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// the new type rational
struct rational {
// Program : userational. C
// Add two rational numbers .
# include < iostream >
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rational r ;
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// computation and output
std :: cout < < " Sum is " < < r + s < < " .\ n" ;
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// POST : return value is the sum of a and b
rational add ( rational a , rational b)
{
rational result ;
result . n = a.n * b .d + a.d * b.n;
result . d = a.d * b .d;
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n/d
"
3!
"!
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EG>[
8
J
9 ; D >? = 06
/
>6
9.
(=
. ==
9;
(=
. ==
9;
.
9;
8
> W M7 .
8 D 8
< 9
.I 8 <
/8
0I
9
=
9
>
08
/9
/8
C8
C
.
.
9.
8
6
-
8
09
8
6
-
;
U .
>=
.=
J
<
9
( ;=
=
.=
>=
I
=
I
I
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8
>[
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0
6
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.
.
J
9;
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6
8
01
t
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8
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/6
<
<
M
=
8
I
0
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6
/?9
?
C
0
8
>8
2/3 = 4/6
rational
(n, d)
9
E
.
J
H
<
H
6
-
1
J
J
=
.=
M.
7
9
B .; E 8
8
8
.
>9
C8
8
.
.
B.
>8
-=
-.
8
B9 .
9.
>6
(=
J=
9;
-
6
.
M;
.=
9;
=
D
>[
.
/8
8
>? . =
BJ
6
Q
<
.
E Z
>8
>6
7
1
C6
-
M9
/
/9
>? =
I
I
=
I
B X
.=
J
.
J
8
8
1
>6
<
Z
8
/6
/8
add (r, s)
8
9.
J
J
'
C6
1
>8
B.
0
-
=
1
>9
BJ
6
=
;;=
BJ
6
B8
>9
B<
extended_int
-.
8
.
/J
0
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> ;=
8
9.
D >6
/8
76
8
I
P
<
J
9\
.
A
>9
>=
;
=
6
M9
B9
8
:6
/
7
8
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8
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-
1.
/0 .
H
.
8
a
t = (t1 , . . . , tN)
K = 1, . . . , N
;\=
H
.
'
BJ
6
>=
C9
6
I
B
/=
>8
08
J
>9
-
8
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1.
]8
C9
7
B
J
>=
.
6
9
/0 .
/
M9
8
/9
.
>? =
7
79
/
/8
E -
C8
/9
/
M9
C
==
B8
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C
=
t := r + s
-
8Z
>0
9;
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9-
<
/<
9.
1
6
=
/
/?
=
.
0
/
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9.
/8
8
76
6
1.
8
7-
9
8
1
-
-
8
6
int int
0
9
.
\;=
<
;
J
I
>
M.
7
8
/-
8
/6
C8
tK = t K
rational
6
8
6
.
H;
;
B
M
E
B
.
MH
>8
C
>
=
What about other operations?
> =X
0
.
8
8
>6
H
.
>9
9
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int
?> . =
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7
C
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76
8
9
8
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.
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6
/9
7
0
0=
)
B8
B
8
6
C
J
/8 =
.
Q
?.
>6
1
9
1
n/d
C- =
I
point q = p
B
J
8
6
P\
7
}
(0,
:8
/8
.
-
/9
B
>
M
H,
// 1
// 2
D
I
8
6
<
;
8;
1
/6
C8
C
M.
7
8
-
01
6
1.
8
int main ()
{
point p;
p . coord [0] = 1;
p . coord [1] = 2;
BJ
6
I
.=
8;
-
>9
M
M
>9
1.
/0- .
6
I
struct point {
double coord [2];
};
[
<
>6 = 8
9
.> D M ;;
0
6
>8
.
9.
C8
-
a
b
add (r, s)
return
-
.
BJ
6
C
6
8
1.
H
# include < iostream >
MH
/8
9.
I
0
M.
7
8
-Z
/0- .
9
6
-
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C8
E
t=t
-
I
J
9
7;
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C9
.
Z E
/
C
\
>- =
1
8
.
J
8
<
9.
9
-
/6
B-
8
rational t = add (r , s );
t
D /8
;
J
C
/6
8
=
C8
C
C
rational
H
6
8
09
8
6
9.
9
( ;=
.=
=
>8
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J=
4.1.4 Initialization and assignment
B
8
.
:
.
s
J
^
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J
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8
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H
I
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O
t.n
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6
Z
6
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8
09
8
.
B/8
C8
CX
M9
<
.
. D M/6
J E 8
>= /
>9
. == C6
<
9;
(=
. >=
D J
9.
= 1
8
= >6
. > =X
J
8
6
1
I 1
>6
C6
/.
>= 7
9
;= - .
/8
.
6
1
>9
/9
/
?
0 W M9
/6
8
>9 =
C
point q = p;
std :: cout < < q. coord [0] < < " "
< < q. coord [1] < < "\n" ;
8
B1 .
M
/9
/6
.
X
:
=
8
/9
8
.
>[
q
9
6
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9.
9
( ;=
.=
1
76
add (r, s)
t
/
C
=
.=
>=
>9
1
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I
t
7
E <
8
.
add
1
=
.;
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C6
-
-
08
>
0
add
-
I
/6
-
9;
add (r, s)
]8
08
J
.=
.
(n, d)
!=
)
return 0;
4.1.5 User-defined operators
rational t = add (r , s );
=
'
Y
'
"
<
J
J
I
.
=
1
1
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>
0
.
.=
I
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/8
1
9
>=
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8
J
J=
9 ; E 9 ;;
;
1
8
.
M.
7
8
6
.
<
<
C
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8
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1
8
0
6
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/
8
.
J
1.
-
J
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;
7
.
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.
J
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H
B 9-
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9.
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76
9;
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8;
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8
8
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C
9
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9.
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1
6
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9
8
8
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a
E 8
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7
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76
9;
=
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.
>=
J
4.1.6 Details
8
E ;;
.
I
=
I
I
// POST : return value is true if and only if a == b
bool operator == ( rational a , rational b)
{
return a .n * b.d == a.d * b .n;
}
C9
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8
6
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9
0;
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8
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08
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1
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6
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8
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8.
9;
7;
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1
8
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6
.
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0
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>9
1
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7
J
1
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6
78
C9
8
09
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:
08
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9-
.
.
;
J
.
J
8
6
.
.
J
8
.
J
<
<
;
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J
I
E
;
M
.
J
P;
.
J
Q
/8
8.
7;
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C9
9;
1
8
6
76
1
I
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W
<=
9;
1
6
8
6
8
CX
8
-
08
.
9.
D >6 =
B
>8
C8
C=
7
:
M
>8
B;
6
>? =
.
.
J
J
-
9
.
.
J
H
=
.
>[
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8
C9
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76
>=
-.
8
9
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6
BI
.
8
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M;
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0
D >6 =
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> H=
8
9-
8
C9
C9
-
8
I
6;
;
%
I
]8
C9
7
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(
"
&
!
#
'
6
8
9-
1
1=
9;
8
9
7-
M;
7
C- =
=
>? <=
9
6
:6
/8
/6
9.
/8
7
B;
6
>? = D ;
8
6;
.
1;
8
6
01
8
8
9;
<
J
9=
;
9
D H
;
B8
:9
1
;
>9
8
I
-
1.
/0 .
8
.
J
:9
>? =
>9
/6
J
1;
8
C9
>9
.=
0J
6
.
/1
8
9
>? . =
S 9-
M
I
.=
.
>=
>6
>
. DC9
J B
8
1
/
D C9
/?
78
.
>=
>6
06
1
M
/9
>
08
9
6
:6
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6
-
.
1
>
9.
6
.
9.
/6
/8
76
I
.=
9;
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>
0
>=
/6
9-
.
J
=
D >6 =
^
9.
/8
76
J
<
;
9;
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8N
M.
7
8
E -
1
7
>
/
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8
1
>8
/<
08
.
8
8
1
/
>9
/8
<
9
9. D .
/6
:8
B
/8
-= 8
>6
1
>9
>
6
I 6
.
B .=
1
=. >?9 J
J
/9
8
.
H=
/. 8J
/9
M
76 E
I /8
/6
9.
C
/6
9
)
7; /9
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C9 D M . =
8.
76
>9
;
6
9
>? <=
/8
H
1
<
J
=
>6
]8
1
.
.
.
>9 =
/8
1
9
8
<
>=
8
8
>-
8
>6
6
7
;=
J
;
6
8
.
;
J
9;
1
8
.
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0
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1
8
.
8.
M;
7
<
<
;
C- =
>9
/
B
6
D >
0
>6
-
9
6
:6
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M.
7
B 8J=
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8
:8
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C
.
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8
C9
>[
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7
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8
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8
B.
0
D =
M.
6
1=
9
: .=
-
-
9
8
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7
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E
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8
.
9
>9
1
8
B9 .
B
9.
>-
N
<
<
J
J
<
;
9;
M.
7
D
8
-
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8
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=
I
I
9=
;
9
H;
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D
-
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.
=
J=
M
B
8
.
.
9.
6
J
J
/6
9.
/8
76
-
=
> H=
/9
>
\
M<
9
8
/9 ;
E
.
J
Q
<
;
P
>8
.
6\
8
9
6
:6
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S8
8
8
B
;
0=
6
76
/8
9.
S/6
B-
/9
:.
>8
C8
>
B
;
6.
>? J=
>
8
>=
71 =
/
7
>[
7
C=
.
J
\
.
1
8
-
.
6
>8
.
.
I
J
J
J=
6\
/6
9.
/8
76
8
C9
8
8
-
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0
B9 .
-
8
>9
C-
O
8
E ;;
M.
7
8
B 9-
/8
6.
/6
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B6
.
9.
8
9
8
8
-
-
/6
/8
-
6
-.
6
76
C
.
9.
O
H;
J
=
I
7
J
"
I
H
;=
;;
J
&
&
!
#
8
1
>9
'
.
-.
08
/6
I
] <=
>8
7`
8
.
>=
X
Z
E -
>6
=
0
?
-
8
9
>?9
8
6
8
L
<
.
KL
=
;
J
=
I
8I
.
09
M8
/
<
J=
H
<
<
9\
S8
:8
1
9
8
8
8
9.
/9
J
>=
>9
1
.
9.
6
1;
8
.
O
9;
>6
>
/
09
>8
.
O
H
J
J
1
=
=
8
C- =
/9
-
>6
J
<
:=
8
.
J
/8 =
M? ;
8
>6
B8
/8
>9
J
<
=
9.
6.
=
=
]W
>
6.
9.
.
J
8
>=
8
-
0<
>9
>6
>
. = E >6
9.
1
/8
76
8
.=
CJ
/9
9-
/J
8
8
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6.
8
8
>9
D ;=
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E
C8
B9 .
8
B9
8.
>9
?6 .
/9
7
/8
>
6
01
.
.
Q
9;
=
>
9.
6.
9;
>6
J
=
>6 =
.
=
.
6
:9
6=
0
1.
78
.
.
.
/6
9.
/8
76
9.
>9
1.
9
8
-
8
9
J
J
I
(- =
J
E
>
1
0
J
1
I
.=
9;
>6
>
9;
;
>=
1
8
O
H
1
I
.=
9;
>6
>
0
&
>=
" !!
8
6
-
9;
>9
6 . D /8 a
9. 8
7
=
=
.
>6
M ; >6
J
^
8
1
9
>= 1
>9
C6
7C
8
0 H >16
1= 8
/ . >8
9; 0
08 D 9
I
I /.
>
. 8
;
1. / I
= /8 I
.
= 6
>6
<= 79
^
.
7
6
>= 9<
.
8
1
J
. = /9
>9
8
1
I
07
>= 9
79
1.
/
7
7
.
;=
4
J
1 D 69.
9. 8
*
D >6 =
>9
>- = " 76
B J 18
6
= /8
. 9.
:8
C8
D
/8
> = /6
7
=
C- =
/
.
<
>P
09
.
-
>9
r + s
8
>6
-
>8
78
>6
M;
>9
!
J
0W
.
;=
J
J
I
E
8
C8
B-
J
I
B
-
#
&
( ?
6
9!
#
.
Overloading resolution.
'
/8
8
7
C6
1
8
<
;
=
D C9
9;
1
-
-
-
8
1
6
.
rational
/
/6
/6
// POST : returns a * a
extended_int square ( extended_int a );
/
add
9.
.=
<=
<
0- =
9;
/8
>8
// POST : returns a * a
rational square ( rational a );
a
7-
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
/8
76
>6
9
.
/8
76
<
<
;
I
09
J D . D 8I
8
/6
;
76
B
0
H
/8
8
.
9.
>
J
/6
:9
6.
= 8
>
. 8
6
06
/9
.H
M;
>9 = 8
9
8
. 0<
J
MH 8
0-8
I
%
79
;
7
9
>8
76
>? <=
" ! /8
9.
= . /6
$
. #
:6
6\ "
/8
>8
# U
6
8
8
9
6
1.
>6
>
.
.
J
I
extended_int
+
6
D
J
J
>? =
>? <=
;
06
B8
6
1
>
0
double
8
M.
7
8
M
.
> H=
/9
8
C9
.
:6
/8
9
6
:6
/8
.=
J=
int
/?
=
I
I
>8
<
1
>6
>
0
B
rational t = p * q - r * s;
7
J
6
8
9-
1
O
>W
8
>9
8
6
>? <=
;
rational t = subtract ( multiply (p , q ) , multiply (r , s ));
J=
1.
.
J=
E
9
6
??
>6
>
08
-
8
;
6.
>-
.
>[
E "
9.
D =
> [ >6
operator
1
E
]W
J
rational
9
I
8
.
.
square
/<
78
B;
6
>9
6;
-
9
-
/
78
6
1
>6
>
08
8
-
8
8
I
.
.
.
H
H;
H
=
?
O
H
J
.
=
9.
8
<=
M8 ;
C
H
8
C=
J
J=
I
.W
>.
6.
C-
6
8
>8
8
6
I
>? =
C9
6
:H
0=
6
1 D
>6
-= >=
< 7
/8
/9
.
J 0 .=
1
8
]8
;
/9
7
/
>- =
8
-= 1
8
>6
.
J
>8
^8
0
9
>6
9-
/8
0
6
-
C8
J
.=
.
add
O
E
;
J=
6\
6
-
.
6
1
6
'
;
> ;=
8
8
1
9
78
/
MH
B
>
>9
8
D '
B
C9
/
?/
>[
rational t = r + s ;
rational t = r + s ; // equivalent to rational t = operator + ( r , s );
// POST : return value is the difference of a and b
rational operator - ( rational a , rational b );
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Exercise 95
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extended_int
&
// POST : return value is the difference of a and b
Z_7 operator - ( Z_7 a , Z_7 b );
4
// POST : return value is true if and only if a >= b
bool operator >= ( rational a , rational b );
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bool operator <= ( rational a , rational b );
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-
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foo(1.0f, 1.0)
8
B
foo(1, 1u)
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foo(1.0, 1)
08
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void foo ( double , double )
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void foo ( unsigned int , int )
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.
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Exercise 98
4.2 Type Variants
foo(1u, 1.0f)
4.2.1 Reference types
rational
rational operator += ( rational a , rational b ) {
a .n = a. n * b.d + a. d * b.n ;
a .d *= b .d;
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}
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int main ()
{
int j = 5;
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increment
int & increment ( int & i )
{
return ++ i;
}
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// POST : b has been added to a ; return value is the new value of a
rational & operator += ( rational & a , rational b)
{
a .n = a. n * b.d + a. d * b.n ;
a .d *= b .d;
return a ;
}
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int i = 3;
int & j = foo (i );
// j refers to expired object
std :: cout < < j < < "\n" ; // undefined behavior
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Rational numbers: addition assignment.
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Reference Guideline:
7
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Rational numbers: input and output.
add
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r + s
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int & foo ( int i)
{
return i ;
}
std :: cout < < " Sum is " < < t.n < < "/" < < t.d < < "\n" ;
std :: cout < < " Sum is " < < t < < "\ n";
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational r)
{
return o < < r.n < < " /" < < r .d;
}
std :: cin > > r;
// POST : r has been read from i
// PRE : i starts with a rational number of the form " n /d"
std :: istream & operator > > ( std :: istream & i , rational & r)
{
char c ; // separating character , e .g . ’/ ’
return i > > r.n > > c > > r.d ;
}
1/2
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2
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// POST : return value is the sum of a and b
rational operator + ( rational a , rational b );
b
b
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational a );
b
b
c
c
c
c
// POST : a has been read from i
// PRE : i starts with a rational number of the form " n /d"
std :: istream & operator > > ( std :: istream & i , rational & a );
d
c
f
e
d
c
b
// Program : userational2.C
// Add two rational numbers .
# include < iostream >
# include < IFM / rational .h >
c
+ "!
"
#
(
namespace ifm {
h
h
f
e
d
$#
e
c
b
b
c
d
e
// Program : rational .C
// Define operations on the type rational
# include < IFM / rational .h >
b
f
b
b
std :: cout < < " Rational number s :\ n";
ifm :: rational s ;
std :: cin > > s;
b
}
Program 32:
b
b
b
h
int main ()
{
// input
std :: cout < < " Rational number r :\ n";
ifm :: rational r ;
std :: cin > > r;
b
b
b
// computation and output
std :: cout < < " Sum is " < < r + s < < " .\ n" ;
b
c
return 0;
b
b
c
d
e
f
c
b
b
h
b
namespace ifm {
b
// Program : rational .h
// Define a type for rational numbers , and declare
// operations on it .
# include < iostream >
b
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational a)
{
return o < < a .n < < " /" < < a .d;
}
c
c
h
c
f
e
d
c
b
b
b
b
+ "!
"
#
(
Program 31:
$#
b
b
c
}
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
rational result ;
result .n = a. n * b.d + a .d * b. n;
result .d = a. d * b.d ;
return result ;
}
d
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operator+
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c
// the new type rational
// POST : a has been read from i
// PRE : i starts with a rational number of the form " n /d"
std :: istream & operator > > ( std :: istream & i , rational & a)
{
char c ; // separating character , e. g . ’/ ’
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int d ; // INV : d != 0
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a, b, c
.
operator+
B
a + b + c
rational
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1
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// POST : return value is the sum of a and b
rational operator + ( rational & a , rational & b )
{
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
.
8
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7
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9;
1
4.2.5 Const-types
6
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Rational number r:
1/2
Rational number s:
1/3
Sum is 5/6.
Y
D '
Program 33:
/
6
C9
/
/?
}
<
E
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M;
return i > > a .n > > c > > a.d ;
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}
rational
// POST : return value is -a
rational operator - ( rational a)
{
a .n = - a .n;
return a ;
}
rational operator - ( rational & a)
{
a .n = - a .n;
return a ;
}
rational operator - ( const rational & a)
{
a .n = - a .n ; // error : a was promised to be constant
return a ;
}
const
%
2
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1
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int n = 5;
const int & i = n ;
int & j = n ;
i = 6;
j = 6;
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[
E 4.2.6 What exactly is constant?
-
8
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const int n = 5;
int i & = n ; // error : const - qualification is discarded
i = 6;
C
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const int n = 5;
n = 6;
8
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const int n = 5
8
Definition.
!
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const
!
"#
const
i becomes a non - modifiable alias of n
j becomes a modifiable alias of n
error : n is modified through const - reference
ok : n receives value 6
4.2.7 Const-references
// POST : return value is the sum of a and b
rational operator + ( const rational & a , const rational & b ) {
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
const int & i = 3;
2
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8
8
E
>8
M1
/
1=
8
/6
>6
. EGJ
J
=
8
8
J
D >=
/
C
:
09
>9
;
9.
8
=
9
>6
0H D
6
. . C=
J 7
87
M;
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1
>6
J E -8
J
.=
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8
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>8
1
8
M.
7
8
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J
>8
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>
[
=
MI
0
6
7
.
9.
[
I
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J -
M.
7
8
=
>-
>
/
08
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8
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. >8
:
9
J
9;
<= - = 1
=
8
< /8 _ D 1
M.
9
- D 7
. >8
8
6
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8
1
.
8
78
.
= 8J
J
;
I
08
9
.
1
I
J
8
>
/
08
I
1
08
.
>
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.
=
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. = MB 8
>
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1
M. /
9;
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7
;
08
< ]8
.
8
= 7
/
>
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8
8
MH 9
9; -=
/
>
6
0
8
H
=
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J >9
1
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E 8
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;
B
8
8
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8
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1
;
E .
8
<
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M ;;
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.
9.
8
6.
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>8
D 1
8
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8
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B
>=
M.
7
8
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08
-
>
/
08
8
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1
>9
/
J
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9;
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7
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8
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M8 ;
9
-
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M.
7
< D 8
E 8
6
1
>?6
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9
M ;;
B
-
1.
8
C8
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8
=
8
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6.
=
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1
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H 9 ;; 9 H 8
7 E 8; 6
.
>6
I 8I
J
J
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:9
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8
1
0
>6
E '
= C9
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E Y% /6
D
. U . 18
8
E
Z 6I
.
. M. E J= D 7
. /8
M
\
>[
0
J
X
6
MB 8
7
B8
08
1
.
I 9; /
>
J=
; C6
H
8
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8
7
. 1. M H
= /
<
.
= :
0- =
/8
.
=
>6
8
C6
09
/
=
;
>=
= 9. M.
8/
>
- D = 7
8
I
8
>6
>9
=
8= >
1
>9 = M
/9 ;
. 6
9;
B
8
J
.
M
;
/I
/8
9
.
M< >
06
C6 E M 9
C6
8
.
0
[
1
>
=
\
=
9
.
8
8
.
. 8
1 D I 8/
9
J
<
J 91
M
8
. /19
I
1
/6
I /
>6
= 8
/6
.
8
0
C
6
MI
I
<
7
6
U . 0
/6
I /
>
C
9;
M . B6
.
8
7
:
7
<
J
9
>9
/ D - = >8
08
/9 D
0;
Z
.
=
I
. C9
. >
>
?08
.
9
I /6
D 9
1
8 . ?0 J 1 .
>
6
1=
=
8
/8
<
>6
1
<
S
. = 68
>8
9J 7
9
>6
. = U9
.= .= 9 ; 9
I
M
I
;=
/H
/9
9
/6
S
>9
M>
const
>6
.
.
J
H;
=
>9
-
I
- D
U .
M.
7
8
>6
1
6
H
>?8 =
9-
E
;
9
:;
/9
1
6
=
9
-
=
I
;
J
.
?> . =
1
>[
]8
08
const
:9
6
>9
8
C-
9.
8
9
/9
J D
/9
8
H;
8
>6
1
9
6
08
6
9.
=
/8
>6
.
.
<
H
9;
1
8
9
( ;=
- . U .Z
= /
>6
8
/8 I
> X >8
C9
1
8
M8 ; / I
C6
0
>=
8
=
.=
6
8
-
9-
8
const
>9
8
-
.
9.
:9
8
B/8
9I
.
M;
>6
8
1
=
const rational
/6
C
>9
7-
8
76
=
9.
;
-
1
>6
/
H;
9
/9
J
J
J
7
const
.
;
70 . =
-
08 .
-
-
D 8
6
01
8
9
:/9
:
.
;
8
>=
.
9.
.
9;
1
8
7
6
=
/8
6.
>9
>[
M.
7
8
E -
M
8
J
0,
-.
;\=
D >6
.
M.
7
8
6
0I
I
>
1.
>
/
08
9
/-
/9
8
79
-
K
9;
M9
U .
M.
7
8
C
>6
8
8
8
;
<
8
6
-
1
8
6
6
.
.
M.
7
8
>
9
?--
>9
=
I
H,
<
9
( ;=
.=
:
I
=
.
M.
7
8
8
-
8
-
J
J
B
9
=
=
1
8
.
=
.=
/J
8
/8 I
>8
1
8
M.
7
8
EG-
>6
>
1
>6
>=
<=
/8 _
>8
1
8
-
8
=
<=
J
;
J
J
9\
C<
M9
;
<
?> . =
0
.
.
.
.
.
.
9.
8
09
>9
-
6
>=
1
6
8
-
C6
/
7
8
>9
>6
=
8
/6
C
.
-
8
J=
:;
09
;
>
B8
B
S
/I
C6
8
D 8
>[
B9
8
8
-
8
:09
.
H
1
9
8
6.
U .
M.
7
E 8
>6
>
-.
H
8
^ Z
I
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.
D J
9.
.=
M.
7
C8
0
:6
9
=
-.
>6
1
8
-
.
=
J=
J
E 8
G
1
>8
/8
9
-
8
.
[
_
<=
9\
C<
>9
8
-=
H;
-
6
7-
-
:8
/8
>8
M.
7
B 8-
C
.
.
.
J
;
H;
6
C<
>9
8
9
/9
9;
1
6
9
9-
8.
/8
9
7;
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C9
/6
8
8
-
06
.
;
J
=
E
A
.
;
H
J
.
=
I
=
S J=
-
-
6
B ;;
6
6
8
/01
9-
8
O
E Z
E '
>6
1.
8
8
8
-
I D 91 \
-.
=
H,
9;
1
8
>6
1.
8
=
>8
8
6
>8
X
9
<
J
<
<
J
P
;
=
6
1
6
-
8
8
8
>?9
>
C9
/
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78
;
>6 =
E .
78
C6
-
9.
8
J
8
9
-
9
8
8
09
D
;
J
8
09
.
.
J
1
;
;
J
/08
<
`
C .=
8
C9
-
9
J
=
J=
)
6
M6
0
-
>9
/6
7
.
.
8
C=
/6
:8
>8
C
-
79
/
/?9
7-
.
.
<
Z
6
6
>9
0
8
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J
J
<
J
9
D H
;
8
:
.
.
.
.
9.
6
B6
B
M
B
B
\
<
;
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6
>
.
.
=
=
U
J
=
I
I
1.
:
;
)
=
=
6
/
8
76
/
7
.
.
:9
/9
9.
1
9;
>6
0
9
0H
6
.
1
>6
-.
=
W
=
U
C
1
C9
/9
>6
7
>
I
I
=
I
=
.=
9
0;
/6
9
9
> J= D ;H
\ 8
6
8
1
.=
M.
7
8
6
1
J
9
0H
6
.
.
J
8
79
7<
9
N
.
> J=
\
0
:8
/8
>8
6
1
8
/6
M .;
M
=
I
X
Z
X
CZ
J
M .;
9.
/
J
)
?- ;
8
.
6
6
.
?- =
8
09
>
/
08
/.
1.
>6
>
09
I
MH
/
8
/8 I
>8
1
8
;
9;
1
<
<
.
9.
-
>8
7
1
C6
>6
/6
>[
J
=
=
B .=
M<
=
I
.
.
.
9.
>8
-
]8
:8
>8
>
>6
/6
8
J
;
9;
C-
6
-
-
H
6
>6
>
08
.
>=
/8
8.
C9
/9
7
1.
>6
>
9
0;
D 88
E
C
/6
8
6
-
-
8
1
1
9
9. D 8.
/8
.
J
8
/6
1
/8
I
I
I
.=
8.
C9
/9
>
.
9;
>6
7
>
0
;
9
U ;;
MH
U
C
/6
9
6
8
:8
09
7
0
.
6\
6
6
/ <=
8
1
.=
S>6
. E 8
;
<
<
>=
8
-
>6
9
.I
/8
D 9
;;
=X
M I M0
0
6 D
6
:
08
/6
>
.
<
9;
01
7
8
D .
/8
/
<
.=
06 \
6
>8
C9
/
;
J
/<
8
>9
7
/9
7
:
.
06 \
6
6
:9
#
84
D .=
1
B-
>9
8
/8 I
>8
1
8
C8
/
< D >6
J
J
J
.=
.
.
8.
6
6\
/8
>
-
9;
/=
8
<
Q
/9
:09
B
B
B
-
8
09
C8
-
8
1
9
U ;;
MH
U
:
09
;
/8
>
/<
8
-
9.
.
9.
>9
D .=
1
.
J
8
9
<
/<
8
6
I
.
J
8:
/<
8
/8 I
.
6
- E
C6
>9
>6
1
.I
<
M =I
:
:
>8
6
-
`
; D -;
6
=
X
.
=
−
-
I
8.
>8
J=
8
.
:;
09
;
0
8
I
>=
9.
8
-
.
t
/1
>6
1
/6
.
=
;;
;
J
J
9;
1
/8
C
.
1.
>6
>
08
>8
=
J
I
/
const rational & operator + ( const rational & a , const rational & b ) {
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
=
=
.
8
/<
8
/8 I
>8
B 18
>
/
08
J
=
9;
>=
-
8
C6
1
8
H
B<
:
J
9
<
/<
8
-
8
-
6
B<
.
:;
09
;
8
6
-
=
result
/6
<
8
>W
8
<
H
8
J
:9
>
0
/8
.
J
J
Z
=
/I
8
8
<
>9 E
-
/6
9
9;
operator+
1
>6
U .
M.
7
8
.;
9.
.
O
=
=
M
C8
C
result
>=
-
08
-
8
1
>8
-
8
/8 I
>8
1
/8
08
/-
Y
A
<
A
X
const rational&
Const Guideline:
8.
1
>6
J=
.
'
1.
8
8
>9
>?- =
E
E
>6
>6
I
=
]8
7
/=
8
<
6
H,
D 8 E
1.
E
>9 E
0<
>9
6
-
9
1.
8
8
8
-
const
M?8 ;
J
H
rational
const
.=
.
9;
8
8
4.2.8 Const-types as return types.
<
<
8
>W
I
return
const
0J
6
.
const
6
8
:6
C6
6
M9
7
.
.
6.
8
6
>-
.
<
<;
J
-
/8
8.
C9
/9
78
:;
09
;
>? =
>9
>=
>- =
1
8
.
J
8
:9
>? =
D const
_
=
.
8
8
7-
D
M
-
>8
.
/=
C8
08
/
/6
C8
:6
/8
9
9-
6
.
<=
J
H
J
I
E J
9-
/1
:8
>8
/6
/8 _
>8
M .;
8
:9
8
M9
C
C9
/
/?
78
DC
;
.=
9.
/6
7
1
>9
-
C8
=
. E -
M.
.
7
D J
=
E 8
[
.
I
J
/8
= 8
/9
09
I 8
>
-.
<
;;=
C9
-0 =
>8
.
.
9; 9.
M . >6 =
7
B8
/6
J
/8
9
8
/0- .
B1 . - =
7
= /
. 8
J
/8 I
C/
9; 9H
;C
;
8
/8
8
C9
/9
7
const
> ;=
8
6
I
1
6
<
E 8
E
6
C9
/
?/
78
6
operator.
J
<
.
.
J
I
.
J
=
8
/8
6
>9
>6
.
.
J
E <
0
D
B
9;
>6
-
.
.=
6
"
)
C9
/
?/
/6
8
/6
C8
C
8
]8
C9
7
.
J
;
=
.
>8
/6
\
>- =
1
8
.
J
8
7
J
8Z
/8
-
>? =
/9
>
=
=
1=
>8
8
/8
-
M;
M
:
>6
9.
(=
B
M
J
. ==
.
=
%
>=
8
8
<= D 8 <
9.
M8 ;
/
0
8
8
C
6.
C=
>
]8
7
/=
8
<
6
H,
8
1
.
8
6
1=
9.
8
<
.
6
-
9-
9-
9
B.
8
9
U ;;
MH
U
:
09
;
1
6
1\
9
H
;
9;
.=
9
M ;;
>8
-
-
8
8
;
:;
09
;
>9
/8
8.
<Z
/9
7
J
9;
1
8
.
C9
D
8
S 6-
7
/
70
=
H
<
D =
8
B
B/8
6.
>9
D 8
1
8
6
-
8
8
76
>
9
<
/<
8
-
X
.
M.
7
8
6
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8
C9
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7
`
J
J
I
I
=
M
.
.
<
J
J=
\;=
>9
<
>
1
8
.
8;
6
/8
7
1
C6
>6
>
6
01
.
;=
9
M H;
/
>9
=
>W
6
/9
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8
/9
-
8
=
I
.=
H;
<
B<
M
:
-
/6
/
/
:8
/8
/6
8
9
-
1
08
:8
8
>6
U ;
>=
U
.
.
9;
8
8
/6
8
/8
-
const
4.2.9 When to use const?
const
const
X
Y
2
&
&
'
" !!
"
+ #
"
&
4
&
%
%
#
"
"
2
)
"
%
"# $
(a, b)
+
%
&
%
( 2
"
+ @
Z
X
Z
X
)
!
@
"
+
3
)
"
"#
#
+
%
#
&
&4
#
'
$
&
"'
"#(
"
)
%
)
%
#%
%
!
( &4
"
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"
)
%
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+
&
%
)
$ !
)
$
%
4
&
"
&
'
#
#
%
%
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) !
"
( +
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!
#2
( )
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+
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&
!
2
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)
#
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&
(
"
)
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#
(
#
)
*
4
+
!
&
#%
&
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#2
=
U
;
J
1
H
<
&
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)
#
6
8
8
.
<
J
>9
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76
/
J
6
C=
7
0
.
9.
/9
-
C9
/
/?
7
>=
/6
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9;
.=
=
6
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8
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>8
1
8
M.
7
8
^-
%
%
+
.=
8
8
.
<
J
>9
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76
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$ !
)
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#
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#
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&
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#
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J
6
C=
7
0
.
9.
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-
C9
/
/?
6
>=
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9;
1
>9
-
#
.
J
.
X
&
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%
9;
"!
"
#
"
-
8
(=
C
/6
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>
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"'
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)
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