1.1 Digital Systems
1.2 Binary Numbers
1.3 Number‐Base Conversions
1.5 Complements of Numbers
1.6 Signed Binary Numbers
1.7 Binary Codes
1.8 Binary Storage and Registers
1.9 Binary Logic

1.3 N U M B E R ‐ B A S E  C O N V E R S I O N S

Representations of a number in a different radix are said to be equivalent if they have
the same decimal representation. For example, (0011) 8 and (1001) 2 are equivalent—both
have decimal value 9. The conversion of a number in base r to decimal is done by
expanding the number in a power series and adding all the terms as shown previously.
We now present a general procedure for the reverse operation of converting a decimal
number to a number in base r. If the number includes a radix point, it is necessary to
separate the number into an integer part and a fraction part, since each part must be
converted differently. The conversion of a decimal integer to a number in base r is done
by dividing the number and all successive quotients by r and accumulating the remain-
ders. This procedure is best illustrated by example.
EXAMPLE 1.1
Convert decimal 41 to binary. First, 41 is divided by 2 to give an integer quotient of 20
and a remainder of 1
2. Then the quotient is again divided by 2 to give a new quotient and
remainder. The process is continued until the integer quotient becomes 0. The coefficients
of the desired binary number are obtained from the remainders as follows:
Integer
Quotient
Remainder Coefficient
41>2 = 20 + 1
2
a0 = 1
20>2 = 10 + 0 a1 = 0
10>2 = 5 + 0 a2 = 0
5>2 = 2 + 1
2
a3 = 1
2>2 = 1 + 0 a4 = 0
1>2 = 0 + 1
2
a5 = 1
Therefore, the answer is (41)10 = (a5a4a3a2a1a0)2 = (101001)2.