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.