Higher Engineering Mathematics, Sixth Edition

92 Higher Engineering Mathematics

Using Table 10.1 to convert this binary number to an

octal number gives 363. 428 and 363. (^428)
= 3 × 82 + 6 × 81 + 3 × 80 + 4 × 8 −^1 + 2 × 8 −^2
= 192 + 48 + 3 + 0. 5 + 0. 03125
=243.53125 10
Now try the following exercise
Exercise 41 Further problems on
conversion between decimal and binary
numbers via octal
In Problems 1 to 3, convert the decimal numbers
given to binary numbers, via octal.

1. (a) 343 (b) 572 (c) 1265
   [

(a) 1010101112 (b) (^10001111002)
(c) (^100111100012)
]

2. (a) 0.46875 (b) 0.6875 (c) 0.71875
   [

(a) 0. 011112 (b) 0. (^10112)
(c) 0. (^101112)
]

3. (a) 247.09375 (b) 514.4375 (c) 1716.78125
   ⎡
   ⎢
   ⎣

(a) 11110111. (^000112)
(b) 1000000010. (^01112)
(c) 11010110100. (^110012)
⎤
⎥
⎦

4. Convert the binary numbers given to decimal
   numbers via octal.
   (a) 111.011 1 (b) 101 001.01
   (c) 1 110 011 011 010.001 1
   [

(a) 7. 437510 (b) 41. (^2510)
(c) 7386. (^187510)
]

10.4 Hexadecimal numbers

The hexadecimal system is particularly important in

computer programming, since four bits (each consist-

ing of a one or zero) can be succinctly expressed using

a singlehexadecimal digit.Two hexadecimal digitsrep-

resent numbers from 0 to 255, a common range used,

for example, to specify colours. Thus, in the HTML

language of the web, colours are specified using three

pairs of hexadecimal digits RRGGBB, where RR is the

amount of red, GG the amount of green, and BB the

amount of blue.

Ahexadecimal numbering systemhas a radix of

16 and uses the following 16 distinct digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F

‘A’ corresponds to 10 in the decimal system, B to 11,

C to 12, and so on.

(a) Converting from hexadecimal to decimal:

For example

1A 16 =^1 ×^161 +A×^160

= 1 × 161 + 10 × 1

= 16 + 10 = 26

i.e. 1A 16 = (^2610)
Similarly, 2E 16 = 2 × 161 +E× 160
= 2 × 161 + 14 × 160
= 32 + 14 = (^4610)
and 1BF 16 = 1 × 162 +B× 161 +F× 160
= 1 × 162 + 11 × 161 + 15 × 160
= 256 + 176 + 15 = (^44710)
Table 10.2 compares decimal, binary, octal and hexa-
decimal numbers and shows, for example, that
2310 = 101112 = 278 = (^1716)
Problem 15. Convert the following hexadecimal
numbers into their decimal equivalents:
(a) 7A 16 (b) 3F 16
(a)7A 16 = 7 × 161 +A× 160 = 7 × 16 + 10 × 1
= 112 + 10 = 122
Thus7A 16 = (^12210)
(b)3F 16 = 3 × 161 +F× 160 = 3 × 16 + 15 × 1
= 48 + 15 = 63
Thus3F 16 = (^6310)