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1.3 Number Base Conversation in Computer by KShare

1.1 Digital Systems
1.2 Binary Numbers
1.3 Number‐Base Conversions
1.4 Octal and Hexadecimal Numbers
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.

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