1. Chapter 2 Representing and Manipulating Information

2. Figure 2.11: Unsigned number examples for w=4. unsigned-values.ppt
23 = 8
22 = 4
21 = 2
20 = 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
[0001]
[0101]
[1011]
[1111]
Figure 2.11: Unsigned number examples for w=4.
unsigned-values.ppt

3. Figure 2.12: two's-complement number examples for w=4.
– 23 = –8
22 = 4
21 = 2
20 = 1 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8
[0001]
[0101]
[1011]
[1111]
Figure 2.12: two's-complement number examples for w=4.
twocomp-values.ppt

4. – 23 = –8
23 = 8
22 = 4
21 = 2
20 = 1 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
[1011]
+16
[1111]
+16
Figure 2.15: Comparing unsigned and two's-complement representations for w=4.
unsigned-twocomp-values.ppt

5. 2w–1 2w +2w–1 –2w–1 Unsigned Two’s complement
2w–1 2w
+2w–1
–2w–1
Unsigned
Two’s
complement
Figure 2.16: Conversion from two's complement to unsigned.
t2u.ppt

6. 2w 2w–1 +2w–1 –2w–1 Unsigned Two’s complement
–2w–1
Figure 2.17: Conversion from unsigned to two's complement.
u2t.ppt

7. Figure 2.19: Examples of sign extension from w=3 to w=4.
– 23 = –8
– 22 = –4
22 = 4
21 = 2
20 = 1 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8
[101]
[1101]
[111]
[1111]
Figure 2.19: Examples of sign extension from w=3 to w=4.
extend-values.ppt

8. 2w+1 2w x + y x +u y Overflow Normal
Figure 2.21: Relation between integer addition and unsigned addition.
uadd-ovf.ppt
Normal

9. +2w +2w –1 +2w –1 –2w –1 –2w –1 –2w x + y Case 4 x +t y Case 3 Case 2
Positive overflow
Case 4
x +t y
+2w –1
+2w –1
Case 3
Normal
Case 2
–2w –1
–2w –1
Case 1
Negative overflow
–2w
Figure 2.23: Relation between integer and two's-complement addition.
tadd-ovf.ppt

10. 2m 2m–1 4 • • • 2 1 bm bm–1 • • • b2 b1 b0 . b–1 b–2 b–3 • • • b–n+1
1/2
1/4
• • •
1/8
Figure 2.30: Fractional binary representation.
fractional-binary.ppt
1/2n–1
1/2n

11. Figure 2.31: Standard floating-point formats. fp-formats.ppt
Single precision 31 30 23 22 s
exp
frac
Double precision 63 62 52 51 32 s
exp
frac (51:32) 31
frac (31:0)
Figure 2.31: Standard floating-point formats.
fp-formats.ppt

12. 1. Normalized s
 0 &  255 f
2. Denormalized s f
3a. Infinity s 1 1 1 1 1 1 1 1
3b. NaN s 1 1 1 1 1 1 1 1
 0
Figure 2.32: Categories of single-precision, floating-point values.
fp-cases.ppt

13. Figure 2.33A: Representable values for six-bit floating-point format.
fp-values.ppt

14. –
–10 –5 +5
+10 +
Denormalized
Normalized
Infinity –1
–0.8
–0.6
–0.4
–0.2
+0.2
+0.4
+0.6
+0.8 +1
Denormalized
Normalized
Infinity +0 –0
Figure 2.33B: Representable values for six-bit floating-point format.
fp-values.ppt