binary system applications

Now that we know the basics of Binary Number system and the basics of Boolean Logic we can concentrate over the application part of Binary Numbers.

Binary Adder

The simplest of all application is the Binary Number Adder. It adds two binary numbers and yields a binary result.

It basically produces the sum of two numbers (A and B) and takes care of the carry (if any). The following table shows the Truth Table of the Binary Numbers-Mathematically, a Full Adder’s equation can be written as:

[(A XOR B) XOR Cin]

[(A.B) + (Cin.(A XOR B)]

A Binary (Full) Adder can be constructed using various AND & OR gates. The circuit diagram is as follows:

Diagrams Showing Circuit Diagram of a Full Adder and Block Diagram working of a Full Adder

The output is obtained at two separate output pins- S (Sum) and Cout (Carry Out). If Cout is High (has a True Value;1) then this means the addition of two Single Bit numbers have resulted into 3rd Bit (as is evident from the Truth Table).

Similarly we can construct a Full Subtractor by simply inverting one of the inputs (using NOT gate) and by feeding them to the given circuit.

Binary Multiplier

Binary Multipliers have several uses and it has paved way to more complex systems like computers and other arithmetic systems.

We are going to discuss a 2 bit by 2 bit binary multiplier. The primary design of a multiplier consists basically of adders only.

Using long multiplication, a product of two N-bit numbers can be expressed as the sum of N N-bit partial products, which are then added to produce a 2N-bit product.

The partial products can be computed from the fact that ai – bj = ai AND bj.

The complexity of the multiplier is in adding the partial products.

In most of the complex adders the Partial products are added in pairs using several binary adders. The following diagram shows how a 2 bit by 2 bit Binary multiplier can be realized:

Figure showing Circuit Diagram of a 2 bit by 2 bit Binary Multiplier

The circuit above uses IC 74283 which is a 4 Bit Adder.

Each bit of one of the number’s is multiplied with the each bit of the other number using an AND gate on by one. The product of each output is shifted 1 bit ahead (excluding the result from the first product). The resulting products are then later added using a 4 Bit Adder. And we get the result.

We are not going to discuss the multiplier in more detail because its working is at this stage beyond or comprehension.

Digital Binary Clock

One of the many interesting applications of Binary Numbers is a digital clock.

A Digital Binary Clock consists of LEDs arranged separately for Hours, Minutes and Seconds as shown below-

The three components of Hours, Minutes and Seconds are separately controlled by timing circuits.

A LED in ON (lighted) state symbolizes a Binary 1 and on the contrary a LED in OFF state symbolizes a Binary 0.

There are three separate adders for each three components. The circuit diagram is built in such a manner that after each second, the adder attached with component depicting seconds is incremented by 1. If this addition has resulted in any carry then it is forwarded to the Minutes component. The adders of all three components are built in such a manner that the total result is never exceeded by 60 (in the case of Hour’s component it is 24), if it exceeds the adder results in a carry and the sum is reset to 0. This carry is shifted to the just next in line adder. For example if the second’s adder resulted in a carry, the carry is given to the minute’s adder. However the case of Hour’s adder the carry is neglected and the count is simply set to 0.

The corresponding output is then sent to the LED’s displaying seconds as a Binary Output and we get the current time in Binary Numbers.

Conclusion

In this Term Paper we have discussed briefly about the Binary Numbers and their applications. The modern day applications are far more complex than the ones discussed in this Term Paper. They involve very complex circuits and can do a variety of jobs. There are devices like computers, mobile phones, video games handhelds, etc. which are far more complex but all work on these principles.

By no means can we underestimate the power of this number system. All the modern advances and even the future technologies will be based on this number system only.

In the end, I would only like to say that we have only made a small step towards a huge world of knowledge and science waiting to be explored.