4-Bit Computer

Description: This article provides a simple introduction to binary and logic, explaining how they are interconnected with electronics to create both simple and complex computers. It highlights that digital computers represent two states—on and off, or zero and one—thus counting in base 2 rather than base 10. In base 2, numbers are counted in ones, twos, fours, eights, etc. For example, the number 2 in base 10 is represented as 10 in base 2. The article discusses how to represent addition in base 10 using a truth table and introduces Boolean operations, commonly known as logic gates in electronics. Three basic types of logic gates are described: the AND gate, which outputs one only when both inputs are one; the OR gate, which outputs one when either input is one; and the NOT gate (or inverter), which outputs the opposite of its input. The logic required for a half-adder is explained, which can compute the sum of two single-bit numbers, and how two half-adders can be combined to form a full-adder, capable of processing inputs A, B, and a carry bit. The article also describes a four-bit adder, known as a ripple carry adder, which can add larger binary numbers. Finally, it outlines how logic gates can be constructed using transistors, detailing the operation of NOT and AND gates, and provides a visual representation of a completed full-adder circuit.

The operation of electronic devices such as computers relies heavily on binary counting and logic operations. Digital computers utilize a binary system, which consists of only two states: on (1) and off (0). This binary representation allows for simple yet effective computation, as it simplifies the design of electronic circuits. The transition from base 10 to base 2 is crucial, as it dictates how numbers are represented and manipulated within the computer’s architecture.

To illustrate the addition process in binary, truth tables serve as a fundamental tool. A truth table outlines all possible input combinations and their corresponding outputs, providing a clear reference for logic operations. For instance, when adding two single-bit binary numbers, the outputs can be derived using Boolean algebra, which is the mathematical foundation for digital logic design.

Logic gates, the building blocks of digital circuits, perform basic logical functions that are essential for computation. The AND gate requires both inputs to be high (1) for the output to be high, while the OR gate requires at least one input to be high. The NOT gate inverts the input signal, providing a crucial function in creating more complex circuits.

The half-adder circuit is a significant development in binary addition, as it can compute the sum and carry of two single-bit inputs. However, to handle larger numbers, multiple half-adders can be cascaded into a full-adder configuration. A full-adder accepts two inputs and a carry-in, producing a sum and a carry-out, which allows for the addition of multi-bit binary numbers.

The ripple carry adder is an extension of this concept, designed to add two 4-bit binary numbers. The carry-out from one full-adder feeds into the carry-in of the next, creating a chain reaction that processes the carry bits across the entire adder. This method, while straightforward, can introduce delays due to the sequential nature of carry propagation.

Transistor-based implementations of logic gates demonstrate the practical application of these concepts. The NOT gate, for instance, utilizes a single transistor to control the flow of current, effectively inverting the input signal. The AND gate requires two transistors, ensuring that current flows to the output only when both inputs are high.

In conclusion, the synthesis of logic gates and the structured approach to binary arithmetic form the foundation of digital electronics. Understanding these principles is vital for designing complex circuits and systems that underpin modern computing technology.If you`ve ever wondered how electronic devices like computers can count, this article gives a simple introduction to binary and logic and shows how they are tied together with electronics to make both simple and complex computers. Please note: There are a couple of mistakes on the slides in the youtube video which are shown corrected below.

Thanks to reader [Veini] for taking the time to review the logic diagrams! To begin with let`s have a look at some fundamentals. Since digital computers can only represent two states, on and off or zero and one, there are only two numbers available; therefore they have to count in base 2, not base 10 as we would do. However, it`s very similar, instead of ones, tens, hundreds and thousands, base 2 counts in ones, twos, fours, eights and so on.

So, for example 2 in base 10 is one- zero in base 2. When we add numbers in base ten we carry over any digits which are greater than 9 into the next magnitude of units, so nine plus one equals zero carry one, or ten. It is useful to represent this in what`s known as a truth table. Here you can see the 4 possible values of the inputs A and B, and the four possible outputs represented by Sum and Carry.

In order to represent the logic` required to get from the possible range of inputs to the desired outputs we use Boolean operations, or as they are more commonly called in electronics, logic gates. Here are the three basic types of logic gates which I`ve chosen because they are the simplest gates to make from transistors.

You can make all other types of gates by combining these three. An AND gate outputs one only when both its inputs are one. An OR gate outputs one when either input is one. Finally a NOT gate (or inverter as it is sometimes called) outputs the opposite of its input, so if the input is one the output is zero and vice-versa. So, going back to our truth table, let`s look at the logic required to get Sum` based on the inputs A and B.

Here we want the logic to output one only when one input is one and not the other, this is known as an exclusive OR gate. We can do this by simply using two AND gates with NOT gates on opposing inputs. If either gate outputs a 1 the result is 1 via the final OR gate. To get the whole truth table we simply add the two logic circuits together. This logic is called a half-adder` due to the fact that it is only capable of working on single bit numbers, since you cannot input the carry bit, you can`t cascade them together to work on larger binary numbers.

To solve this we combine two half-adders together to make a full-adder. This logic takes A, B and a carry as input and outputs the sum and carry. If you followed along with the half adder it`s pretty easy to see how this works from the logic diagram. Now the simple full-adder logic circuits can be combined to allow bigger binary numbers to be added together.

This picture shows a four bit adder, in fact, due to the way the carry bit ripples` down, this is known as a ripple carry adder. Since both the A and B inputs are now 4 bits we can add together 1111 and 1111 or 15 plus 15 in base 10 to get a five bit result.

Now let`s take a quick look at how we build logic gates using transistors. First up is the NOT gate. Here if the input is 1 it causes the electricity to flow from the collector to the emitter (top to bottom). Since the electricity will always follow the path of least resistance the output will be zero. If the input is zero, the transistor prevents the flow from collector to emitter, so the electricity flows out of the output causing it to be one.

Next up is the AND gate. This requires two transistors, the inputs are on the bases and only if both inputs are one can the electricity flow to the output, making it a one also. Here you can see a picture of a completed full-adder, the A, B and carry inputs are on the left and the sum and carry outputs are on the right.

By making more of these it is possi