Number Systems

                   The study of number systems is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a digital computer. It is one of the most basic topics in digital electronics. In this chapter we will discuss different number systems
commonly used to represent data. We will begin the discussion with the decimal number system.
Although it is not important from the viewpoint of digital electronics, a brief outline of this will be
given to explain some of the underlying concepts used in other number systems. This will then be
followed by the more commonly used number systems such as the binary, octal and hexadecimal
number systems.

1.1 Analogue Versus Digital :

                      

                 There are two basic ways of representing the numerical values of the various physical quantities with which we constantly deal in our day-to-day lives. One of the ways, referred to as analogue, is to express the numerical value of the quantity as a continuous range of values between the two expected
extreme values. For example, the temperature of an oven settable anywhere from 0 to 100 °C may be
measured to be 65 °C or 64.96 °C or 64.958 °C or even 64.9579 °C and so on, depending upon the
accuracy of the measuring instrument. Similarly, voltage across a certain component in an electronic
circuit may be measured as 6.5 V or 6.49 V or 6.487 V or 6.4869 V. The underlying concept in this
mode of representation is that variation in the numerical value of the quantity is continuous and could
have any of the infinite theoretically possible values between the two extremes.
The other possible way, referred to as digital, represents the numerical value of the quantity in steps
of discrete values. The numerical values are mostly represented using binary numbers. For example,
the temperature of the oven may be represented in steps of 1 °C as 64 °C, 65 °C, 66 °C and so on.
To summarize, while an analogue representation gives a continuous output, a digital representation
produces a discrete output. Analogue systems contain devices that process or work on various physical
quantities represented in analogue form. Digital systems contain devices that process the physical
quantities represented in digital form.

               Digital techniques and systems have the advantages of being relatively much easier to design and
having higher accuracy, programmability, noise immunity, easier storage of data and ease of fabrication
in integrated circuit form, leading to availability of more complex functions in a smaller size. The
real world, however, is analogue. Most physical quantities – position, velocity, acceleration, force,
pressure, temperature and flowrate, for example – are analogue in nature. That is why analogue
variables representing these quantities need to be digitized or discretized at the input if we want to
benefit from the features and facilities that come with the use of digital techniques. In a typical system
dealing with analogue inputs and outputs, analogue variables are digitized at the input with the help
of an analogue-to-digital converter block and reconverted back to analogue form at the output using a
digital-to-analogue converter block. Analogue-to-digital and digital-to-analogue converter circuits are
discussed at length in the latter part of the book. In the following sections we will discuss various
number systems commonly used for digital representation of data.