We should all be familiar with the decimal (or base-ten) number system. Recall that each digit within any given decimal number is associated with a weight. Furthermore, the value of that number is the sum of the digits after each has been multiplied by its weight.
To illustrate let us consider Table 2-1 and assume that the number 654.52, written (654.52)10 to specify base 10, is being represented. Note that the digits range form 0,1,…,9.
Hundreds |
Tens |
Units |
One tenth |
One hundredth |
||
Weights |
102 |
101 |
100 |
10-1 |
10-2 |
|
Digits |
6 |
5 |
4 |
5 |
2 |
|
Weighted Value |
600 |
50 |
4 |
0.5 |
0.02 |
Total 654.52 |
Table 2-1 Decimal Number System
Just as the decimal system with its ten digits is a base-ten system, the binary number system with its 2 digits, ‘0’ and ‘1’ is a base-two system. Table 2-2 shows the weighting for the binary number system up to 2 decimal places before and 3 places after the binary point (.) note the similarity to the decimal system.
Weights |
22 |
21 |
20 |
. |
2-1 |
2-2 |
Table 2-2 Binary Weights
 | The least significant bit (LSB) is the rightmost binary digit which has the lowest binary weight of a given number. |
| The most significant bit (MSB) is the leftmost binary digit which has the highest binary weight of a given number. |
| Counting in binary is similar to the decimal number system. The LSB is begins with zero (0) and is incremented until the maximum digit value is reached. The adjacent bit positions are then filled appropriately as the iterative counting process continues. |
To express the value of a given binary number as its decimal equivalent we just need to sum the digits after each has been multiplied by its associated weight.
Convert (110101) 2 to decimal form.
Binary Weights |
25 |
24 |
23 |
22 |
21 |
20 |
|
Weight Value |
32 |
16 |
8 |
4 |
2 |
1 |
|
Binary Number |
1 |
1 |
0 |
1 |
0 |
1 |
|
Decimal Value |
32 |
16 |
0 |
4 |
0 |
1 |
Total (53)10 |
Convert (0.1011) 2 to decimal form.
Binary
Weights
2-1
2-2
2-3
2-4
Weight
Value
0.5
0.25
0.125
0.0625
Binary
Number
1
0
1
1
Decimal
Value
0.5
0
0.125
0.0625
Total
(0.6875)10
One way to find the binary number that is equivalent to a given decimal number is to determine the set of binary weights whose sum is equal to the decimal number. For example (24) 10 is equivalent to 24 + 23 i.e. 16 +8, as illustrated below. Thus by placing a one in these weigh positions and zeros elsewhere we realise that (24) 10 = 1*24 + 1*23 + 0*22 + 0*21 + 0*20. Extracting the coefficients of the 2x terms we obtain
(24) 10 = (11000) 2. This is referred to as the sum-of-weights method.
Binary Weights
24
23
22
21
20
Weight
Value
16
8
4
2
1
Binary
Number
1
1
0
0
0
Alternatively, we may use the more systematic approach called the Repeated-Division-by-2 method shown in the example below.
Converting (18) 10 to binary
Divide the quotient by two and record the remainder.
Repeat step (a) until the quotient is equal to zero (0).
The first remainder produced is the LSB in the binary number and the last remainder (R) the MSB. Accordingly, the binary number is then written (from left to right) with the MSB occurring first
2
18
2
9
R 0 (LSB)
2
4
R 1
2
2
R 0
2
1
R 0
0
R 1 (MSB)
Therefore, (18) 10 = (10010) 2
The techniques used to convert decimal fractions to binary are similar to the methods demonstrated previously to convert decimal whole numbers. We may either use the sum-of –weights method or in this case, the repeated multiplication–by-2 method. In the multiplication–by-2 method we repeatedly multiply the fraction by two, and record the carry, until the fraction product is zero. The first carry produced us the MSB, while the last carry is the LSM. Remember that the binary point precedes the MSB. To illustrate lets consider the conversion of (0.3125) 10 to binary.
0.3125 * 2
Carry
0.625 * 2
0 (MSB)
0.25 * 2
1
0.50 * 2
0
0.00
1 (LSB)
(0.3125) 10 = (0.0101) 2
Copyright © Adrian Als , 1999
This page was last modified: Wednesday, April 12, 2000
