Page 72 - Engineering Digital Design

2.6 SIGNED BINARY NUMBERS 43

0.464 x 2 0.928 0
0.928 x 2 0.856 1
0.856x2 0.712 1
0.712x2 0.424 1 0.654 10 = 0.10100111 2
0.424 x 2 0.848 0 £ max = 2~ 8
EXAMPLE 2.8 0.654i 0 ->• W 8 terminated at 4 digits:

•N s xr F I
0.654 x 8 0.232 5
0.232 x 8 0.856 1 0.654 10 = 0.5166 8
0.856x8 0.848 6 with error bounds
4
3
0.848 x 8 0.784 6 0 < e < 7 x 8~ = 1.71 x 10~ by Eq. (2.9)
EXAMPLE 2.9 Let 0.5166 8 -»• N 2 be rounded to 8 bits and let 0.5166 8 ->• N\Q be rounded
to 4 decimal places:

2
1
3
0.5166 8 = 5 x 8" + 1 x 8~ + 6 x 8~ + 6 x 8~ 4
= 0.625000 + 0.015625 + 0.011718 + 0.001465
= 0.6538io rounded to 4 decimal places; e 10 < 10~ 4

•N s x r F I
0.6538 x 2 0.3076 1
0.3076 x 2 0.6152 0
0.6152 x 2 0.2304 1
0.2304 x 2 0.4608 0
0.4608 x 2 0.9216 0
0.9216 x 2 0.8432 1
0.8432 x 2 0.6864 1
0.6864 x 2 0.3728 1 0.5166 8 = 0.10100111 2 (compare with Example 2.7)
8
4
0.3728 x 2 0.7457 0 e 10 < 10~ + 2~ = 0.0040
EXAMPLE 2.10 0.10100111 2 -» WBCH
•A 1
0.10100111 2 =0.1010 0111 =O.A7 BC H

2.6 SIGNED BINARY NUMBERS

To this point only unsigned numbers (assumed to be positive) have been considered. How-
ever, both positive and negative numbers must be used in computers. Several schemes have
been devised for dealing with negative numbers in computers, but only four are commonly
used:

• Signed-magnitude representation
• Radix complement representation