Introductions to Ceiling
Introduction to the rounding and congruence functions
General
The rounding and congruence functions have a long history that is closely related to the history of number theory.
Many calculations use rounding of the floating-point and rational numbers to the closest smaller or larger integers.
J. Nemorarius (1237) was one of the first mathematicians to use the quotient of two numbers m and n in a modern
sense, but the word quotient appeared for the first time around 1250 in the writings of Meister Gernadus.
Special notations for rounding and congruence functions were introduced much later. C. F. Gauss (1801) suggested
the symbol mod (k = m mod n) for the notation of the property that the ratio Hm - kL  n is an integer. He observed
that k and m are the congruent modulo n. The number n is called modulus.
C. F. Gauss (1808) and J. Liouville (1838) widely used the floor and round functions in their investigations. They
and other mathematicians used different and sometimes confusing notations for those functions. The modern
notations of dzt and `zp for floor and ceiling functions, respectively, were suggested by K. E. Iverson (1962). The
notation dzp for the rounding function was proposed by J. Hastad (1988).
Definitions of the rounding and congruence functions
The rounding and congruence functions include seven basic functions. They all deal with the separation of integer
or fractional parts from real and complex numbers: the floor function (entire part function) dzt, the nearest integer
function (round) dzp, the ceiling function (least integer) `zp, the integer part intHzL, the fractional part fracHzL, the
modulo function (congruence) m mod n, and the integer part of the quotient (quotient or integer division)
quotientHm, nL.
The floor function (entire function) dzt can be considered as the basic function of this group. The other six functions can be uniquely defined through the floor function.
Floor
For real z, the floor function dzt is the greatest integer less than or equal to z.
For arbitrary complex z, the function dzt can be described (or defined) by the following formulas:
dxt Š n ; x Î R ß n Î Z ß n £ x < n + 1
dzt Š dReHzLt + ä dImHzLt.
Examples:
d3.2t Š 3, d3t Š 3, d-0.2t Š -1, d-2.3t Š -3, e 3 u Š 0, d-Πt Š -4, e-4 -
e 2 u Š 2, e 2 u Š 3.
5
Round
7
2
5
3
äu Š -4 - 2 ä,
http://functions.wolfram.com
2
For real z, the rounding function dzp is the integer closest to z (if z ¹ ± 2 , ± 2 , ¼).
1
3
For arbitrary z, the round function dzp can be described (or defined) by the following formulas:
dxp Š n ; x Î R í n Î Z í x - n¤ <
dzp Š dReHzLp + ä dImHzLp
n+
n+
1
2
1
2
Š n ;
n
2
1
2
ÎZ
Š n + 1 ;
n+1
2
Î Z.
Examples: d3.2p Š 3, d3p Š 3, d-0.2p Š 0, d-2.3p Š -2, e 3 q Š 1, d-Πp Š -3, e-4 2
e 2 q Š 4.
äq Š -4 - 2 ä, e 2 q Š 2,
5
3
5
7
Ceiling
For real z, the ceiling function `zp is the smallest integer greater than or equal to z.
For arbitrary z, the function `zp can be described (or defined) by the following formulas:
`xp Š n ; x Î R ß n Î Z ß n - 1 < x £ n
`zp Š `ReHzLp + ä `ImHzLp.
Examples: `3.2p Š 4, `3p Š 3, `-0.2p Š 0, `-2.3p Š -2, a 3 q Š 1, `-Πp Š -3, a-4 2
a 2 q Š 4.
5
3
äq Š -4 - ä, a 2 q Š 3,
7
Integer part
For real z, the function integer part intHzL is the integer part of z.
For arbitrary z, the function intHzL can be described (or defined) by the following formulas:
intHxL Š n ; x Î R ì n Î Z ì 0 £ sgnHxL Hx - nL < 1 ì x ¹ 0
intHzL Š intHReHzLL + ä intHImHzLL.
Examples:
intH3.2L Š 3,
intH-ΠL Š -3, intI-4 -
5
3
intH3L Š 3,
intH-0.2L Š 0,
äM Š -4 - ä, intI 2 M Š 2, intI 2 M Š 3.
5
intH-2.3L Š -2,
intI 3 M Š 0,
2
7
Fractional part
For real z, the function fractional part fracHzL is the fractional part of z.
For arbitrary z, the function fracHzL can be described (or defined) by the following formulas:
fracHxL Š x - n ; x Î R ì n Î Z ì 0 £ sgnHxL Hx - nL < 1 ì x ¹ 0
5
http://functions.wolfram.com
3
fracHzL Š fracHReHzLL + ä fracHImHzLL.
fracH3.2L Š 0.2,
Examples:
fracH-ΠL Š 3 - Π, fracI-4 -
5
3
fracH3L Š 0,
äM Š -
2ä
,
3
fracH-0.2L Š -0.2,
fracI 2 M Š 2 , fracI 2 M Š 2 .
5
1
7
fracH-2.3L Š -0.3,
fracI 3 M Š 3 ,
2
2
1
Mod
For complex n and m, the mod function m mod n is the remainder of the division of m by n. The sign of m mod n for
real m, n is always the same as the sign of n.
The mod function m mod n can be described (or defined) by the following formula:
m
m mod n Š m - n
n
.
The functional property m mod n Š n I n mod 1M Š n I n - e n uM makes the behavior of m mod n similar to the
m
behavior of e n u.
m
m
m
Examples: 5 mod 2 Š 1, 8 mod 3 Š 2, -5 mod 3 Š 1, H7 ΠL mod 3 Š -21 + 7 Π, H27 - 3 äL mod 4 Š 3 + ä,
fracH-ΠL Š 3 - Π, H2.7 - 3 äL mod 5 Š 2.7 + 2 ä.
Quotient
For complex n and m, the integer part of the quotient (quotient) function quotientHm, nL is the integer quotient of m
and n.
The quotient function quotientHm, nL can be described (or defined) by the following formula:
quotientHm, nL Š
Examples:
m
n
.
quotientH5, 2L Š 2,
quotientH13, 3L Š 4,
quotientH-4, 3L Š -2,
quotientHΠ, 2L Š 1,
quotientH27 - 3 ä, 5L Š 5 - ä, quotientH-Π, 2L Š -2, quotientH2.7 - 3 ä, 5L Š -ä.
Connections within the group of rounding and congruence functions and with other
function groups
Representations through related functions
The rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL have numerous
representations through related functions, which are shown in the following tables, where the symbol ΧA HaL means
the characteristic function of a set A (having the value 1 when its argument a is an element of the specified set A,
and a value of 0 otherwise):
http://functions.wolfram.com
4
x ; x Î R
Floor
Hor Hm, nL ; m Î R ß n Î RL
Round
dxt Š fx - 2 r ;
1
Floor
x+1
2
dxt Š fx - 2 r + 1 ;
1
Round
dxp Š fx +
dxp Š fx -
1
v
2
1
v
2
;
;
dxt Š fx 2 x-1
4
ÏZ
2 x-1
4
ÎZ
dxp Š fx + 2 v - ΧZ J
1
Ceiling
1
r+
2
x+1
2
ÎZ
x+1
ΧZ J 2 N
2 x-1
N
4
`xp Š dxt + 1 ; x Ï Z
`xp Š dxt ; x Î Z
`xp Š dxt - ΘH ΧZ H xL - 1L + 1
`xp Š fx + 2 r ;
1
`xp Š fx +
x+1
2
1
r-1
2
;
`xp Š fx + 2 r - ΧZ J
1
IntegerPart
ÏZ
intHxL Š dxt ; x > 0 Þ x Î Z
intHxL Š dxt + 1 ; x < 0 ì x Ï Z
intHxL Š dxt + 1 - sgnH ΧZ H xL + ΘHxLL
ÏZ
x+1
2
ÎZ
x+1
N
2
intHxL Š fx - 2 r ; x ³ 0 í
1
intHxL Š fx - 2 r + 1 ; x < 0 ë
Mod
intHxL Š fx - 2 r + 1 + ΧZ J
x+1
2
ÎZ
x+1
N - sgnH ΧZ HxL + ΘHxLL
2
fracHxL Š x - dxt ; x > 0 Þ x Î Z
fracHxL Š x - fx - 2 r ; x ³ 0 í 2 Ï Z
fracHxL Š x - dxt - 1 ; x < 0 ì x Ï Z
1
x+1
fracHxL Š x - fx - 2 r -1 ; x < 0 ë 2 Î Z
fracHxL Š x - dxt - 1 + sgnH ΧZ H xL + ΘHxLL
1
x+1
fracHxL Š x - fx - 2 r - 1 - ΧZ J 2 N + sgnH ΧZ HxL + ΘHxLL
1
m mod n Š m - n f n v
m
x+1
m mod n Š m - n f n - 2 r ;
m
1
quotient Hm, nL Š f n v
m
m+n
2n
m mod n Š m - n - n f n - 2 r ;
m
Quotient
ÏZ
1
1
FractionalPart
x+1
2
m mod n Š m - n J ΧZ J
1
m+n
m
N+fn
2n
quotientHm, nL Š f n - 2 r ;
m
quotientHm, nL Š
quotientHm, nL Š
m
fn
m
fn
1
-
m+n
2n
1
r+1
2
1
r+
2
;
ÏZ
m+n
2n
ÎZ
- 2 rN
1
ÏZ
m+n
2n
ÎZ
m+n
ΧZ J 2 n N
http://functions.wolfram.com
5
x ; x Î R
Mod
Hor Hm, nL ; m Î R ß n Î RL
Floor
Round
Quotient
dxt Š x - x mod 1
dxp Š x - fracJx +
1
1
N+ 2
2
dxp Š x - fracJx + 2 N 1
1
2
;
;
dxp Š x - Jx + 2 N mod 1 +
1
Ceiling
IntegerPart
FractionalPart
Mod
Quotient
`xp Š x + -x mod 1
1
2
dxt Š quotientHx, 1L
2 x-1
4
ÏZ
2 x-1
4
ÎZ
- ΧZ J
2 x-1
N
4
dxp Š quotientJx + 2 , 1N ;
1
2 x-1
4
dxp Š quotientJx + 2 , 1N - 1 ;
1
dxp Š quotientJx + 2 , 1N - ΧZ J
1
`xp Š -quotientH-x, 1L
ÏZ
2 x-1
4
ÎZ
2 x-1
N
4
intHxL Š x - x mod 1 ; x > 0 Þ x Î Z
intHxL Š quotientHx, 1L ; x > 0 Þ x Î Z
intHxL Š x + 1 - x mod 1 ; x < 0 ì x Ï Z
intHxL Š quotientHx, 1L + 1 ; x < 0 ì x Ï Z
intHxL Š x + 1 - x mod 1 - sgnH ΧZ HxL + ΘHxLL intHxL Š quotientHx, 1L + 1 - sgnH ΧZ H xL + ΘHxLL
fracHxL Š x mod 1 ; x > 0 Þ x Î Z
fracHxL Š x mod 1 - 1 ; x < 0 ì x Ï Z
fracHxL Š x mod 1 - 1 + sgnH ΧZ HxL + ΘHxLL
quotient Hm, nL Š
m-m mod n
n
fracHxL Š x - quotientHx, 1L ; x > 0 Þ x Î Z
fracHxL Š x - quotientHx, 1L - 1 ; x < 0 ì x Ï Z
fracHxL Š x - quotientHx, 1L - 1 + sgnH ΧZ H xL + ΘHxLL
m mod n Š m - n quotient Hm, nL
http://functions.wolfram.com
z Hor Hm, nLL
6
Floor
Round
fz -
Floor
ä
Round
dzp Š fz +
ä ΧZ J
ΧZ J
Ceiling
IntegerPart
dzt Š `zp + ΘH ΧZ H ReHzLL - 1L ä ΘH- ΧZ H ImHzLLL - 1
dzt Š -`-zp
ImHzL+1
ΧZ J 2 N
dzp Š bz -
2 ImHzL-1
N4
ä ΧZ J
2 ReHzL-1
N
4
`zp Š dzt - ΘH ΧZ H ReHzLL - 1L +
ä ΘH- ΧZ H ImHzLLL + 1
`zp Š -d-zt
int HzL Š dzt +
1 + ä - sgnH ΧZ HReHzLL +
ΘHReHzLLL ä sgn H ΧZ HImHzLL + ΘHImHzLLL
m mod n Š m - n f n v
m
`zp Š fz +
ä ΧZ J
1+ä
r
2
ImHzL+1
N
2
int HzL Š fz ΧZ J
- ΧZ J
ReHzL+1
N+ä
2
ΧZ J
1+ä
r-1-ä2
ReHzL+1
ΧZ J 2 N - ä
ImHzL+1
ΧZ J 2 N +
sgnH ΧZ HReHzLL + ΘHReHzLLL +
ä sgn H ΧZ HImHzLL + ΘHImHzLLL
m mod n Š m + n Jf
1+ä
2
m
- nrm
ΧZ J 2 JReJ n N + 1NN m
ä ΧZ J 2 JImJ n N + 1NNN
1
quotient Hm, nL Š f n v
ImHzL+1
N2
sgnH ΧZ HReHzLL + ΘHReHzLLL ä sgn H ΧZ HImHzLL + ΘHImHzLLL
frac HzL Š z - fz -
m
quotient Hm, nL Š f n m
1+ä
r+
2
ΧZ J 2 JReJ n N + 1NN +
1
m
ä ΧZ J 2 JImJ n N + 1NN
1
z Hor Hm, nLL
Floor
Round
Ceiling
IntegerPart
Mod
dzt Š z - z mod 1
dzp Š
1+ä
2
+z-J
1+ä
2
`zp Š z + -z mod 1
+ zN mod 1 - ä ΧZ J
2 ImHzL-1
N4
int HzL Š z + 1 + ä - z mod 1 - ä sgnH ΧZ HImHzLL +
ΘHImHzLLL - sgn H ΧZ HReHzLL + ΘHReHzLLL
m
ΧZ J
2 ReHzL-1
N
4
FractionalPart frac HzL Š z mod 1 - 1 - ä + ä sgnH ΧZ HImHzLL + ΘHImHzLLL +
sgnH ΧZ HReHzLL + ΘHReHzLLL
Mod
Quotient
quotient Hm, nL Š
m-m mod n
n
1+ä
r+
2
ΧZ J
2 ImHzL+1
N
4
2 ReHzL+1
N+
4
ReHzL+1
N2
1+ä
r+1+ä+
2
1
Quotient
Ceiling
ReHzL+1
ΧZ J 2 N +
1+ä
v2
FractionalPart frac HzL Š z - dzt - 1 - ä +
sgnH ΧZ HReHzLL + ΘHReHzLLL +
ä sgn H ΧZ HImHzLL + ΘHImHzLLL
Mod
1+ä
r+
2
intHzL Š `zp - sgnH ΧZ HReHzLL + ΘHReHzLLL +
ä H-sgnH ΧZ HImHzLL + ΘHImHzLLL ΘH- ΧZ HImHzLLL +
ΘH ΧZ HReHzLL - 1L + 1L
fracHzL Š z - `zp + sgnH ΧZ HReHzLL +
ΘHReHzLLL - ä H1 - sgnH ΧZ HImHzLL +
ΘHImHzLLL - ΘH- ΧZ HImHzLLL +
ΘH ΧZ HReHzLL - 1LL
m mod n Š m + n - n b n r m
n ΘJ ΧZ J ReJ n NN - 1N + n ä ΘJ- ΧZ J ImJ n
m
m
m mod n Š m + n b- n r
m
quotient Hm, nL Š b n r + ΘJ ΧZ J ReJ n NN - 1N
m
m
ä ΘJ- ΧZ J ImJ n NNN - 1
m
quotientHm, nL Š -b- n r
m
Quotient
dzt Š quotient Hz, 1L
dzp Š quotientJz +
1+ä
,
2
1N - ΧZ J
`zp Š -quotient H-z, 1L
2 ReHzL-1
N-ä
4
ΧZ
int HzL Š quotientHz, 1L + 1 + ä - ä sgnH ΧZ HImHzLL
ΘHImHzLLL - sgn H ΧZ HReHzLL + ΘHReHzLLL
frac HzL Š z - quotientHz, 1L - 1 - ä + ä sgnH ΧZ HIm
sgnH ΧZ HReHzLL + ΘHReHzLLL
m mod n Š m - n quotient Hm, nL
The rounding and congruence functions dzt, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL can also be represented
through elementary functions by the following formulas:
http://functions.wolfram.com
dzt Š z +
`zp Š z +
7
tan-1 Hcot HΠ zLL
Π
tan-1 HcotHΠ zLL
Π
+
2
1
2
tan-1 Hcot HΠ zLL
intHzL Š z +
fracHzL Š -
1
-
m mod n Š
Π
n
2
-
n
Π
quotient Hm, nL Š
+ sgnHΘHzLL -
tan-1 cot
m
n
+
1
Π
; z Î R ì z Ï Z
- sgnHΘHzLL +
Π
tan-1 Hcot HΠ zLL
; z Î R ì z Ï Z
Πm
n
tan-1 cot
;
m
n
Πm
n
; z Î R ì z Ï Z
1
2
1
2
; z Î R ì z Ï Z
ÎRí
m
;
m
-
1
2
n
n
ÏZ
ÎRí
m
n
Ï Z.
The best-known properties and formulas of the number theory functions
Simple values at zero
The rounding and congruence functions dzt, dzp, `zp, intHzL, and fracHzL have zero values at zero:
d0t Š 0
d0p Š 0
`0p Š 0
intH0L Š 0
fracH0L Š 0.
Specific values for specialized variables
The values of five rounding and congruence functions dzt, dzp, `zp, intHzL, and fracHzL at some fixed points or for
specialized variables and infinities are shown in the following table:
http://functions.wolfram.com
8
dzt
0
1
-1
ä
-ä
dzp
0
1
-1
ä
-ä
`zp
0
1
-1
ä
-ä
intHzL
0
1
-1
ä
-ä
fracHzL
0
0
0
0
0
0
0
1
0
1
2
-2
-1
0
0
0
-2
ä
2
0
0
ä
0
ä
2
-2
-ä
0
0
0
-2
3
2
1
2
2
1
1
2
-2
-2
-2
-1
-1
-2
3ä
2
ä
2ä
2ä
ä
ä
2
-2 ä
-2 ä
-ä
-ä
-2
23
10
2
2
3
2
3
10
-3
-Π
-3
-4
-3
-3
-3
-3
-3
-3
0
3-Π
- 10
-3
-3
-2
-2
- 10
-3.4
-4
-3
-3
-3
-0, 4
23
10
2-3ä
2-3ä
3-2ä
2-2ä
3
10
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
fracH¥L Î H0, 1L
frac H-¥L Î H-1, 0L
frac Hä ¥L Î H0, äL
frac H-ä ¥L Î H-ä, 0L
Ž L Î H0, 1L
frac H¥
intHnL Š n
intHä nL Š ä n
fracHnL Š 0
fracHä nL Š 0
¥
-¥
ä¥
-ä ¥
Ž
¥
fracH¥L Î H0, 1L
frac H-¥L Î H-1, 0L
frac Hä ¥L Î H0, äL
frac H-ä ¥L Î H-ä, 0L
Ž L Î H0, 1L
frac H¥
z
0
1
-1
ä
-ä
1
2
1
ä
3
-
3ä
2
27
-äã
¥
-¥
ä¥
-ä ¥
Ž
¥
n ; n Î Z
ä n ; n Î Z
dnt Š n
dä nt Š ä n
dnp Š n
dä np Š ä n
`np Š n
`ä np Š n
1
ä
1
ä
7
- Hã - 2 L ä
x + ä y ;
dx + ä yt Š dx + ä yp Š
`x + ä yp Š int Hx + ä yL Š
frac Hx + ä yL Š
xÎRìyÎR
dxt + ä dyt
dxp + ä dyp
`xp + ä `yp
intHxL + ä int HyL
fracHxL + ä frac HyL
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
¥
-¥
ä¥
-ä ¥
Ž
¥
The values of mod function m mod n, and quotientHm, nL at some fixed points or for specialized variables are shown
here:
http://functions.wolfram.com
9
m\n 1
0
0
2
0
3 4 5 6 7 8 9 10 n
0 0 0 0 0 0 0 0 0 mod n Š 0 ; n ¹ 0
1
0
1
1 1
1 1 1 1 1 1
2
3
4
5
6
7
8
9
10
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
2
0
1
2
0
1
2
0
1
2
3
4
0
1
2
3
4
0
m
m mod 1 Š 0 ;
mÎZ
2
3
0
1
2
3
0
1
2
2
3
4
5
0
1
2
3
4
2
3
4
5
6
0
1
2
3
2
3
4
5
6
7
0
1
2
2
3
4
5
6
7
8
0
1
1 mod n Š n + 1 ; -n Î N+
1 mod n Š 1 ; n Î Z ß n > 1
2 mod n Š 2 ; n Î Z ß n > 2
3 mod n Š 3 ; n Î Z ß n > 3
4 mod n Š 4 ; n Î Z ß n > 4
5 mod n Š 5 ; n Î Z ì n > 5
6 mod n Š 6 ; n Î Z ß n > 6
7 mod n Š 7 ; n Î Z ß n > 7
8 mod n Š 8 ; n Î Z ß n > 8
9 mod n Š 9 ; n Î Z ß n > 9
10 mod n Š 10 ; n Î Z ß n > 10
2
3
4
5
6
7
8
9
0
n mod n Š 0
H2 nL mod n Š 0
m
m\n 1
0
0
2
0
3 4 5 6 7 8 9 10 n
0 0 0 0 0 0 0 0 quotient H0, nL Š 0 ; n ¹ 0
1
1
0
0 0
0 0 0 0 0 0
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
1
2
2
3
3
4
4
5
0
1
1
1
2
2
2
3
3
0
0
0
1
1
1
1
1
2
m
quotientHm, 1L Š
m ; m Î Z
0
0
1
1
1
1
2
2
2
0
0
0
0
1
1
1
1
1
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
quotientH1, nL Š -1 ; n Î Z ß n < 0
quotientH1, nL Š 0 ; n Î Z ß n > 1
quotientH2, nL Š 0 ; n Î Z ß n > 2
quotientH3, nL Š 0 ; n Î Z ß n > 3
quotientH4, nL Š 0 ; n Î Z ß n > 4
quotientH5, nL Š 0 ; n Î Z ì n > 5
quotientH6, nL Š 0 ; n Î Z ß n > 6
quotientH7, nL Š 0 ; n Î Z ß n > 7
quotientH8, nL Š 0 ; n Î Z ß n > 8
quotientH9, nL Š 0 ; n Î Z ß n > 9
quotientH10, nL Š 0 ; n Î Z ß n > 10
quotient Hn, nL Š 1
quotient H2 n, nL Š 2
m
m mod n Š m ; m Î N ß n Î Z ß m < n
m mod n Š m - n ; m Î N+ ì n Î N+ ì n £ m < 2 n
m mod n Š m - k n ; m Î N+ ì n Î N+ ì k Î N+ ì k n £ m < Hk + 1L n
Hp - 1L! mod p Š p - 1 ; p Î P
2 p-1
mod p3 Š 1 ; p Î P ì p > 3
p-1
B2 n ¤ mod 1 Š ∆ n+1
2
mod 1,0
+ H-1Ln
â
2 n+1
k=3
1
k
ΧZ
quotientHm, nL Š 0 ; m Î N ì n Î N+ ì m < n
2n
k-1
ΧP HkL +
1
2
mod 1
http://functions.wolfram.com
10
quotientHm, nL Š 1 ; m Î N+ ì n Î N+ ì n £ m < 2 n
quotientHm, nL Š k ; m Î N+ ì n Î N+ ì k Î N+ ì k n £ m < Hk + 1L n
quotientHHp - 1L!, pL Š
quotient
Hp - 1L! + 1 - p
1
2 p-1
, p3 Š
p-1
p3
p
; p Î P
2 p-1
- 1 ; p Î P ì p > 3.
p-1
Analyticity
All seven rounding and congruence functions (floor function dzt, round function dzp, ceiling function `zp, integer
part intHzL, fractional part fracHzL, mod function m mod n, and the quotient function quotientHm, nL) are not analytical
functions. They are defined for all complex values of their arguments z Î C and Hm, nL Î C2 . The functions dzt, dzp,
`zp, and intHzL are piecewise constant functions and the functions fracHzL, m mod n, and quotientHm, nL are piecewise
continuous functions.
Periodicity
The rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, and quotientHm, nL are not periodic functions.
m mod n is a periodic function with respect to m with period n:
Hm + nL mod n ‡ m mod n
Hm + k nL mod n Š m mod n ; k Î Z.
Parity and symmetry
Four rounding and congruence functions (round function dzp, integer part intHzL, fractional part fracHzL, and mod
function m mod n) are odd functions. The quotient function quotientHm, nL is an even function:
d-zp Š -dzp
intH-zL Š -intHzL
fracH-zL Š -frac HzL
-m mod -n Š -Hm mod nL
quotientH-m, -nL Š quotientHm, nL.
The rounding and congruence functions dzt, dzp, `zp, intHzL, and fracHzL have the following mirror symmetry:
dz t Š dzt - ä H1 - ΧZ HImHzLLL
dz p Š dzp
`z p Š `zp + ä H1 - ΧZ HImHzLLL
intHz L Š intHzL
fracHz L Š fracHzL.
http://functions.wolfram.com
11
Sets of discontinuity
The floor and ceiling functions dzt and `zp are piecewise constant functions with unit jumps on the lines
ReHzL = k ê ImHzL = l ; k, l Î Z.
The functions dzt (and `zp) are continuous from the right (from the left) on the intervals Hk - ä ¥, k + ä ¥L, k Î Z, and from
above (from below) on the intervals Hä k - ¥, ä k + ¥L, k Î Z.
The function dzp is a piecewise constant function with unit jumps on the lines ReHzL +
The function dzp is continuous from the right on the intervals I2 k intervals I2 k +
1
2
- ä ¥, 2 k +
1
2
1
2
- ä ¥, 2 k -
1
2
1
2
= k ë ImHzL +
1
2
= l ; k, l Î Z.
+ ä ¥M, k Î Z, and from the left on the
+ ä ¥M, k Î Z.
The function dzp is continuous from above on the intervals I-¥ + 2 ä k - 2 , ¥ + 2 ä k - 2 M, k Î Z, and from below on the
intervals I-¥ + 2 ä k + 2 , ¥ + 2 ä k + 2 M, k Î Z.
ä
ä
ä
ä
The function intHzL (and frac HzL) is a piecewise constant (continuous) function with unit jumps on the lines
ReHzL = k ê ImHzL = l ; k, l Î Z, k ¹ 0, l ¹ 0.
The functions intHzL and frac HzL are continuous from the right on the intervals Hk - ä ¥, k + ä ¥L, k Î N+ , and from the left
on the intervals H-k - ä ¥, -k + ä ¥L, k Î N+ .
The functions intHzL and frac HzL are continuous from above on the intervals H-¥ + ä k, ¥ + ä kL, k Î N+ , and from below on
the intervals H-¥ - ä k, ¥ - ä kL, k Î N+ .
The functions m mod n and quotient Hm, nL are piecewise continuous functions with jumps on the curves
ReI n M = k ë ImI n M = l ; k, l Î Z. The functional properties m mod n Š n I n mod 1M Š n I n - e n uM and
m
m
m
m
m
quotientHm, nL Š quotientI n , 1M Š e n u make the behavior of that functions similar to the behavior of floor function
e n u.
m
m
m
The previous described properties can be described in more detail by the formulas from the following table:
http://functions.wolfram.com
12
z ±Ε Hor m ± ΕL ; Ε ® +0
Floor
limΕ®+0 dz ± Εt Š dzt ReHzL Î Z
Round
Ceiling
limΕ®+0 dz ± Εp Š dzp ;
1
4
limΕ®+0 dz ± ä Εt Š dzt -
1
4
limΕ®+0 dz ± ä Εp Š dzp ;
;
1±1
2
limΕ®+0 `z ± ä Εp Š `zp +
ReHzL Î Z
IntegerPart
Quotient
H2 ImHzL ± 1L Î Z
H2 ImHzL ¡ 1L Î Z
1
4
1±1
2
ä ;
ImHzL Î Z
limΕ®+0 intHz ± ΕL Š intHzL ; ±ReHzL Î N+
limΕ®+0 intHz ± ΕL Š intHzL ± 1 ; ¡ReHzL Î N+
limΕ®+0 intHz ± ä ΕL Š intHzL ; ±ImHzL Î N+
limΕ®+0 intHz ± ä ΕL Š intHzL ± ä ; ¡ImHzL Î N+
limΕ®+0 Hm ± ΕL mod n Š m mod n + n
limΕ®+0 Hm ± ä ΕL mod n Š m mod n + ä n
FractionalPart limΕ®+0 fracHz ± ΕL Š fracHzL ; ±ReHzL Î N+
limΕ®+0 fracHz ± ΕL Š fracHzL ¡ 1 ; ¡ReHzL Î N+
Mod
1
4
limΕ®+0 dz ± ä Εp Š dzp ± ä ;
ReH2 z ¡ 1L Î Z
ä ;
1¡1
2
ImHzL Î Z
H2 ReHzL ± 1L Î Z
limΕ®+0 dz ± Εp Š dzp ± 1 ;
limΕ®+0 `z ± Εp Š `zp +
z ±ä Ε Hor m ± ä ΕL ; Ε ® +0
;
1¡1
2
m
ReJ n N
Î Zín Î Rín >0
1¡1
2
limΕ®+0 fracHz ± ä ΕL Š fracHzL ; ±ImHzL Î N+
limΕ®+0 fracHz ± ä ΕL Š fracHzL ¡ ä ; ¡ImHzL Î N+
;
limΕ®+0 quotientHm ± Ε, nL Š quotientHm, nL ReJ n N Î Z í n Î R í n > 0
1¡1
2
;
m
ImJ n N
Î Zín Î Rín >0
1¡1
2
;
limΕ®+0 quotientHm ± ä Ε, nL Š quotientHm, nL - ä
m
ImJ n N Î Z í n Î R í n > 0
1¡1
2
;
m
Series representations
The rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL have the following
series representations:
x
Floor
Round
Ceiling
IntegerPart
dxt Š x -
Quotient
+ Π Ú¥
k=1
1
dxp Š x + Π Ú¥
k=1
1
`xp Š x +
1
2
intHxL Š x +
FractionalPart fracHxL Š
Mod
1
2
+
1
Π
1
-Π
1
Π
sinH2 Π k xL
k
H-1Lk sinH2 Π k xL
k
sinH2 Π k xL
Ú¥
k=1
k
sinH2 Π k xL
Ú¥
k=1
k
sinH2 Π k xL
Ú¥
k=1
k
m
n
; x Î R ì x Ï Z
; x Î R í x +
1
2
; x Î R ì x Ï Z
- ΘHxL +
+ ΘHxL -
1
2
1
2
fnvŠ
m
ÏZ
; x Î R ì x Ï Z
; x Î R ì x Ï Z
m
n
f n + 2r Š
m
m
bnr
1
Š
m
intJ n N
m
n
m
n
+
Š
m
n
m
fracJ n N
1
2n
+
Š
1
2
; m Î Z í
2Πkm
Πk
1
n-1
Úk=1 sinJ n N cotJ n N + 2
; m Î Z í
n-1
+
1
2n
-
2Πkm
Πk
1
N cotJ n N - 2
n
Úk=1 sinJ
1
2
1
2n
Úk=1 sinJ
n-1
m
1
sgnJ n N + 2 n
m
1
sgnJ n N - 2 n
m mod n Š
n
2
m mod n Š
n
2
2Πkm
Πk
1
N cotJ n N + 2
n
-
n
Π
-
1
2
1
Ú¥
k=1 k
2Πkm
Πk
n-1
Úk=1 sinJ n N cotJ n N
2Πkm
Πk
n-1
Úk=1 sinJ n N cotJ n N
2Πkm
sinJ n N
;
m
n
2Πkm
Πk
n-1
Úk=1 sinJ n N cotJ n N
quotientHm, nL Š
m
n
quotientHm, nL Š
m
n
+ Π Ú¥
k=1
1
+
1
2n
; m Î
1
k
sinJ
Úk=1 sinJ
n-1
ÎQí
; m
m
n
Ï
; m Î Z í
2Πkm
1
N- 2
n
;
m
n
Î
2Πkm
Πk
1
N cotJ n N - 2
n
Transformations and argument simplifications (arguments involving basic arithmetic operations)
;
The values of rounding and congruence functions dzt, dzp, `zp, intHzL, and fracHzL at the points -z, ±ä z,
z + n ; n Î Z can also be represented by the following formulas:
http://functions.wolfram.com
13
dzt
z
-z
äz
-ä z
z+n
dzp
d-zp Š - dzp
dä zp Š ä dzp
d-ä zp Š -ä dzp
dz + n
`zp
intHzL
d-zt Š -dzt ; ReHzL Î Z ß ImHzL Î Z
dä zt Š ä dzt + ΧZ HImHzLL - 1 d-ä zt Š -ä dzt - ä H1 - ΧZ HReHzLLL dz + n
d-zt Š -dzt - sgnH ReHzL¤L - ä sgnH ImHzL¤L ;
dä zt Š d-ImHzLt + ä dReHzLt d-ä zt Š dImHzLt + ä d-ReHzLt
ReHzL Ï Z ì ImHzL Ï Z
d-zt Š -dzt + ΧZ HzL - 1 ; z Î R
d-zt Š -dzt - ä H1 - ΧZ HImHzLLL sgnH ImHzL¤L H1 - ΧZ HReHzLLL sgnH ReHzL¤L
ReH
`-zp Š -`zp ; ReHzL Î Z ß ImHzL Î Z
`ä zp Š ä `zp - ΧZ HImHzLL + 1 `-ä zp Š -ä `zp + ä H1 - ΧZ HReHzLLL `z + n
`-zp Š -`zp + ä sgnH ImHzL¤L + sgnH ReHzL¤L ;
`ä zp Š -dImHzLt + ä `ReHzLp `-ä zp Š `ImHzLp - ä dReHzLt
ReHzL Ï Z ì ImHzL Ï Z
`-zp Š -`zp + ä H1 - ΧZ HImHzLLL sgnH ImHzL¤L +
H1 - ΧZ HReHzLLL sgnH ReHzL¤L
int H-zL Š -int HzL
frac HzL frac H-zL Š -frac HzL
int Hä zL Š ä int HzL
int H-ä zL Š -ä int HzL
int Hn
frac Hä zL Š ä frac HzL
frac H-ä zL Š -ä frac HzL
frac Hn
The values of the functions m mod n and quotientHm, nL at the points H±m, -nL, H-m, nL, Hä m, ä nL, H±ä m, nL,
Hm, ±ä nL, and I n , 1M have the following representations:
m
Hm, nL
m mod n
H-m, -nL H-mL mod H-nL Š -Hm mod nL
Hm, -nL
m mod -n Š m mod n + ΧZ J n N n - n ; m Î R ß n Î R
m
m
ä n J1 - ΧZ JImJ n NNN sgn J¢ImJ n N¦N
m
m
m
n
ÏZ
-m mod n Š - ΧZ J n N n + n - m mod n ; m Î R ß n Î R
m
-m mod n Š -Hm mod nL + n J1 - ΧZ JReJ n NNN sgnJ¢ReJ n N¦N +
m
Hä m, ä nL
ä n J1 - ΧZ JImJ n NNN sgnJ¢ImJ n N¦N
Hä m, nL
m
Hä mL mod Hä nL Š ä Hm mod nL
Hä mL mod n Š n - n
m
ΧZ JImJ n NN + ä Hm mod nL
H-ä m, nL H-ä mL mod n Š -ä n J ΧZ JReJ n NN - 1N - ä Hm mod nL
m
Hm, ä nL
m mod Hä nL Š m mod n + J ΧZ JReJ n NN - 1N n
m
Hm, -ä nL m mod H-ä nL Š m mod n + J ΧZ JImJ n NN - 1N ä n
m
J n , 1N
m
m
n
mod 1 Š
m mod n
n
quotientHm, -nL Š -quotientHm, nL - J1 - ΧZ JReJ n NN
m
ä J1 - ΧZ JImJ n NNN sgnJ¢ImJ n N¦N
m
-m mod n Š n - m mod n ; m Î R í n Î R í
m
quotient Hm, -nL Š -quotientHm, nL + ΧZ J n N - 1 ; m
m
m mod -n Š m mod n - n J1 - ΧZ JReJ n NNN sgnJ¢ReJ n N¦N m
H-m, nL
quotientHm, nL
quotient H-m, -nL Š quotient Hm, nL
m
m
quotientH-m, nL Š ΧZ J n N - 1 - quotientHm, nL ; m Î
m
quotientH-m, nL Š -quotientHm, nL - J1 - ΧZ JReJ n N
m
ä J1 - ΧZ JImJ n NNN sgnJ¢ImJ n N¦N
m
m
quotient Hä m, ä nL Š quotient Hm, nL
quotient Hä m, nL Š ä quotientHm, nL + ΧZ JImJ n NN - 1
m
quotient H-ä m, nL Š -ä quotientHm, nL + ä J ΧZ JReJ n N
m
quotient Hm, ä nL Š -ä quotientHm, nL + ä J ΧZ JReJ n NN m
quotient Hm, -ä nL Š ΧZ JImJ n NN + ä quotientHm, nL - 1
m
quotient J n , 1N Š quotient Hm, nL
m
Transformations and argument simplifications (arguments involving related functions)
http://functions.wolfram.com
14
Compositions of rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL with
the rounding and congruence functions in many cases lead to very simple zero results:
Floor
ddztt Š dzt
dz - dztt Š 0
Round
ddzpt Š dzp
Ceiling
d`zpt Š `zp
IntegerPart
Round
ddztp Š dzt
d`zpp Š `zp
dintHzLp Š int HzL
Ceiling
`dztp Š dzt
ddzpp Š dzp
dz - dzpp Š 0
`dzpp Š dzp
`intHzLp Š int HzL
IntegerPart
int HdztL Š dzt
int HdzpL Š dzp
``zpp Š `zp
`z - `zpp Š 0
int H`zpL Š `zp
frac H`zpL Š 0
frac HintHzLL Š 0
z
Floor
FractionalPart frac HdztL Š 0
dmt mod n Š dmt - n f
Mod
dmt mod 1 Š 0
Quotient
dn xt
n
`n xp
n
fracHdzpL Š 0
quotient Hdmt, nL Š
dmt
v
n
dmp mod n Š dmp - n f
dmp mod 1 Š 0
dmt
f n v
quotient Hdmp, nL Š
quotientHdmt, 1L Š dmt
dmp
v
n
dintHzLt Š int HzL
int HintHzLL Š intHzL
intHz - intHzLL Š 0
`mp mod n Š `mp - n f
`mp mod 1 Š 0
dmp
f n v
quotient H`mp, nL Š
quotientHdmp, 1L Š dmp
`mp
v
n
`mp
f n v
quotientH`mp, 1L Š `mp
intHmL mod n Š intHmL - n
intHmL mod 1 Š 0
quotient HintHmL, nL Š f
quotientHintHmL, 1L Š intHmL
Š dxt ; x Î R ß n Î Z
Š `xp ; x Î R ß n Î Z.
Hm, nL
Floor
Round
Mod
dm mod nt Š
Quotient
m
fm - n f n vv
dquotientHm, nLt Š f n v
m
dm mod 1t Š 0
f n - quotientHm, nLv Š 0
m
dm mod np Š fm - n f n vr
dquotientHm, nLp Š f n v
m
m
Ceiling
`m mod np Š bm - n f n vr
`quotientHm, nLp Š f n v
IntegerPart
intHm mod nL Š intJm - n f n vN
m
m
m
FractionalPart fracHm mod nL Š fracJm - n f n vN
m
Mod
Hm mod nL mod n Š m mod n
Hm mod nL mod n Š m - n f n v
m
Quotient
quotient Hm mod n, nL Š f
quotientHm mod 1, 1L Š 0
Addition formulas
m mod n
v
n
int HquotientHm, nLL Š f n v
m
frac HquotientHm, nLL Š 0
quotientHm, nL mod n Š f n v - n f n f n vv
m
1
m
quotientHm, 1L mod 1 Š 0
quotient HquotientHm, nL, nL Š f n f n vv
1
quotientHquotientHm, 1L, 1L Š dmt
intH
n
m
The rounding and congruence functions dzt, dzp, `zp, intHzL, and fracHzL satisfy the following addition formulas:
http://functions.wolfram.com
15
z+n
z1 + z2
dz + nt Š dzt + n ; n Î Z
Floor
Round
dz1 + z2 t Š dz1 + z2 - dz1 t - dz2 tt + dz1 t + dz2 t
dz + np Š dzp + n ; n Î Z
í ReHzL +
1
2
Ï Z í ImHzL +
1
2
ÏZ
Ceiling
`z + np Š `zp + n ; n Î Z
`z1 + z2 p Š `z1 p + `z2 p + `z1 + z2 - `z1 p - `z2 pp
IntegerPart
intHz + nL Š intHzL + n - ΘHz + nL + ΘHzL ; n Î Z ì z Ï Z
FractionalPart fracHz + nL Š fracHzL + ΘHz + nL - ΘHzL ; n Î Z ì z Ï Z
Hm + k nL mod n Š m mod n ; k Î Z
quotientHm + k n, nL Š quotientHm, nL + k ; k Î Z.
Multiple arguments
The rounding and congruence functions dzt, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL have the following
relations for multiple arguments:
n z Ior k m M
dn zt Š n dzt + Ún-1
k=0 k ΘJz mod 1 - n N J1 - ΘJz mod 1 k
Floor
Round
Ceiling
IntegerPart
`n zp Š n `zp - Ún-1
k=0 k ΘJ- n - z mod 1 + 1N J1 - ΘJk
k+1
NN
n
k+1
n
; n Î N ß z Î R
- z mod 1 + 1NN ; n Î N ß z Î R
int Hn zL Š intHzL n + n sgnH ΧZ H zL + ΘHzLL - sgnH ΧZ H n zL + ΘHzLL +
n-1
Úk=0 k ΘJz mod 1 - n N J1 - ΘJz mod 1 -
k+1
NN - n + 1
n
; n Î N ß z Î R
n-1
Úk=0 k ΘJz mod 1 - n N J1 - ΘJz mod 1 -
k+1
NN + n - 1
n
; n Î N ß z Î R
k
FractionalPart frac Hn zL Š fracHzL n - n sgnH ΧZ H zL + ΘHzLL + sgnH ΧZ H n zL + ΘHzLL k
Hk mL mod n Š k Hm mod nL - n Úk-1
j=0 j ΘJ n - k - quotientHm, nLN J1 - ΘJ n m
Mod
j
quotientHm, nLNN ; k Î N í
Quotient
m
m
n
quotient Hk m, nL Š k quotientHm, nL + Úk-1
j=0 j JΘJ n mod 1 - k N J1 - ΘJ n mod 1 m
j
m
Sums of the floor and ceiling functions dzt and dzp satisfy the following relations:
dz1 t + dz2 t Š dz1 + z2 t - dz1 + z2 - dz1 t - dz2 tt
n-1
k=0
km+x
n
Šâ
m-1
k=0
kn+x
m
; x Î R ì n Î N+ ì m Î N+
`z1 p + `z2 p Š `z1 + z2 p - `z1 + z2 - `z1 p - `z2 pp
â
n-1
k=0
x-km
n
k
Šâ
m-1
k=0
x-kn
m
-
ÎR
Sums of the direct function
â
j+1
; x Î R ì n Î N+ ì m Î N+ .
Identities
All rounding and congruence functions satisfy numerous identities, for example:
j+1
k
NNN ; k Î N í
m
n
ÎR
http://functions.wolfram.com
n
+
3
n+
e n +
n+2
6
1
2
+
+
1
2
n+4
6
Š
16
n
n+3
Š g w+
; n Î Z
2
6
n+
1
4
+
1
2
; n Î Z
n + 1 u Š e 4 n + 2 u ; n Î Z
Ha + cL mod n Š Hb + dL mod n ; a mod n Š b mod n ì c mod n Š d mod n ì a Î R ì b Î R ì c Î R ì d Î R ì n Î R.
Complex characteristics
Complex characteristics (real and imaginary parts ReHzL and ImHzL, absolute value z¤, argument ArgHzL, complex

conjugate z, and signum sgnHzL) of the rounding and congruence functions can be represented in the forms shown in
the following tables:
z
Re
Im
Abs
Arg
Floor
Re HdztL Š dReHzLt
Im HdztL Š dImHzLt
dzt¤ Š
dImHzLt2 + dReHzLt2
dzp¤ Š
dReHzLp2 + dImHzLp2
Abs
Arg
Conjugate
Sign
IntegerPart
Re HintHzLL Š int HReHzLL
Im HintHzLL Š int HImHzLL
`zp¤ Š
`ImHzLp2 + `ReHzLp2
intHzL¤ Š
2
2
intHImHzLL + intHReHzLL
Arg HdzpL Š
tan-1 HdReHzLp, dImHzLpL
Arg H`zpL Š
Arg HintHzLL Š
tan-1 H`ReHzLp, `ImHzLpL
tan-1 HintHReHzLL, intHImHzLLL
sgn HdztL Š
sgn HdzpL Š
sgn H`zpL Š
dzt
dzt¤
dzp Š dReHzLp - ä dImHzLp
x + ä y ;
Floor
xÎRìyÎR
Re
Im
Ceiling
Re H`zpL Š `ReHzLp
Im H`zpL Š `ImHzLp
Arg HdztL Š
tan-1 HdReHzLt, dImHzLtL
Conjugate dzt Š dReHzLt - ä dImHzLt
Sign
Round
ReHdzpL Š dReHzLp
Im HdzpL Š dImHzLp
Re Hdx + ä ytL Š dxt
Im Hdx + ä ytL Š dyt
dx + ä yt¤ Š
dxt2 + dyt2
Arg Hdx + ä ytL Š
tan-1 Hdxt, dytL
dx + ä yt Š dxt - ä dyt
sgn Hdx + ä ytL Š
dx+ä yt
dx+ä yt¤
dzp
dzp¤
`zp Š `ReHzLp + ä `ImHzLp intHzL Š intHReHzLL - ä int HImHz
sgn HintHzLL Š
`zp
`zp¤
intHzL
intHzL¤
Round
Ceiling
IntegerPart
ReHdx + ä ypL Š dxp
Im Hdx + ä ypL Š dyp
ReH`x + ä ypL Š `xp
Im H`x + ä ypL Š `yp
Re HintHx + ä yLL Š int Hx
Im HintHx + ä yLL Š int Hy
dx + ä yp¤ Š
dxp2 + dyp2
Arg Hdx + ä ypL Š
tan-1 Hdxp, dypL
dx + ä yp Š dxp - ä dyq
sgn H`x + ä ypL Š
dxp+ä dyp
dxp2 + dyp2
`x + ä yp¤ Š
`xp2 + `yp2
Arg H`x + ä ypL Š
tan-1 H`xp, `ypL
`x + ä yp Š `xp - ä `yp
sgn H`x + ä ypL Š
`x+ä yp
`x+ä yp¤
intHx + ä yL¤ Š
2
intHxL
Arg HintHx + ä yLL Š
tan-1 HintHxL, intHyLL
intHx + ä yL Š intHxL - ä
sgn HintHx + ä yLL Š
int
int
2
http://functions.wolfram.com
Hm, nL
Re
Im
Abs
17
Mod
Re Hm mod nL Š ReHmL + f
f
ImHmL ImHnL+ReHmL ReHnL
f
ImHmL ReHnL-ImHnL ReHmL
2
ImHnL +ReHnL
ImHmL ImHnL+ReHmL ReHnL
v ReHnL
ImHnL2 +ReHnL2
ImHnL2 +ReHnL2
2
2
ImHnL +ReHnL
ImHmL ImHnL+ReHmL ReHnL
2
ImHnL +ReHnL
Arg Hm mod nL Š tan JReHmL + f
-1
f
f
ImHmL ImHnL+ReHmL ReHnL
ImHnL2 +ReHnL2
ImHmL ReHnL-ImHnL ReHmL
Conjugate m mod n Š ReHmL + f
ImHnL2 +ReHnL2
ä JImHmL - f
2
ImHmL ImHnL+ReHmL ReHnL
2
2
ImHnL +ReHnL
sgn Hm mod nL Š Jf
ImHmL ReHnL-ImHnL ReHmL
ä JImHmL - f
ImHnL2 +ReHnL2
ImHnL2 +ReHnL2
2
v ImHnL -
ImHmL ImHnL+ReHmL ReHnL
ImHnL2 +ReHnL2
v ImHnL - f
ImHmL ImHnL+ReHmL ReHnL
2
2
ImHnL +ReHnL
v ImHnL - f
v ReHnL -
ImHnL +ReHnL
ImHnL2 +ReHnL2
ImHmL ImHnL+ReHmL ReHnL
ImHnL2 +ReHnL2
ImHmL ReHnL-ImHnL ReHmL
ImHnL2 +ReHnL2
f
v ReHnL +
v ImHnL + ReHmL - f
ImHnL +ReHnL
2
2
ImHnL +ReHnL
ImHnL2 +ReHnL2
ImHmL ImHnL+ReHmL ReHnL
ImHnL2 +ReHnL2
v-äf
sgn HquotientHm, nLL Š
v ReHnLNN “
ImHmL ImHnL+ReHmL ReHnL
ImHmL ImHnL+ReHmL ReHnL
quotientHm, nL Š
v ReHnLN
ImHnL2 +ReHnL2
sgnHm mod nL Š sgnHnL ; m Î R ß n Î R
2
v +
ReHmL ReHnL+ImHmL ImHnL 2
tan-1 Jf
v ImHnL -
ImHmL ReHnL-ImHnL ReHmL
ImHnL +ReHnL
2
v
Arg HquotientHm, nLL Š
m
sgnHquotientHm, nLL Š sgnJ n
v ReHnLN OO
2
Derivatives of the rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and quotientHm, nL can
be evaluated in the classical and distributional sense. In the last case, all variables should be real and results include
the Dirac delta function. All rounding and congruence functions also have fractional derivatives. All these derivatives can be represented as shown in the following tables:
Hin classical senseL
¶
¶x
Hin distributional sense for real xL
Floor
¶dzt
¶z
Š0
¶dxt
¶x
Round
¶dzp
¶z
Š0
¶dxp
¶x
Ceiling
¶`zp
¶z
Š0
¶`xp
¶x
IntegerPart
¶intHzL
¶z
FractionalPart
¶fracHxL
¶x
Š0
Š1
Š Ú¥
k=-¥ ∆Hx - kL
Š Ú¥
k=-¥ ∆Jx - k + 2 N
1
Š Ú¥
k=-¥ ∆ Hx - kL
¶intHxL
¶x
¶fracHxL
¶x
Š Ú¥
k=-¥ ∆k,0 ∆Hx - kL
Š x - Ú¥
k=-¥ ∆k,0 ∆ Hx - kL
¶Α
¶zΑ
Hin fractional senseL
¶Α dzt
¶zΑ
¶Α dzp
¶zΑ
¶Α `zp
¶zΑ
Š
Š
Š
¶Α intHzL
¶zΑ
¶Α fracHzL
¶zΑ
Im
Im
ä
Differentiation
¶
¶z
v
ImH
2
Jf
ImHnL2 +ReHnL2
f
ImHmL ImHnL+ReHmL ReHnL
ImHmL ReHnL-ImHnL ReHmL
K- KJImHmL - f
v ImHnL - f
v ReHnLN +
2
2
2
2
ImHmL ReHnL-ImHnL ReHmL
ImHmL ReHnL-ImHnL ReHmL
v
quotientHm, nL¤ Š
v ImHnL +
2
v ImHnL + ReHmL - f
ImHmL ImHnL+ReHmL ReHnL
ImHnL2 +ReHnL2
2
v ImHnL - f
ImHnL2 +ReHnL2
f
ImHnL +ReHnL
ImHnL2 +ReHnL2
ImHmL ReHnL-ImHnL ReHmL
ReHmL ReHnL+ImHmL ImHnL
Im HquotientHm, nLL Š
ImHmL ReHnL-ImHnL ReHmL
ImHmL ReHnL-ImHnL ReHmL
v ReHnLN
f
v ImHnL -
v ReHnLN O
v ReHnL, ImHmL - f
Re HquotientHm, nLL Š
v ImHnL -
v ReHnLN + Jf
2
Quotient
v ImHnL -
ImHmL ImHnL+ReHmL ReHnL
ImHmL ReHnL-ImHnL ReHmL
ReHmL - f
Sign
ImHnL2 +ReHnL2
m mod n¤ Š - KJImHmL - f
2
f
Arg
v ReHnL
Im Hm mod nL Š ImHmL - f
ImHnL2 +ReHnL2
ImHmL ReHnL-ImHnL ReHmL
dzt z-Α
GH1-ΑL
dzp z-Α
GH1-ΑL
`zp z-Α
GH1-ΑL
Š
Š
intHzL z-Α
GH1-ΑL
Α z1-Α
GH2-ΑL
+
fracHzL z-Α
GH1-ΑL
2
http://functions.wolfram.com
18
¶
¶m
Hin classical
senseL
Mod
Quotient
¶Hm mod nL
¶m
Š1
¶quotientHm,nL
¶m
Š0
¶
¶n
¶
¶m
Hin classical
senseL
¶Hm mod nL
¶n
m
-f n v
Š
¶quotientHm,nL
¶n
Š0
¶
¶n
Hin distributional sense
for real m, nL
Hin distributional sense for real
m, nL
¶Hm mod nL
Š m+
¶m
¥
Úk=-¥ ∆Hm - k nL
¶Hm mod nL
¶n
¶quotientHm,nL
¶quotientHm,nL
Š
sgnHnL Ú¥
k=-¥ ∆Hm - k nL
¶m
Š sgnHnL JintJ n N m
m
Ú¥
k=-¥ ∆k,0 ∆J n
n2
m
-
¶n
sgnHmL m
n2
- kNN
¶Α
¶mΑ
Hin fractional senseL
¶Α Hm mod nL
¶mΑ
Α m1-Α
GH2-ΑL
+
Š
Hm mod nL mGH1-ΑL
¶Α quotientHm,nL
Š
Ú¥
k=-¥ ∆k,0 ∆ J n - kN
m
¶mΑ
Š
quotientHm,nL m-Α
GH1-ΑL
Indefinite integration
Simple indefinite integrals of the rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and
quotientHm, nL have the following representations:
Ù f HzL â z
Floor
Round
Ceiling
IntegerPart
Ù dzt â z Š z dzt
Ù dzp â z Š z dzp
Ù `zp â z Š z `zp
Ù intHzL â z Š z intHzL
FractionalPart Ù fracHzL â z Š z fracHzL Mod
Quotient
z2
2
Ù z mod n â z Š z Hz mod nL Ù m mod z â z Š
1
2
z2
2
z Hm + m mod zL
Ù quotientHz, nL â z Š z quotientHz, nL
Ù quotientHm, zL â z Š z quotientHm, zL
Definite integration
Some definite integrals of the rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and
quotientHm, nL can be evaluated and are shown in the following table:
http://functions.wolfram.com
aÎR
Floor
19
Ù0 f HtL â t ; n Î N
Ù0 f HtL â t
n
Ù0 dtt â t Š
n
a
Ù0 dtt â t Š
n Hn-1L
2
a
Α-1
f HtL â t
Ù0 t
a
1
2
H2 a - dat - 1L dat
Α-1
Ù0 t dtt â t Š
a
1
Α
Hdat aΑ - ΖH-ΑL + ΖH-Α, d
Round
Ù0 dtp â t Š
n2
2
Ù0 dtp â t Š
1
2
H2 a - dapL dar
Α-1
Ù0 t dtp â t Š
Ceiling
Ù0 `tp â t Š
n Hn+1L
2
Ù0 `tp â t Š
1
2
H2 a - `ap + 1L `ap
Α-1
Ù0 t `tp â t Š
IntegerPart
Ù0 intHtL â t Š
n
n
n
a
n Hn-1L
2
FractionalPart Ù0 fracHtL â t Š
n
Mod
a
n
2
; n Î N Ù0 intHtL â t Š
; n Î N
a
1
2
Ù0 fracHtL â t Š
a
H2 a - intHaL - 1L int HaL
1
2
IfracHaL - fracHaL + aM
2
1
2
IHa mod nL - n Ha mod nL + a nM
2
1
a+m-m mod a
Š 2 J-ΨH1L J
N m2 + a m +
a
Ù0 t mod n â t Š
a
Ù0 m mod t â t
a
a Hm mod aLN
Quotient
Ù0 quotientHt, nL â t Š
a
a
1
Α
Jfa + 2 v aΑ 1
H1 - 2-Α L ΖH-ΑL + ΖJ-
a
`ap aΑ -ΖH-ΑL+ΖH-Α
Α-1
Ù0 t intHtL â t Š
a
Α
intHaL aΑ -ΖH-ΑL
Α-1
Ù0 t fracHtL â t Š
a
Α
aΑ+1
Α+1
-
1
Α
Α-1
Ù0 t Ht mod nL â t Š
a
aΑ+1
Α+1
n - n quotientHa, nLL
1
Α
-
nΑ+1 ΖH-ΑL + nΑ+1
ReHΑL > -1
quotientHa, nL H2 a -
HH
ΖH-ΑL + ΖH-Α, a ReHΑL > -1
a Α-1
Ù0 t
1
2
Hm mod tL â t Š
1
Α2 +Α
mΑ+1 Α ΖJΑ + 1,
Α-1
Ù0 t quotientHt, nL â t Š
a
a Α-1
Ù0 t
Jm
a+m
1
Α
Hquotient
nΑ HΖH-Α, quotientHa,
quotientHm, tL â t Š
quotientHm,aL aΑ +mΑ ΖHΑ,quotientHm
Α
ReHΑL > 1
Integral transforms
L
All Fourier transforms of the rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL, m mod n, and
quotientHm, nL can be evaluated in a distributional sense and include the Dirac delta function:
http://functions.wolfram.com
20
Ft @ f HtLD HzL
Floor
Π
2
-
∆HzL +
Ft @dtpD HzL Š
Ft @`tpD HzL Š
2Π
Ú¥
k=1
ä
2Π
ä
Π
2
∆HzL +
∆H2 k Π-zL-∆H2 Π k+zL
Ú¥
k=1
k
Ú¥
k=1
ä
ä
2Π z
2Π
Π
2
∆ HzL
ä
Fct @`tpD HzL Š
Fst @dttD HzL Š
2 Π ∆ HzL
ä
-ä
2 Π ∆ HzL
¢
Π
2
1
k
J∆J
2kΠ
n
Π
2
ä
2Π
n
1
+ zNN
∆ HzL ¢
- zN - ∆J
2Πk
n
∆HzL +
cotI 2 M -
1
1
Iz cotI 2 M - 2M +
z
2 Π z2
1
Π
2
In z cotI
Π
2
nz
M2
1
cotI
2Π z
nz
M2
2Π
2Π
Floor
Round
Ceiling
IntegerPart
Lt @dttD HzL Š
2Π
Lt @dtpD HzL Š
Lt @`tpD HzL Š
1
Hãz -1L z
Mod
Quotient
1
z2
Lt @t mod nD HzL Š
1
z2
z
N
ãz -1
J1 -
Lt @quotientHt, nLD HzL Š
; ReHzL > 0
nz
N
ãn z -1
1
Hãn z -1L z
z
1
; ReHzL > 0
J1 -
; ReHzL < -1
ΖH-zL
Mt @dtpD HzL Š - z JΖ J-z, 2 N + 2-z N ; ReHzL < -1
; ReHzL > 0
ãz
Hãz -1L z
FractionalPart Lt @fracHtLD HzL Š
Mt @dttD HzL Š -
; ReHzL > 0
Hãz -1L z
ãz2
Lt @intHtLD HzL Š
Mt @ f HtLD HzL
; ReHzL > 0
1
Hãz -1L z
3
Mt @intHtLD HzL Š Lt @fracHtLD HzL Š
1
z2
; ReHn zL > 0 Mt @t mod nD HzL Š
; ReHzL < -1
ΖH-zL
J1 z
z
N
ãz -1
; -1 < ReHzL < 0
nz+1 ΖH-zL
Mt @m mod tD HzL Š -
z
mz+1
; ReHzL > 0
ΖHz+1L
; ReHn zL > 0 Mt @quotientHt, nLD HzL Š -
z+1
Mt @quotientHm, tLD HzL Š
nz
mz
ΖH-zL
z
ΖHzL
z
; -1 < ReHzL < 0
; ReHzL < -1
; ReHzL > 1
Ú¥
k=
Fst @quotientH
Laplace and Mellin integral transforms of the rounding and congruence functions dzt, dzp, `zp, intHzL, fracHzL,
m mod n, and quotientHm, nL can be evaluated in the classical sense:
Lt @ f HtLD HzL
Ú¥
k=
Fst @t mod nD
1
∆HzL
Ú¥
k=
Fst @fracHtLD Hz
n
n ∆HzL
Fct @quotientHt, nLD HzL Š -
+ zNN
2Π
1
∆HzL
2 Π z2
Ú¥
k=
Fst @intHtLD HzL
∆HzL
2M +
Π
2
Fst @`tpD HzL Š
2Π
z
2Π z
Fct @t mod nD HzL Š
∆HzL 2Πk
- zN - ∆J n
cot I 2 M
z
2Π z
Fct @fracHtLD HzL Š
-
2kΠ
J∆J n
1
∆HzL -
1
2Π z
Π
2
Π
2
Ú¥
k=
Fst @dtpD HzL Š
z
¢
∆H2 k Π-zL-∆H2 Π k+zL
k
1
Ú¥
k=1 k
Ú¥
k=1
Fst @ f HtLD HzL
2Π
csc I 2 M
1
2Π z
Fct @intHtLD HzL Š -
∆H2 k Π-zL-∆H2 Π k+zL
k
Ft @quotientHt, nLD HzL Š 2Π
Fct @dtpD HzL Š -
-
+
Ft @t mod nD HzL Š n
än
Quotient
Ú¥
k=1
-ä
2Π z
FractionalPart Ft @fracHtLD HzL Š
2Π
H-1Lk H∆H2 k Π-zL-∆H2 Π k+zLL
k
¢
Ft @intHtLD HzL Š -
2Π
Mod
Fct @dttD HzL Š -
-
z
cotI 2 M -
1
1
2 Π ∆ HzL
ä
IntegerPart
Fct @ f HtLD HzL
¢
ä
Ceiling
2Π
2 Π ∆ HzL
ä
Round
∆H2 k Π-zL-∆H2 Π k+zL
Ú¥
k=1
k
ä
Ú¥
k=
http://functions.wolfram.com
21
Summation
Sometimes finite and infinite sums including rounding and congruence functions have rather simple representations, for example:
â
y
k=0
â
p-1
k=1
â
n-1
k=1
k
y
+ x Š dx y + H`yp - yL dx + 1tt ; x Î R ì y Î R ì y > 0
kq
p
k
n
p-1 p-1
l=1 k=1
p-1
â
2
k=1
p
â xk=0
n-1
k=0
â
¥
n=1
Hp - 1L Hq - 1L ; p Î N+ ì q Î N+ ì gcdHp, qL Š 1
-
n
kl
km
1
2
k=1
k
y
q
Š
n
1
4
k
k
Š
2
Š
1
4n
â cot
n-1
k=1
kΠ
n
mkΠ
cot
n
; m Î N+ ì n Î N+ ì gcdHm, nL Š 1
Hp - 1L Hq - 1L ; p Î N+ ì q Î N+ ì gcdHp, qL Š 1
Š n-m-
pm n-m
2
n
n
1
Š `x y - `x - 1p Hy + `-ypLp ; x Î R ì y Î R ì 0 < x < 1 ì 0 < y < 1
p m
fmv
pk
-
Hp - 2L ; p Î P
2
2
+â
mk
-
n
2
p-1
Š
p
1
q-1
qk
y
â
2
k
-
ââ
1
Š
k
Hk - 1L Hkm - 1L
pm
; n Î N+ ì m Î N+ ì p Î N+
1 n
p m
; k Î N+ ì m Î N+ .
Zeros
Zeros of rounding and congruence functions are given as follows:
dzt Š 0 ; 0 £ ReHzL < 1 ß 0 £ ImHzL < 1
dzp Š 0 ; -
1
2
£ ReHzL £
1
2
í-
1
2
£ ImHzL £
`zp Š 0 ; -1 < ReHzL £ 0 ß -1 < ImHzL £ 0
1
2
intHzL Š 0 ; ReHzL¤ < 1 ß ImHzL¤ < 1
fracHzL Š 0 ; ReHzL Î Z Þ ImHzL Î Z
m mod n Š 0 ; m Š 0 ß n ¹ 0
quotientHm, nL Š 0 ; 0 £ Re
m
n
< 1 í 0 £ Im
m
n
< 1.
http://functions.wolfram.com
Applications of the rounding and congruence functions
All rounding and congruence functions are used throughout mathematics, the exact sciences, and engineering.
22
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Copyright
This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas
involving the special functions of mathematics. For a key to the notations used here, see
http://functions.wolfram.com/Notations/.
Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for
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[email protected].
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23