, Topic I
1. 1 Real numbers ,
Algebraic manipulations
Number field :
•
Real numbers ,
IR : All numbers I e.
go.li
2 it , } ,
- I - . .
I
•
Integers , 2 : positive ,
negative integers
and zero f- . .
-3 ,
-2 ,
-1
,
a , I ,
2,3 - . .
}
•
Non -
negative integers , IN :
positive integers
and zero
f o
,
I , 2,3 . - .
}
Operations :
•
Add att defined for
- subtract a- to all red a and I
at
Multiply
•
-
Division £ defined for all real a and b- ,
except
tota
Comparisons :
acts a > hi , ast A7T att
, , ,
shorthand :
"
•
a = a. a -
a - . .
a for all real a
,
HE IN and N 70
-
n times
. [ =/ for a f- 0
X x-
y
As when a -_ a
I = a
a× f- is undefined .
Ix I
?
=
× '
i. do is undefined yet
a×
-
i. = a =
I
Summations
I -2
, proof by induction
Summations :
¥2 nlhtll 12h41 )
'
j =
Mtn ) 1+2+3-1 .tn ? ,
i. = . .
6
M In ) = 1+2+3-1 . . . th n
hlhtl )
I
Mint htlh 1) 1- In -21 -1 + I j =
2
j= ,
= -
.
. .
a. Min) = Intl ) -14+111-4+1 ) + . . . -1 ( htt )
n times
hlhtl )
i. Min ) =
2
z
Summation Notation :
II
.
flk )
•
fill + f- 121 + f- 13 ) + . . .
+ flkl is written as ,
.
More
generally we define , for a ,t integer acts
II. aflkl =
flat + flat 't + . .
.
+ fitt
properties
• :
of label
Independent
:
1
eflkl =
IIe till =
¥Ea f- I
' '
k l called
, , is a
dummy
hmm
index
② off
peel terms :
II.afoot =
flkl + fitt
=
flat -1 . .
.
+ fib -
It
3 We can shift the label :
II. f- 1kt =
II ,
fit - thi ) =
f Im -
a) = final + flattest . . .
+ f- latte -
a)
= flat -14111 . - .
=
f- 155-55) + fltb-551
+ f- 114 f- . -
tf 165-55)
, 22
j⇐
' '
't
'
Example pin ) I + + t n
j
=
• : = o . . .
works for
only
Proof by induction :
integers
Suppose we want to
prove that same result is true for all +
integers ,
h .
If it can be shown that the
result holds for n=I and also that ,
for all N71 , if the result is true for n = N , then it must also
be true for n = Ntl , then follows that the result is true for all
positive n
Inland )
:=€=a for
'
Qin )
k equal Intl )
negative integer
=
Prove that pin ) is to :
any non n
Question
-
.
:
i. = stands for a
definition
0
' z
when n =
0 pie) = [ok = a = a
Q a) =
to 1410+1110-11 ) = 0 i. 1710 ) =
Q lol i . True for h= 0
i. Pll) = QU )
True for h =/
Assume it is true for n=N
i. PIN ) = QIN )
when n= Ntl when u = Ntl
PINHI = ET bi
① 1µg =
LNHI 121*1+11 IN -11+1 )
k= I
'
6
= FINI t Will =
to IN -11 ) 12N -13 ) IN -14
'
=
QINI + IN -111
2
=
IN IZNHIINH ) 1- IN -111 i. PINT ) = QI Ntl)
= '
Ntl ) ( IN lsNH ) + IN-111 ) since true for n=o ,
then it is t ru e for all IN
=
INH ) I NL2N-1II -161N -111 )
=
to INH ) I ZN2-1TN -16 )
= to IN -11112 # 3) IN -12 )
General template
:
all E IN
Claim f- In )= gin ) for n
proof : .
show f- lot =
gio )
-
show that for n
any
i
if f- In ) =
gun) ,
then final =
glntl )
1.3 Binomial theorem
Binaural coefficients :
i.
Definition Factorial function : n ! = 1×2×3 × . . . X In 2)
-
X In -
1) X h ,
h E IN
ht
0 ! =
I In -11 ! =
n i . It -11 ! = = I = 0 !
Factorial function is
only defined for -heintegemm
•
•
Factorial function is never zero
Binomial coefficient
2.
Definition :
For all h .
k E IN , h 7k ,
-
h
(1) with)
I
Ck
'
n choose k =
!
=
> .
Pascal 's triangle :
.to/--lnn/-- I
n!
proof :
(2) =
d. in -4 ! =
I
h !
(1) =
h ! In -
n ) !
=
I
i. 141=111
•
properties of Pascal 's
triangle
:
①
symmetry :
121=11 )
, h !
proof :
II ) =
k :( n -
KI !
(Ik )
n !
=
i. in -
ntkl !
=
4-kl ! k !
= II )
② I 1) = III ) + 1hL ) ,
For all in > k -
the number is the sum of two numbers above
k > o
In-1 ) ! In-11 !
Proof :
RHS =
µ -11 ! In -1 htt ) !
-
X k ! In -
I -
kl !
In -11 ! In -
1) !
As In k ) ! (n k-11 ! In k )
=
+
-
-
=
1k -11 ! 1h kl ! k ! In tkl !
-
- -
In -11 ! In -11 !
= +
1k -11 ! In -
tell ! Intel klk-11 ! In k -11 ! -
kin -11 ! + In-11 ! In k ) -
=
klk-11 ! In k -11 !
-
In - k)
= h In -11 !
talk-11 ! In -
k -
1) ! In k -
)
!
%)
n
=
In )
=
I box hand
k ! - k
II
-
The little on the
right
'
of proof
'
side means end
Binaural Theorem :
i. Binaural Theorem :
For any non -
negative integer IN
, nza
-1121
"
171 't -111)a" th
"
't
" " " ' "
(att ) =
a + a a + . . .
t . .
. + to
=
If (1) a
"-
kfk
-
Binaural coefficient
Question :
Express (text 't IHX 15 as a
polynomial in ×
'
HX15 5) 1+5×+10×2 -110×7+5×4 -1×5 )
'
4- X ) + I =
(I -5×+10×2 - lax -15×4 -
+ + I
= 21 It 10×2+5×4 )
induction
proof by
:
:
a.
• when n = 0
,
II. "kt_
'
4th ) =
I
,
IL) a 111111111=1
True for h = 0
Just to be sure
,
when u =/
(att )
'
= att ,
LEO 1k ) atktk =
4) a
'
b-
°
+ 11 )a°t
'
=
att
i. True for n= I
•
Assume it is true for h=n
¥
k
1k ) and
"
i. lattt =
.
b-
•
sub n = h t I
n
""
(att ) = late ) ( att )
=
4th ) ¥411)a"ktk expand the bracket
get
¥4111 ⇐ 111 antebkti
"'
th
"
-
+
+In-n+IfI
= a
k
an "tk
k
>
LEO (1) 12,1 "zk
"
-
-
=
+
a
-
shift the variables
the same
€ 1L) and € 11,1 and"zk+
"" " k htt
↳ > =
a + b- + b-
,
2 ,
"'
EI,[ (1) II ) )
""
*
= a
"'
+ + ,
an thet
property ②
111=1111 -11nF )
) and"zk
'
£2
htt
= and +
,
I + b-
II 1h11 anti kzk
-
=
1. 1 Real numbers ,
Algebraic manipulations
Number field :
•
Real numbers ,
IR : All numbers I e.
go.li
2 it , } ,
- I - . .
I
•
Integers , 2 : positive ,
negative integers
and zero f- . .
-3 ,
-2 ,
-1
,
a , I ,
2,3 - . .
}
•
Non -
negative integers , IN :
positive integers
and zero
f o
,
I , 2,3 . - .
}
Operations :
•
Add att defined for
- subtract a- to all red a and I
at
Multiply
•
-
Division £ defined for all real a and b- ,
except
tota
Comparisons :
acts a > hi , ast A7T att
, , ,
shorthand :
"
•
a = a. a -
a - . .
a for all real a
,
HE IN and N 70
-
n times
. [ =/ for a f- 0
X x-
y
As when a -_ a
I = a
a× f- is undefined .
Ix I
?
=
× '
i. do is undefined yet
a×
-
i. = a =
I
Summations
I -2
, proof by induction
Summations :
¥2 nlhtll 12h41 )
'
j =
Mtn ) 1+2+3-1 .tn ? ,
i. = . .
6
M In ) = 1+2+3-1 . . . th n
hlhtl )
I
Mint htlh 1) 1- In -21 -1 + I j =
2
j= ,
= -
.
. .
a. Min) = Intl ) -14+111-4+1 ) + . . . -1 ( htt )
n times
hlhtl )
i. Min ) =
2
z
Summation Notation :
II
.
flk )
•
fill + f- 121 + f- 13 ) + . . .
+ flkl is written as ,
.
More
generally we define , for a ,t integer acts
II. aflkl =
flat + flat 't + . .
.
+ fitt
properties
• :
of label
Independent
:
1
eflkl =
IIe till =
¥Ea f- I
' '
k l called
, , is a
dummy
hmm
index
② off
peel terms :
II.afoot =
flkl + fitt
=
flat -1 . .
.
+ fib -
It
3 We can shift the label :
II. f- 1kt =
II ,
fit - thi ) =
f Im -
a) = final + flattest . . .
+ f- latte -
a)
= flat -14111 . - .
=
f- 155-55) + fltb-551
+ f- 114 f- . -
tf 165-55)
, 22
j⇐
' '
't
'
Example pin ) I + + t n
j
=
• : = o . . .
works for
only
Proof by induction :
integers
Suppose we want to
prove that same result is true for all +
integers ,
h .
If it can be shown that the
result holds for n=I and also that ,
for all N71 , if the result is true for n = N , then it must also
be true for n = Ntl , then follows that the result is true for all
positive n
Inland )
:=€=a for
'
Qin )
k equal Intl )
negative integer
=
Prove that pin ) is to :
any non n
Question
-
.
:
i. = stands for a
definition
0
' z
when n =
0 pie) = [ok = a = a
Q a) =
to 1410+1110-11 ) = 0 i. 1710 ) =
Q lol i . True for h= 0
i. Pll) = QU )
True for h =/
Assume it is true for n=N
i. PIN ) = QIN )
when n= Ntl when u = Ntl
PINHI = ET bi
① 1µg =
LNHI 121*1+11 IN -11+1 )
k= I
'
6
= FINI t Will =
to IN -11 ) 12N -13 ) IN -14
'
=
QINI + IN -111
2
=
IN IZNHIINH ) 1- IN -111 i. PINT ) = QI Ntl)
= '
Ntl ) ( IN lsNH ) + IN-111 ) since true for n=o ,
then it is t ru e for all IN
=
INH ) I NL2N-1II -161N -111 )
=
to INH ) I ZN2-1TN -16 )
= to IN -11112 # 3) IN -12 )
General template
:
all E IN
Claim f- In )= gin ) for n
proof : .
show f- lot =
gio )
-
show that for n
any
i
if f- In ) =
gun) ,
then final =
glntl )
1.3 Binomial theorem
Binaural coefficients :
i.
Definition Factorial function : n ! = 1×2×3 × . . . X In 2)
-
X In -
1) X h ,
h E IN
ht
0 ! =
I In -11 ! =
n i . It -11 ! = = I = 0 !
Factorial function is
only defined for -heintegemm
•
•
Factorial function is never zero
Binomial coefficient
2.
Definition :
For all h .
k E IN , h 7k ,
-
h
(1) with)
I
Ck
'
n choose k =
!
=
> .
Pascal 's triangle :
.to/--lnn/-- I
n!
proof :
(2) =
d. in -4 ! =
I
h !
(1) =
h ! In -
n ) !
=
I
i. 141=111
•
properties of Pascal 's
triangle
:
①
symmetry :
121=11 )
, h !
proof :
II ) =
k :( n -
KI !
(Ik )
n !
=
i. in -
ntkl !
=
4-kl ! k !
= II )
② I 1) = III ) + 1hL ) ,
For all in > k -
the number is the sum of two numbers above
k > o
In-1 ) ! In-11 !
Proof :
RHS =
µ -11 ! In -1 htt ) !
-
X k ! In -
I -
kl !
In -11 ! In -
1) !
As In k ) ! (n k-11 ! In k )
=
+
-
-
=
1k -11 ! 1h kl ! k ! In tkl !
-
- -
In -11 ! In -11 !
= +
1k -11 ! In -
tell ! Intel klk-11 ! In k -11 ! -
kin -11 ! + In-11 ! In k ) -
=
klk-11 ! In k -11 !
-
In - k)
= h In -11 !
talk-11 ! In -
k -
1) ! In k -
)
!
%)
n
=
In )
=
I box hand
k ! - k
II
-
The little on the
right
'
of proof
'
side means end
Binaural Theorem :
i. Binaural Theorem :
For any non -
negative integer IN
, nza
-1121
"
171 't -111)a" th
"
't
" " " ' "
(att ) =
a + a a + . . .
t . .
. + to
=
If (1) a
"-
kfk
-
Binaural coefficient
Question :
Express (text 't IHX 15 as a
polynomial in ×
'
HX15 5) 1+5×+10×2 -110×7+5×4 -1×5 )
'
4- X ) + I =
(I -5×+10×2 - lax -15×4 -
+ + I
= 21 It 10×2+5×4 )
induction
proof by
:
:
a.
• when n = 0
,
II. "kt_
'
4th ) =
I
,
IL) a 111111111=1
True for h = 0
Just to be sure
,
when u =/
(att )
'
= att ,
LEO 1k ) atktk =
4) a
'
b-
°
+ 11 )a°t
'
=
att
i. True for n= I
•
Assume it is true for h=n
¥
k
1k ) and
"
i. lattt =
.
b-
•
sub n = h t I
n
""
(att ) = late ) ( att )
=
4th ) ¥411)a"ktk expand the bracket
get
¥4111 ⇐ 111 antebkti
"'
th
"
-
+
+In-n+IfI
= a
k
an "tk
k
>
LEO (1) 12,1 "zk
"
-
-
=
+
a
-
shift the variables
the same
€ 1L) and € 11,1 and"zk+
"" " k htt
↳ > =
a + b- + b-
,
2 ,
"'
EI,[ (1) II ) )
""
*
= a
"'
+ + ,
an thet
property ②
111=1111 -11nF )
) and"zk
'
£2
htt
= and +
,
I + b-
II 1h11 anti kzk
-
=
