Download Representing numbers in different bases

Representing numbers in
different bases
In base r
D = an-1 an-2 … a0 . a-1 a-2 …
In base 10:
N = an-1 * rn-1 + an-2 * rn-2 + … + a0 +
a-1*r-1 + a-2*r-2 + …
Representing numbers in
different bases
Convert:
(0.41)10 to ()4
Representing numbers in
different bases
0.41
= an-1 * 4n-1 + an-2 * 4n-2 + … + a0 +
a-1*4-1
0.41
+ a-2 * 4-2 + …
= a-1*4-1 + a-2* 4-2 + …
=0
Representing numbers in
different bases
0.41
= a-1*4-1
+ a-2 * 4-2 + …
4
1.64
= a-1
+ a-2 * 4-1 + a-3
<1
a-1 = 1
* 4-2
Complement to Base r
Definition:
Number
D
n=4
(1101)2
n=2
(12)10
n digits
xxxxxxxx
m digits
.
r-complement
yyyyyy
2-complement
10-complement
rn - D
24
10000-1101 =0011
102
100-12 = 88
Complement-1 to Base r
Definition:
Number
D
n=3 m=2
(1101.11)2
n=1
(12)10
n digits
xxxxxxxx
m digits
.
yyyyyy
1-complement
9-complement
(r-1) complement
rn-r-m - D
1111.11-1101.11 =0010.00
99-12 = 87
Another representation of 2
complement
BCD Weight: 2n-1
2-complement
Weight:
-2n-1
an-1 an-2 … a0 . a-1 a-2 …
BCD Coding
Two complement
1101 =
- 0011
-23 + 22 + 1
-3
Calculating the r complement
r complement
(r-1) complement
rn-r-m - D
+r-m
rn - D
+1
0011
Number (base 2): 1101
1-complement:
0010
0 in complement to 1
Number
1-complement
00000
11111
Two representations to 0!
Complement to 1 vs. 2
Calculation
Zero preserntation
1-Complement 2-Complement
Easy
Harder
Dual
Singe
We usually use 2-complement
Subtraction using 1-complement
M–N
M>N-1
Carry
M + 2n-N-1 = 2n+(M-N-1)
M<N-1
<0
>0
2n+(M-N-1)
2n+(M-N-1)
Carry exists
Add it to the result
No Carry
Take the complement and put (-)
-[ 2n – (2n+(M-N-1)) -1 ]
(M-N)
-(N-M)
Example I
3
-5
No Carry
0011
+1010
101
-010 = -2
Example II
3
-2
Carry
1
011
+101
000
1
001
Changing number of bits
Given a number in 2 complement with n bits
What is the representation with m>n bits ?
Changing number of bits
0011
1011
00 0011
11 1011
Binary Multiplication
1101
X 0011
1101
1101
100111
13
X 03
39
2-Complement multiplication
-3
X
5
Carry
1
1101
X
0101
111101
00000
1101
110001
2-Complement multiplication
-3
X
-5
1101
X
1011
?????
2-Complement multiplication
-3
X
-5
Remember:
Last digit has
negative weight
1101
X
1011
1111101
111101
00000
0011
0001111
=15