082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

12 Chapter 1 Review of algebraic techniques
Convertingfrombinarytodecimal
Considerthebinarynumber110101 2
.Asthebaseis2thismeansthataswemovefrom
right toleft the position of each digitrepresents an increasing power of2asfollows:
110101 2
= 1(2 5 ) +1(2 4 ) +0(2 3 ) +1(2 2 ) +0(2 1 ) +1(2 0 )
= 1(32) +1(16) +0(8) +1(4) +0(2) +1(1)
=32+16+4+1
= 53 10
Hence 110101 2
and 53 10
areequivalent.
Example1.12 Convert the following todecimal: (a) 1111 2
(b)101010 2
Solution (a) 1111 2
= 1(2 3 ) +1(2 2 ) +1(2 1 ) +1(2 0 )
= 1(8) +1(4) +1(2) +1(1)
=8+4+2+1
= 15 10
(b) 101010 2
= 1(2 5 ) +0(2 4 ) +1(2 3 ) +0(2 2 ) +1(2 1 ) +0(2 0 )
=1(32)+0+1(8)+0+1(2)+0
=32+8+2
= 42 10
Convertingdecimaltobinary
Wenowlookatsomeexamplesofconvertingnumbersinbase10tonumbersinbase2,
that is from decimal to binary. We make use of Table 1.1, which shows various powers
of 2,when converting from decimal tobinary. Table 1.1 may be extended as necessary.
Table1.1
Powers of2.
2 0 1 2 4 16 2 8 256
2 1 2 2 5 32 2 9 512
2 2 4 2 6 64 2 10 1024
2 3 8 2 7 128 2 11 2048
Example1.13 Convert 83 10
toabinary number.
Solution We need to express 83 10
as the sum of a set of numbers, each of which is a power of 2.
From Table 1.1 we see that 64 is the highest number in the table that does not exceed
the given number of 83. We write
83=64+19
Wenowfocusonthe19.FromTable1.1,16isthehighestnumberthatdoesnotexceed
19. So wewrite
19=16+3