Common Rules For Weighted Number Systems
Technicians often encounter values expressed in binary, octal, or even the hexadecimal number systems. In our fifth installment of The Practicing Technician’s Series, today’s discussion focuses on characteristics that are common to all of these number systems.
The first thing to understand is that these are all weighted number systems which use columns of digits. As with the decimal system, the names of these other number systems denote the base or “radix” of the system. The base of a number system can be described as the number of states that can be represented by a single digit or column. For instance, the base 10 “decimal” system can represent 10 distinct states in a single digit place holder using symbols 0 – 9. Each of these number systems behave the same way with respect to the weighting of the columns used to represent groups when expressing multi-digit values. As you add columns to the left of the least significant digit, you increase the exponent you are raising the radix to by one.
What is meant by Position Weighting ?
X^{3} | X^{2} | X^{1} | X^{0} | |
Binary (base 2) | 8 | 4 | 2 | 1 |
Octal (base 8) | 512 | 64 | 8 | 1 |
Decimal (base 10) | 1000 | 100 | 10 | 1 |
Hexadecimal (base 16) | 4096 | 256 | 16 | 1 |
When evaluating a value in a given number system, the above table illustrates the weight of the digits in each column for the given number system. We work with the decimal system daily, and it follows these rules as well. The decimal value 372, is evaluated as 3 groups of 100, 7 groups of 10, and 2 groups of 1. We apply this same general rule to all the above systems. For the octal value of 372, we would have 3 groups of 64, 7 groups of 8, and 2 groups of 1.
The impact of shifting digits to the left and right
As a technician in the increasingly digital age, you may often come across the use of the binary or base 2 number system. As a consequence of the above stated nature of weighted number systems, shifting a given value to the left is the same as multiplying that value by the radix of the number system. Shifting a binary value to the left is the same as multiplying it by 2. Although most often used with binary values, this shifting as a form of multiplication is common to all weighted number systems. Conversely, if a binary value is shifted to the right, it is the same as dividing the value by 2 in the case of binary, or the radix of any other positional weighted number system. Again, we are already familiar with this fact, if not consciously, by our extensive usage of the decimal system in everyday base 10 calculations. It is important to remember that this same basic principal can easily be applied to octal or hexadecimal values as well despite our unfamiliarity with these special purpose number systems.
The Number of available states for a given number of digits
Weighted number systems have other common properties. To calculate the total number of states that can be represented by a given number of digits in a particular number system, we simply raise the radix of the system, to the power of the number of digits we are using. The resulting value represents how many distinct states that are available with that many digits in that particular number system.
The Highest value for a given number of digits
To calculate the highest value that can be represented by a given number of digits in a particular number system, we simply subtract one from the calculation for available states. This is due to the fact that in each system, “0” is one of the states represented.
Working with different weighted number systems may seem a bit intimidating until you remember that these alternate base number systems follow the same rules as the decimal number system which we work with every day.
If you liked this post, check out our previous articles in The Practicing Technician’s Series;
Using the Natural Log or “ln” Function in Circuit Analysis
How to Create Correct Ohm's Law KCL Branch Equations for Nodal Analysis
How to Solve Simultaneous Equations with Multiple Unknowns
Converting Parallel RL Circuits to their “Easier To Work With” Series Equivalents
We hope this has been helpful to you as a practicing or student technician. We are looking for other ideas for this continuing Practicing Technician’s Series. Please let us know what you would like us to write about by sending us your thoughts and questions at support@gbctechtraining.com.