Page 9 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes own value or weight expressed as a power of 2. This is illustrated in Figure 1.2. Here, places to
the left of the binary point (Counterpart of the decimal point) are positive powers of 2 and places
to the right are negative powers of 2.
Figure 1.2: Binary Position Values as Powers of 2

The number 1011.101 is shown represented in the Figure 1.2. To find its equivalent in the decimal
system we simply take the sum of the products of each digit value (0 or 1) and its positional value:
1011.101 = (1 * 2 ) + (0 * 2 ) + (l * 2 ) + (1 * 2 )
1
2
3
0
2
+ (0 * 2 ) + (0 * 2 ) + (1 * 2 )
–1
–2
–3
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625 10
Notice in the preceding operation that subscripts (2 and 10) were used to indicate the base in
which the particular number is expressed. This convention is used to avoid confusion whenever
more than one number system is being employed.
In the binary system, the term binary digit is often abbreviated to the term bit, which we
will use henceforth. Thus, in the number expressed in Figure 1.2 there are 4 bits to the left
of the binary point, representing the integer part of the number, and 3 bits to the right of the
binary point, representing the fractional part. The most significant bit (MSB) is the leftmost bit
(largest weight). The least significant bit. (LSB) is the rightmost bit (smallest weight). These
are indicated in Figure 1.2.
Binary Counting: When we deal with binary numbers, we will usually be restricted to a specific
number of bits. This restriction is based on the circuitry that is used to represent these binary
numbers. Let us use 4-bit binary numbers to illustrate the method for counting in binary.
The sequence (shown in Figure 1.3) begins with all bits at 0; this is called the zero count. For each
successive count, the units (2 ) position toggles; that is, it changes from one binary value to the
0
other. Each time the units bit changes from a 1 to a 0, the twos (2 ) position will toggle (change
1
states). Each time the two’s position changes from 1 to 0, the fours (2 ) position will toggle (change
2
states). Likewise each time the fours position goes from 1 to 0, the eights (2 ) position toggles.
3
This same process would be continued for the higher-order bit positions if the binary number
had more than 4-bits.

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