General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Problems of Sign-Bit Systems
Generally: An n-bit signed binary number system
allows to represent the integers from −2 n−1 + 1 to
+2 n−1 − 1.
2 n−1 −1 positive numbers, 2 n−1 −1 negative numbers,
and the zero
Thus we represent
#({〈〈s〉〉 − | s ∈ B n }) = 2 · (2 n−1 − 1) + 1 = 2 n − 1
numbers all in all
One number must be represented twice
(But there are 2 n strings of length n.)
10 . . . 0 and 00 . . . 0 both represent the zero as
−1 · 0 = 1 · 0.
signed binary Z
0 1 1 1 7
0 1 1 0 6
0 1 0 1 5
0 1 0 0 4
0 0 1 1 3
0 0 1 0 2
0 0 0 1 1
0 0 0 0 0
1 0 0 0 -0
1 0 0 1 -1
1 0 1 0 -2
1 0 1 1 -3
1 1 0 0 -4
1 1 0 1 -5
1 1 1 0 -6
1 1 1 1 -7
We could build arithmetic circuits using this, but there is a more elegant way!
c○: Michael Kohlhase 252
All of these problems could be dealt with in principle, but together they form a nuisance, that at
least prompts us to look for something more elegant. The two’s complement representation also
uses a sign bit, but arranges the lower part of the table in the last slide in the opposite order,
freeing the negative representation of the zero. The technical trick here is to use the sign bit (we
still have to take into account the width n of the representation) not as a mirror, but to translate
the positive representation by subtracting 2 n .
The Two’s Complement Number System
Definition 428 Given the binary string
a = 〈an, . . . , a0〉 ∈ Bn+1 , where n > 1. The
integer represented by a in the (n + 1)-bit two’s
complement, written as 〈〈a〉〉 2s
n , is defined as
〈〈a〉〉 2s
n = −an · 2 n + 〈〈an−1, . . . , a0〉〉
= −an · 2 n n−1
+ ai · 2 i
i=0
Notation 429 Write B 2s
n (z) for the binary string
that represents z in the two’s complement number
system, i.e., 〈〈B 2s
n (z)〉〉 2s
n = z.
c○: Michael Kohlhase 253
2’s compl. Z
0 1 1 1 7
0 1 1 0 6
0 1 0 1 5
0 1 0 0 4
0 0 1 1 3
0 0 1 0 2
0 0 0 1 1
0 0 0 0 0
1 1 1 1 -1
1 1 1 0 -2
1 1 0 1 -3
1 1 0 0 -4
1 0 1 1 -5
1 0 1 0 -6
1 0 0 1 -7
1 0 0 0 -8
We will see that this representation has much better properties than the naive sign-bit representation
we experimented with above. The first set of properties are quite trivial, they just formalize
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