]
Ma
n
n
,
A
.
No
tes
o
n
t
h
e
N
u
m
e
r
al
S
y
s
te
m
o
f
t
h
e
Yo
r
u
b
a
Nati
o
n
.
Jo
u
r
n
a
l
o
f
th
e
A
n
th
r
o
p
o
l
o
g
ica
l
I
n
s
titu
te
o
f G
r
ea
t B
r
ita
in
a
n
d
I
r
ela
n
d
.
1887
:
59
-
64.
[
1
1
]
A
ta
n
d
a,
J
A
.
T
h
e
Yo
r
ù
b
á
p
eo
p
le:
th
eir
o
r
ig
i
n
,
cu
l
tu
r
e
an
d
ci
v
ilizatio
n
.
T
h
e
Yo
r
ù
b
á:
Hi
s
to
r
y
,
C
u
lt
u
r
e
an
d
L
a
n
g
u
a
g
e.
1
9
9
6
:
3
-
3
4
.
[
1
2
]
E
k
u
n
d
a
y
o
,
S.
V
ig
e
s
i
m
al
n
u
m
er
al
d
er
iv
a
tio
n
al
m
o
r
p
h
o
lo
g
y
:
Yo
r
u
b
a
g
r
a
m
m
atic
al
co
m
p
eten
c
e
ep
ito
m
ized
.
An
t
h
r
o
p
o
lo
g
ical
L
i
n
g
u
i
s
tics
.
1
9
7
7
:
436
-
4
5
3
.
[
1
3
]
Hall,
G.
Yo
r
u
b
a
Nu
m
b
er
s
'
.
I
n
Fo
lk
lo
r
e
Fo
r
u
m
,
1
9
7
3
:
1.
[
1
4
]
Ak
i
n
ad
e,
OO,
Od
ej
o
b
i,
OA.
C
o
m
p
u
tatio
n
al
m
o
d
ellin
g
o
f
Yo
r
u
b
a
n
u
m
er
al
s
i
n
a
n
u
m
b
er
-
to
-
te
x
t
co
n
v
er
s
io
n
s
y
s
te
m
.
Jo
u
r
n
a
l o
f La
n
g
u
a
g
e
Mo
d
ellin
g
.
2
0
1
4
;
2
(
1
)
:1
6
7
-
211.
[
1
5
]
S
m
i
t
h
,
DE
,
Kar
p
in
s
k
i,
L
C
.
T
h
e
Hin
d
u
-
A
r
ab
ic
Nu
m
er
al
s
.
C
o
u
r
ier
C
o
r
p
o
r
atio
n
.
2
0
1
3
.
[
1
6
]
B
ab
ar
in
d
e,
O
.
T
h
e
L
i
n
g
u
i
s
tic
An
al
y
s
i
s
o
f
t
h
e
Str
u
ct
u
r
e
o
f
t
h
e
Yo
r
u
b
a
N
u
m
er
als.
Jo
u
r
n
a
l
o
f
Lit
e
r
a
tu
r
e,
La
n
g
u
a
g
es a
n
d
Li
n
g
u
is
tics
.
2
0
1
3
;
1
:7
8
-
88.
[
1
7
]
O
y
etad
e,
B
A
.
Jo
u
r
n
a
l o
f Ni
g
eria
n
La
n
g
u
a
g
e
s
a
n
d
Lit
era
tu
r
es
.
1996;
2
:
1
7
-
46.
[
1
8
]
A
ta
n
d
a,
J
A
.
A
n
I
n
tr
o
d
u
ctio
n
t
o
Yo
r
ù
b
á
His
to
r
y
.
I
b
ad
an
Un
i
v
er
s
it
y
P
r
ess
.
1980
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8938
IJ
-
AI
Vo
l.
5
,
No
.
4
,
Dec
em
b
er
2
0
1
6
:
1
2
7
–
1
3
4
134
[
1
9
]
Aw
o
b
u
lu
y
i,
O
.
T
o
w
ar
d
s
a
t
y
p
o
lo
g
y
o
f
co
ale
s
ce
n
ce
.
Jo
u
r
n
a
l
o
f
W
est
A
f
r
ica
n
La
n
g
u
a
g
e
s
.
1
9
8
7
;
1
7
(
2
)
:
5
-
22.
[
2
0
]
B
ir
d
,
S
.
N
LT
K
:
th
e
N
a
tu
r
a
l
La
n
g
u
a
g
e
T
o
o
lkit.
I
n
P
r
o
ce
ed
in
g
s
o
f
t
h
e
C
O
L
I
NG/
AC
L
o
n
I
n
ter
ac
ti
v
e
p
r
esen
tatio
n
s
ess
io
n
s
.
2
0
0
6
:
69
-
7
2
.
A
s
s
o
ciat
io
n
f
o
r
C
o
m
p
u
t
atio
n
al
L
in
g
u
is
t
ics.
[
2
1
]
B
ir
d
,
S,
Klein
,
E
,
L
o
p
er
,
E
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