Sequential and Combinational Circuits

A Sequential circuit combinational logic circuit that consists of inputs variable (X), logic gates (Computational circuit), and output variable (Z).

Combinational circuit produces an output based on input variable only, but Sequential circuit produces an output based on current input and previous input variables. That means sequential circuits include memory elements which are capable of storing binary information. That binary information defines the state of the sequential circuit at that time. A latch capable of storing one bit of information.

There are two types of input to the combinational logic :

1. External inputs which not controlled by the circuit.
2. Internal inputs which are a function of a previous output states.

Secondary inputs are state variables produced by the storage elements, where as secondary outputs are excitations for the storage elements.

Types of Sequential Circuits – There are two types of sequential circuit :

• Asynchronous sequential circuit – These circuit do not use a clock signal but uses the pulses of the inputs. These circuits are faster than synchronous sequential circuits because there is clock pulse and change their state immediately when there is a change in the input signal. We use asynchronous sequential circuits when speed of operation is important and independent of internal clock pulse.But these circuits are more difficult to design and their output is uncertain.
• Synchronous sequential circuit – These circuit uses clock signal and level inputs (or pulsed) (with restrictions on pulse width and circuit propagation). The output pulse is the same duration as the clock pulse for the clocked sequential circuits. Since they wait for the next clock pulse to arrive to perform the next operation, so these circuits are bit slower compared to asynchronous. Level output changes state at the start of an input pulse and remains in that until the next input or clock pulse.

  We use synchronous sequential circuit in synchronous counters, flip flops, and in the design of MOORE-MEALY state management machines.We use sequential circuits to design Counters, Registers, RAM, MOORE/MEALY Machine and other state retaining machines.

Combinational Circuits

A combinational circuit comprises of logic gates whose outputs at any time are determined directly from the present combination of inputs without any regard to previous inputs.

A combinational circuit performs a specific information-processing operation fully specified logically by a set of Boolean functions.

The basic components of a combinational circuit are: input variables, logic gates, and output variables.

Design procedure of a Combinational Circuit

The design procedure of a combinational circuit involves the following steps:

1. The problem is stated.
2. The total number of available input variables and required output variables is determined.
3. The input and output variables are allocated with letter symbols.
4. The exact truth table that defines the required relationships between inputs and outputs is derived.
5. The simplified Boolean function is obtained from each output.
6. The logic diagram is drawn.

The combinational circuit that performs the addition of two bits is called a half adder and the one that performs the addition of three bits (two significant bits and a previous carry) is a full adder.

 

 

 

Laws of Boolean Algebra

As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively.

A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra.

Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions.

The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.

Examples of these individual laws of Boolean, rules and theorems for Boolean Algebra are given in the following table.

Truth Tables for the Laws of Boolean

Boolean Expression	Description	Boolean Algebra Law or Rule
A + 1 = 1	A in parallel with closed = “CLOSED”	Annulment
A + 0 = A	A in parallel with open = “A”	Identity
A . 1 = A	A in series with closed = “A”	Identity
A . 0 = 0	A in series with open = “OPEN”	Annulment
A + A = A	A in parallel with A = “A”	Idempotent
A . A = A	A in series with A = “A”	Idempotent
NOT A = A	NOT NOT A (double negative) = “A”	Double Negation
A + A = 1	A in parallel with NOT A = “CLOSED”	Complement
A . A = 0	A in series with NOT A = “OPEN”	Complement
A+B = B+A	A in parallel with B = B in parallel with A	Commutative
A.B = B.A	A in series with B = B in series with A	Commutative
A+B = A.B	invert and replace OR with AND	de Morgan’s Theorem
A.B = A+B	invert and replace AND with OR	de Morgan’s Theorem

The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the Distributive Law allowing the factoring of an expression, are the same as in ordinary algebra.

Each of the Boolean Laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given Boolean expression as well as for simplifying complicated digital circuits.

A brief description of the various Laws of Boolean are given below with A representing a variable input.

Description of the Laws of Boolean Algebra

• Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will equal 1
•  
  • A . 0 = 0    A variable AND’ed with 0 is always equal to 0
  • A + 1 = 1    A variable OR’ed with 1 is always equal to 1
•  
• Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal that term
•  
  • A + 0 = A   A variable OR’ed with 0 is always equal to the variable
  • A . 1 = A    A variable AND’ed with 1 is always equal to the variable
•  
• Idempotent Law – An input that is AND´ed or OR´ed with itself is equal to that input
•  
  • A + A = A    A variable OR’ed with itself is always equal to the variable
  • A . A = A    A variable AND’ed with itself is always equal to the variable
•  
• Complement Law – A term AND´ed with its complement equals “0” and a term OR´ed with its complement equals “1”
•  
  • A . A = 0    A variable AND’ed with its complement is always equal to 0
  • A + A = 1    A variable OR’ed with its complement is always equal to 1
•  
• Commutative Law – The order of application of two separate terms is not important
•  
  • A . B = B . A    The order in which two variables are AND’ed makes no difference
  • A + B = B + A    The order in which two variables are OR’ed makes no difference
•  
• Double Negation Law – A term that is inverted twice is equal to the original term
•  
  • A = A     A double complement of a variable is always equal to the variable
•  
• de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems,
•  
• (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example:  A+B = A . B
•  
• (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example:  A.B = A + B

 

Other algebraic Laws of Boolean not detailed above include:

• Boolean Postulates – While not Boolean Laws in their own right, these are a set of Mathematical Laws which can be used in the simplification of Boolean Expressions.
•  
  • 0 . 0 = 0    A 0 AND’ed with itself is always equal to 0
  • 1 . 1 = 1    A 1 AND’ed with itself is always equal to 1
  • 1 . 0 = 0    A 1 AND’ed with a 0 is equal to 0
  • 0 + 0 = 0    A 0 OR’ed with itself is always equal to 0
  • 1 + 1 = 1    A 1 OR’ed with itself is always equal to 1
  • 1 + 0 = 1    A 1 OR’ed with a 0 is equal to 1
  • 1 = 0    The Inverse (Complement) of a 1 is always equal to 0
  • 0 = 1    The Inverse (Complement) of a 0 is always equal to 1
•  
• Distributive Law – This law permits the multiplying or factoring out of an expression.
•  
  • A(B + C) = A.B + A.C    (OR Distributive Law)
  • A + (B.C) = (A + B).(A + C)    (AND Distributive Law)
•  
• Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.
•  
  • A + (A.B) = (A.1) + (A.B) = A(1 + B) = A  (OR Absorption Law)
  • A(A + B) = (A + 0).(A + B) = A + (0.B) = A  (AND Absorption Law)
•  
• Associative Law – This law allows the removal of brackets from an expression and regrouping of the variables.
•  
  • A + (B + C) = (A + B) + C = A + B + C    (OR Associate Law)
  • A(B.C) = (A.B)C = A . B . C    (AND Associate Law)

Boolean Algebra Functions

Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table.

Function	Description	Expression
1.	NULL	0
2.	IDENTITY	1
3.	Input A	A
4.	Input B	B
5.	NOT A	A
6.	NOT B	B
7.	A AND B (AND)	A . B
8.	A AND NOT B	A . B
9.	NOT A AND B	A . B
10.	NOT AND (NAND)	A . B
11.	A OR B (OR)	A + B
12.	A OR NOT B	A + B
13.	NOT A OR B	A + B
14.	NOT OR (NOR)	A + B
15.	Exclusive-OR	A . B + A . B
16.	Exclusive-NOR	A . B + A . B

Laws of Boolean Algebra Example No1

Using the above laws, simplify the following expression:  (A + B)(A + C)

Q =	(A + B).(A + C)
	A.A + A.C + A.B + B.C	– Distributive law
	A + A.C + A.B + B.C	– Idempotent AND law (A.A = A)
	A(1 + C) + A.B + B.C	– Distributive law
	A.1 + A.B + B.C	– Identity OR law (1 + C = 1)
	A(1 + B) + B.C	– Distributive law
	A.1 + B.C	– Identity OR law (1 + B = 1)
Q =	A + (B.C)	– Identity AND law (A.1 = A)

 

Then the expression:  (A + B)(A + C) can be simplified to A + (B.C) as in the Distributive law.

Code Conversion

There are many methods or techniques which can be used to convert code from one format to another. We'll demonstrate here the following

• Binary to BCD Conversion
• BCD to Binary Conversion
• BCD to Excess-3
• Excess-3 to BCD

Binary to BCD Conversion

Steps

• Step 1 -- Convert the binary number to decimal.

• Step 2 -- Convert decimal number to BCD.

Example − convert (11101)₂ to BCD.

Step 1 − Convert to Decimal

Binary Number − 11101₂

Calculating Decimal Equivalent −

Step	Binary Number	Decimal Number
Step 1	111012	((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10
Step 2	111012	(16 + 8 + 4 + 0 + 1)10
Step 3	111012	2910

Binary Number − 11101₂ = Decimal Number − 29₁₀

Step 2 − Convert to BCD

Decimal Number − 29₁₀

Calculating BCD Equivalent. Convert each digit into groups of four binary digits equivalent.

Step	Decimal Number	Conversion
Step 1	2910	00102 10012
Step 2	2910	00101001BCD

Result

(11101)2 =  (00101001)BCD

BCD to Binary Conversion

Steps

• Step 1 -- Convert the BCD number to decimal.

• Step 2 -- Convert decimal to binary.

Example − convert (00101001)_BCD to Binary.

Step 1 - Convert to BCD

BCD Number − (00101001)_BCD

Calculating Decimal Equivalent. Convert each four digit into a group and get decimal equivalent for each group.

Step	BCD Number	Conversion
Step 1	(00101001)BCD	00102 10012
Step 2	(00101001)BCD	210 910
Step 3	(00101001)BCD	2910

BCD Number − (00101001)_BCD = Decimal Number − 29₁₀

Step 2 - Convert to Binary

Used long division method for decimal to binary conversion.

Decimal Number − 29₁₀

Calculating Binary Equivalent −

Step	Operation	Result	Remainder
Step 1	29 / 2	14	1
Step 2	14 / 2	7	0
Step 3	7 / 2	3	1
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the least significant digit (LSD) and the last remainder becomes the most significant digit (MSD).

Decimal Number − 29₁₀ = Binary Number − 11101₂

Result

(00101001)BCD = (11101)2

BCD to Excess-3

Steps

• Step 1 -- Convert BCD to decimal.

• Step 2 -- Add (3)₁₀ to this decimal number.

• Step 3 -- Convert into binary to get excess-3 code.

Example − convert (0110)_BCD to Excess-3.

Step 1 − Convert to decimal

(0110)_BCD = 6₁₀

Step 2 − Add 3 to decimal

(6)₁₀ + (3)₁₀ = (9)₁₀

Step 3 − Convert to Excess-3

(9)₁₀ = (1001)₂

Result

(0110)BCD = (1001)XS-3

Excess-3 to BCD Conversion

Steps

• Step 1 -- Subtract (0011)₂ from each 4 bit of excess-3 digit to obtain the corresponding BCD code.

Example − convert (10011010)_XS-3 to BCD.

Given XS-3 number  = 1 0 0 1 1 0 1 0 
Subtract (0011)2   = 1 0 0 1 0 1 1 1
                    --------------------
               BCD = 0 1 1 0   0 1 1 1

Computer architecture

In computer engineering, computer architecture is a set of rules and methods that describe the functionality, organization, and implementation of computer systems. Some definitions of architecture define it as describing the capabilities and programming model of a computer but not a particular implementation.

Von Neumann Architecture

Von Neumann architecture was first published by John von Neumann in 1945.

His computer architecture design consists of a Control Unit, Arithmetic and Logic Unit (ALU), Memory Unit, Registers and Inputs/Outputs.

Von Neumann architecture is based on the stored-program computer concept, where instruction data and program data are stored in the same memory.  This design is still used in most computers produced today.

Central Processing Unit (CPU)

The Central Processing Unit (CPU) is the electronic circuit responsible for executing the instructions of a computer program.

It is sometimes referred to as the microprocessor or processor.

 

 

The CPU contains the ALU, CU and a variety of registers.

Registers

Registers are high speed storage areas in the CPU.  All data must be stored in a register before it can be processed.

MAR	Memory Address Register	Holds the memory location of data that needs to be accessed
MDR	Memory Data Register	Holds data that is being transferred to or from memory
AC	Accumulator	Where intermediate arithmetic and logic results are stored
PC	Program Counter	Contains the address of the next instruction to be executed
CIR	Current Instruction register	Contains the current instruction during processing

Arithmetic and Logic Unit (ALU)

The ALU allows arithmetic (add, subtract etc) and logic (AND, OR, NOT etc) operations to be carried out.

Control Unit (CU)

The control unit controls the operation of the computer’s ALU, memory and input/output devices, telling them how to respond to the program instructions it has just read and interpreted from the memory unit.

The control unit also provides the timing and control signals required by other computer components.

Buses

Buses are the means by which data is transmitted from one part of a computer to another, connecting all major internal components to the CPU and memory.

A standard CPU system bus is comprised of a control bus,data bus and address bus. 

Address Bus	Carries the addresses of data (but not the data) between the processor and memory
Data Bus	Carries data between the processor, the memory unit and the input/output devices
Control Bus	Carries control signals/commands from the CPU (and status signals from other devices) in order to control and coordinate all the activities within the computer

Memory Unit

The memory unit consists of RAM, sometimes referred to as primary or main memory.  Unlike a hard drive (secondary memory), this memory is fast and also directly accessible by the CPU.

RAM is split into partitions.  Each partition consists of an address and its contents (both in binary form).

The address will uniquely identify every location in the memory.

Loading data from permanent memory (hard drive), into the faster and directly accessible temporary memory (RAM), allows the CPU to operate much quicker.

 

 

Digital Counter

A Digital Counter is obtained by arranging the flip-flops. These are the applications of flip-flops. Other than counting, these are used for measuring the frequency as well as time. These are used to increase the addresses in memory. The operation of these devices depends on the single clock applied.  These are made of flip-fops as basic elements. There are two states included in the flip-flops. The high state is represented by 1 and the low state by 0. The operation of the counter can be made possible by high state-driven flip-flops. Counters consist of modes that are represented by the number of countable states, for example a mod-8 counter. In this, the number of states countable is from 000 to 111 (that is 0 to 7). Hence eight countable states name the counter as a mod-8 counter.

What is Digital Counter?

Definition: The circuit is designed with digital logic to obtain information about the number of events that occurred. This type of digital logic device can be defined as a Counter. The design of counters can be achieved by following various steps.

1. The number of Flip-Flops based on the requirement.
2. Accordingly, the State diagrams/ Excitation Tables are drawn.
3. Further, the expressions are minimized using various techniques.
4. Finally, the logic diagram is designed.

The above steps are especially followed fro the design of the type of counter known as Synchronous Counters.

Types of Digital Counters

The interconnection of the flip-flops results in the classification of the counters.  Although the single clock signal applied to the counters. There is a difference among the operation based on a single cock applied to the flip-flops in the circuit or the signal applied to the main flip-flop.

The types of counters are:

1. Asynchronous Counters
2. Synchronous Counters

The asynchronous counters are also referred to as Ripple counters. The simplest in design among the other counters is the ripple counter. The least quantity of hardware is required for this counter. The flip of one stage reflects the flip in another stage in asynchronous counters. These counters are further described as ‘Serial Counters’.

But in the other type of counter called synchronous counters, each flip-flop in the circuit gets triggered with the same clock at the same time. This is possible by connecting each flip-flop clock line with a single clock. The operation occurred in such types of circuits will be in coordination or synchronization with all the flip-flops present.  Hence these are referred to as Synchronous Counters.

Based on the clock signal applied the counters are classified as synchronous and asynchronous counters. In the Asynchronous type, only the first flip-flop is provided by the main cock. the remaining flip-flops are powered with the output obtained in the previous stage of flip-flops. There is another category of counters referred to as Decade Counters. This counter is able to count the pulses for ten values. The counting is done from 0 to 9.  Even this counting can be extended to 15 states.

What is the difference between Synchronous and Asynchronous Counters?

Based on the connections and the clock applied to the flip-flops there are certain differences between Synchronous and Asynchronous counters.  Some of the differences are listed as follows:

 

Asynchronous Counters	Synchronous Counters
1. In this counter, the main clock is applied to the first flip-flop. Whereas the other flip-flops are provided with the clock from the output of previous stage flip-flops.	1. The arrangement of the flip-flops in this counter is driven by the same clock.
2. The clock is not simultaneous for all the flip-flops in the circuit.	2. The clock is simultaneously provided for all the flip-fops.
3. To the more number of states, the design of this counter is simple to implement.	3. In this counter, as the number of states is increasing the design tends to become complex.
4. The propagation delay is more as the counters are not in sync.	4. The propagation delay will be less in comparison to the asynchronous counters because of the sync among the flip-flops.

In comparison to the asynchronous counters, the operation will be fast in synchronous counters. The frequency required for operation in the synchronous counters is high. In terms of cost, it is low in asynchronous counters.

Digital Counter Applications

These counter circuits are the basic ones in the ‘Digital Electronics’. These counters possess various applications.

1. In the conversions from Analog to Digital, these counters are used.
2.   In the applications of Timers for example Washings machines where we set the time. These counters are used.
3. With the help of these counters, a ‘Digital Triangular Wave Generator’ can be designed.
4. In the application of ‘Digital Clock’ counters are used.
5. A practical example of these devices is seen in the malls, stadiums, or the auditoriums. In the above situations to keep the data on the number of persons. This can be made possible or it will become simple because of these counters

Digital registers

Introduction

A Flip-flop is a 1 bit memory cell which can be used for storing the digital data. To increase the storage capacity in terms of number of bits, you can use a group of flip-flops. Such a group of flip-flops is known as a Register. The n-bit register will consist of n number of flip-flop(s) and it is capable of storing an n-bit word.

The binary-data, in a register, can be transfered within itself from one flip-flop to another. A shift register is a type of register that allows such data transfers. Shift register has 4 modes of operations.

Shift Registers

A shift register is a piece of circuit that you will find inside a microprocessor. It is used to store and modify data. Now this is one of the simplest explanation I can give for a 2nd grade student.

But if you are some one who is studying about computers, this is just not sufficient. So I will have to explain it in a bit more detail. Alright? I will be getting a bit technical here. But it is how you will get a solid understanding of shift register. So you will have to bare it. Okay?

Sub-modules

The sub-modules of digital registers are:

1. Serial-in Serial-out
2. Serial-in Parallel-out
3. Parallel-in Serial-out
4. Parallel-in Parallel-out

Types Of Shift Registers

Based on the way the data is shifted, we have 5 different types of shift registers. They are:

• Left Shift Registers
• Right Shift Registers
• Bidirectional Shift Registers
• Circular Shift Registers &
• Linear Feedback Shift Registers

Now let explain what each of these registers work like:

Left Shift Registers

We have already talked about the left shift register. So I think you are familiar with it by now. If not, let me re-iterate. So in the case of a left shift register, the data is shifted to the left on each clock cycle.

Right Shift Registers

This is just like the left shift register. But here it is shifting the data to the right on each clock cycle.

Bidirectional Shift Registers

In the case of a bidirectional shift register, we can shift the data in both the directions. So you can shift the data either to it’s left or right!

Circular Shift Registers

In the case of a circular shift register, the last output is connected back as input. So your data will not be thrown out. Instead, will be shifted either left or right in a circular fashion!

Linear Feedback Shift Registers

In this type of shift register, the input of one flip flop will be linear output value of the previous flip flop.

Advantages & Disadvantages Of A Shift Register

It is now time to talk about the advantages & disadvantages of using a shift register. So what are they?

Advantages Of A Shift Register

• They are very fast to use.
• Very quick when you want to convert data from serial to parallel or vice versa. They are faster than normal serial to parallel converter circuits.
• They are very simple in design. So you can easily rig up a circuit to create a shift register.
• We can use them to encrypt or decrypt the data.
• We can use them to a delay signal.
• It is used in CDMA to generate Pseudo Noise Sequence Number.
• We can use them to track our data!

Disadvantages Of A Shift Register

While we could see that it has major advantages, shift register has one major disadvantage. That is:

• The strength of the output current coming from a shift register is not so strong.

Binary Arithmetic

Binary arithmetic is essential part of all the digital computers and many other digital system.

Binary Addition

Binary addition is the easiest of the processes to perform. As you'll see with the other operations below, it is essentially the same way you learnt to do addition of decimal numbers by hand (probably many years ago in your early school years). The process is actually easier with binary as we only have 2 digits to worry about, 0 and 1.

The process is that we line the two numbers up (one under the other), then, starting at the far right, add each column, recording the result and possible carry as we go.

Here are the possibilities:

• 0 + 0 = 0
• 1 + 0 = 1
• 1 + 1 = 2 which is 10 in binary which is 0 with a carry of 1
• 1 + 1 + 1 (carry) = 3 which is 11 in binary which is 1 with a carry of 1

The carry is involved whenever we have a result larger than 1 (which is the largest amount we may represent with a single binary digit).

Adding more than two numbers

It is possible to add more than 2 binary numbers in one go but it can soon get unweildly managing the carries. My suggestion is that you add the 1st and 2nd numbers together. Then take the result and add the third number to that. Then take the result and add the 4th etc. This way you may add as many binary numbers as you like and the complexity will never increase. It's a little more work but with practice you will get very quick at it.

Binary Multiplication

Binary multiplication is just about as easy as binary addition. Again it is the same process as we would do with decimal multiplication by hand. Again it is easier as binary only has 0 and 1.

We line the two numbers up (similar to addition). Then we multiply the entire top number by each individual digit of the bottom number. As we move across each digit we pad out the result with 0's to line it up. Finally we add all the results together.

Here are the possibilities:

• 0 * 0 = 0
• 1 * 0 = 0
• 1 * 1 = 1

As you have no doubt noticed, the process is fairly straight forward. If the binary digit on the second row we are multiplying by is a 1 then pad out accordingly and write out the top binary number. If the binary digit on the second row we are multiplying by is a 0 then we can just write out 0's.

Binary Subtraction

With binary subtraction we start to get a little more difficult (But not that difficult). Similar to binary addition, we will work through the numbers, column by column, starting on the far right. Instead of carrying forward however, we will borrow backwards (when necessary).

Here are the possibilities:

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1 we can't do so we borrow 1 from the next column. This makes it 10 - 1 which is 1.

Another approach

The above example is the most convenient way for us to do binary subtraction by hand. There is another approach however and this is the way that computers subtract binary digits. This approach is called Two's Complement.

Let's say we want to compute 1000 ( 8 ) - 11 ( 3 ).

• Step 1: Write the equation out, padding the bottom number with 0's
  1000
  0011 -
• Step 2: Invert the digits of the lower number
  1000
  1100
• Step 3: Add 1 to the lower number
  1000
  1101
• Step 4: Add those two numbers together to get 10101
• Step 5: Remove the leading 1 (and any 0's after it). You are left with 101 ( 5 ).

Binary Division

Binary division is probably the most difficult of the binary equations. Fortunately, it is also made easier by the fact we only have to deal with 1's and 0's.

First off, some terminology. The number we are dividing by is the divisor. The number we are dividing into is the dividend.

The process is as follows:

• Step 1: Create the working portion of the dividend. Starting at the right, keep including digits until we have a number that the divisor will go into.
• Step 2: Work out how many times the divisor goes into the working portion (with binary this is easy as it will always be 1). Write this number above the line (in line with the far right digit of the working number).
• Step 3: Subtract the divisor from the working number. This becomes the beginning of the new working number.
• Step 4: Bring down digits from the dividend and add to the new working number until we have a new working number large enough for the divisor to go into.
• Step 5: Repeat steps 2 to 4 until we are at the end of the dividend.
• Step 6: The result of the final subtraction is the remainder.