test Thursday ,
assignment
posted , Thursday .
Thursday , also
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last week before spring book Is midterm ?
to Roughly :
4.I- 4.3 -5 . I – 5 . 3
His m e n Number theory next week
week after Midterm !
Highly relevant to C S, especially b u t not 801dg
target. lots of projects
no quiz this
integers, of course, here arithmetic functions t , x : 2 4 2 1 2 ,
blot does In Q
G $ eotcdly
reminder r, neon ?
staying within The dish
UFO, then Here
grwith OEren- 1 g.t a-nor+r
(They’re not a field. )
there’s a thot
thing : divide
with renoir .
theorem says Het
F- = equivalently ,
q=m dir lmathbf {dir }
and r– mmod
12=4 , m mod
P ” modulo
172–12.14 t in
particularly
That means
exactly, inin mind!
4416.61cL → PII alb and b/c then a- aq and abq’ for some q,q’fE
” “Since”
and b/c Hen
We need to write ⇐ af for some q. o.bg q,
we’re done .
is the study of reminders in their own right .
in Iarithmetic w e treat different integers modular
con q=qq, and
Modukrorilhndic
specifically ,
es Ate som e
the modulus –
a mod m moot @ and
a.D= – q , q”hirdat↳Ifa= mlq ‘ )
a=p (mod lol if they
que the sane
a= btmlq- oil
if thy bee the
Sene reminder
on division
subtracting ,
” is congruent to b modulo ,a
To justify this further, let’s observe t h t u c a n do arithmetic
if you know
b mod m , you can figure act atb ,
w ith just
ranchers ? That is 31
land m ) and
be b ‘ limed n ) then
(mod m) . for some k and
ab: a’b’ as a t km
The assumption
land in) and ‘
here atb= tot(Helm
b =p’ them
la’t knitb’tlml
m( lat goaG=
. ‘m kl aid’ mod’ ul as desired
+ Lkoleldtkllm LTE
Iet dos this Ween ? It rang feet if
veneindors
some complicated competition,
we can completion ,
in the middle of
neat . tractable .
small, this Watch
computationsmore
as big as it’s built out of
for division (mod 6) but
5¥2 (mod6)
the last digit of 2745 ?
of of 2 2 4,8 I 2,4, 8,612,48
digitspowers : ,6, ,,
2kt’t mod [( 2K
mod 10). 2)
Indeed , since
2k mod lo (mod10)s
(mod 10 ) , this is 2a=2b (wool10), uh,’d
Wet the pattern
Imdb ) ‘ Yw!:D
• Mod7,we love:
of 27 is 2 ,,
212 ! μ( nod lol , k
-11=1 (nod 41 “,
, k:3 (used41
mood lol kIO (nod4) ,
8=1 (mod 7- I
so 2kI 2 Card 7) if k 1 (nod 31
1512=-1 trod 71 )
G 2,41,2,4I,2,41,
4 Inset71 1(nod-7)
k ± 2 (nod 3) kIO Lasd3)
, s o 2745=2
We formalize all of this by introducing the set at
– {0,1m-13with
” EarsIn a-mb= abmodm n,
:En’Em→Em atub= I
interesting
arithmetic
operations
then use followed by mod
the function
x n , Wich
of I modm: →
true tte cool algebraic operations :
} immediate =a a+my=a
operations
(atm b)the
= ath(bthc), sinforxm
} homework
is trade , in Em if a
bat inlet is ,
subtraction ?
know division is not
t.nl ] modm )
d Anda -amod
modin (nod n )
sisfrecfian f
Im×Em→Emis a-nd- a-b
mod m -4=-1.10+6
Z ? We all In a commutative
is just these n
be coefcl with division ? x – 5 = KI
3 card GodG )
2x I 4 Card
x:2Cased61,k )
cold be 4 as well
tomorrow . interestingly ,
arithmetic . ”
arithmetic
XyTn: In× In→ In and= abmodn
hfire lntgerrqrcsentafians
We usually w r i t e
, b’atbmod
in ab ÷ n , atb÷n
notation ,
using ten digits I . . ,
But there’s nothing special
n in write
an 10kt an
ta Iota with .
down (aka”
10″ ‘t write
aiEE for ,,.
” h=anBktanB’t.. ta Do
tgl ( 20010213=2.35+0.3’t 0.3 ‘t 1.540.3 ‘
– Base 3 is called tenor .
2=(49-7),.
— ie{0, ,}
Similarly to
work all Thet well :
ar- 1ktant1″‘t… + a. teas.10aiC-II{03
. Base less space .
(7) . base
= ( 000000011
d i g i t s . )
{false, Computers
1 is essentially folly males ?
that base -1 doesn’t
exponentially
n u me r a l s
( specifically , in
←fit Zz 0,13
true) also
↳: { Eoff , on
very aufcnl
6=28 nor a base 28 “digit” is celled of2.MG#lockll,b=2&Chexadecimell ,
Styerling E
more modern
use ,g= f o r i n s t a n c e
colors . There’s ×
71Gt It + I l
2560 e- 2672+11=2683 .
hexadecimal
×E*s→ colors usually determining insects
s y mb o l s
digits . egrgbfFF.FI, FF
etc , l e k
2 , n e e d
( A- 7 – B ) h e A . 1 6 7 7 . 1 6 1 – 1 3
d i g i t s ! 4,516,718,9 AB SD
pour – her
21+20=(27-271)/6 ‘t
( 1/010011/2=27+26+2’t
oooo 0001 →
I 10013 9 solo→ 2 10103A
Lnininun C F 9 2)
( 1/00111/1001001072
0011 → 3 10119 B
Hoos C 1101 3D
1110 S E 1111 → F
dose to Itsy
289=2.102-1
Converting
I 1/00/000
1001 go (2891.111100101K
::÷:i÷::i . 0 } In .
= 2.100 t 8 .
= ( ;(“”” letflood 101012 + (lool),
0 0 factor for
I ” ” ” ‘ III’s I ,
fheeitfise
100=64+321-4–1.26
+1.25+1.22
-cech then h , tether rise reseat.
289–256+32+1
Less algorithmic process : Find largest multiple of peer of a sneller
or, say 28,9
121241C But
t 2° = 1/00100001/2 289 to base 5 :
+1.542.5 ‘t f – 50
factor, then sum .
t.ynfindge.fi ?
C o u l d d o a l oopi ng S a r a h
5,10, 15,20, 125, 50,75,
efficient & Less obvious : the test digit is the reminder !
ah6kt. . . .
he hee and 90
b (and” ‘t .
ndivbinbasebas late. .- a.)g
ta , Therefore,togetninbaseb: get
100,125,250,0375
a. = h +1=112716
48=6.8 to –
= (120) = (12),
-_ (ah . – a instance,
8=6I-12 Msdubrtxqoneniar
clever way of numbers
large dogs 3) , find a utter:
( For shell box , eg
2 , 4=-1, 2, t, 211,2,I, . . – , so 240=1(mod31 )
(289 ) 289 =
need to take big
( 100100001 ) , o
256 t 32+1 , so
” ” 21=228.225.2 2289=2. 2.
Observe that
(z” )! 2*2122″
= 25.35–875=2 mod
35.2=-122.35
I c a n get
f- 3572=1225
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