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]]>what is the vector analysis? Vector analysis is the mathematical analysis used for understanding the core concepts of electromagnetic field theory. It involves:

- vector algebra
- orthogonal coordinate systems
- vector calculus

Vector algebra discusses about the arithmetic operations on vectors quantities like addition subtraction and multiplication. In vector addition and subtraction, we use the parallelogram rule while for vectors multiplications we have scalar and vector product.

These are the coordinate systems in which coordinates are perpendicular to each other. There is always an angle of 90 degree between the axis. Three types of orthogonal coordinates systems are:

**cartesian or rectangular coordinate systems.****cylindrical coordinate systems**- spherical coordinate systems.

Also read here for more lectures

what is the scalar triple product and vector triple product?

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]]>The post what is the cylindrical coordinate system? appeared first on EE-Vibes.

]]>what is the cylindrical coordinate system? This system helps to analyze the geometry of a **coaxial cable **and we can analyze its electric field intensity or electric field more easily using them. Three coordinates that are used for representing a point in cylindrical are P(р,Ф,Ζ). р is the radial distance of a cylindrical surface, Ф is the angle that a point makes with the x-axis in counter clockwise direction, z is the height of a cylinder. These three coordinates give us constant planes. At top surface of the cylinder, z is constant while р andФ are variables. Similarly at outer cylindrical surface, р is constant while Ф and Ζ are variable. The cross sectional plane obtained by cutting the cylinder vertically gives us a plane on which Ф is constant while р and Ζ vary.

The following picture shows how these cylindrical coordinate systems with their unit vectors are represented.

https://www.youtube.com/watch?v=h-oy0lklXHk

Also read here

https://eevibes.com/electromagnetic-field-theory/what-is-the-cartesian-coordinate-system/

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]]>The post what is the vector algebra? appeared first on EE-Vibes.

]]>what is the vector algebra? vector algebra is the first part of vector analysis where we perform basic algebraic operations on vectors that are :

- addition
- subtraction and
- multiplication.

In order to add two or more vectors, we follow the head to tail rule that is implemented using parallelogram rule. If we want to add any two vectors A and B then it is necessary to join the head of one vector with the tail of another vector and after joining them together, the tail of one vector is joined to the head of another vector as shown below.

The vectors subtraction can be performed just like the vectors addition. Because if you want to perform the operation A-B then it can also be written as A+(-B)=A-B. While performing the vectors subtraction you just need to invert the sign of the vector that is being subtracted and then add it into the first vector. Consider the following example.

Here it can been that two vectors A and B are subtracted. Fist step is to draw the negative of B that will be in opposite direction to B. Then draw a parallel vector of A on -B. Finally they are added using head to tail rule.

There are three types of vector multiplication that are

- simple multiplication
- scalar or dot product
- vector or cross product

In simple product, a vector is multiplied with a constant K which extends its magnitude and results in a new vector B. So, **B**=k**A. **If k is positive then the direction of B is same as that of A and if it is negative, then the direction of B is opposite to that of A.

If two vectors are multiplied such that the resultant is a scalar quantity or a number then it is called scalar or dot product. The scalar product of two vectors A and B is written as

**A.B**=ABCOSʘ

scalar product helps to determine the projection of one vector in the direction of another vector. So the scalar projection of vector A on B is given as

ACOSʘ=A.Ḃ

where Ḃ is the unit vector of vector B.

In order to convert this scalar projection into the vector projection you need to multiply it with unit vector of B again as shown below

ACOSʘ=A.Ḃ.Ḃ

If the multiplication of two vectors results in a vectors result in a vector quantity then it is called vector or cross product. The formula for cross multiplication of A and B is given as

**AxB**=ABSinʘ.an

where an is the unit vector perpendicular to the plane containing A and B.

https://www.youtube.com/watch?v=gbsz9-Jn9So

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]]>What is the cartesian coordinate system?

https://www.youtube.com/watch?v=TJ3WSTu8isY

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]]>what is the scalar triple product and vector triple product?

When we multiply three vectors such that their resultant is a scalar quantity then this is called scalar triple product.

The mathematical expression for it is given as

(AXB). C

The scalar triple product represents the volume of a parallelepiped.

If the product of three vectors results in a vector quantity, then it is called vector triple product. The mathematical representation is

AX(BXC)=B.(AC)-C.(AB)A. This is also called BAC-CAB rule.

https://www.youtube.com/watch?v=Um8S_wlCAHU

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]]>The post what is the scope of a variable? How you define the lifetime of local and global variables in programming? appeared first on EE-Vibes.

]]>what is the scope of a variable? The scope of a variable can be defined as the accessibility of a variable in the program. It means whenever a variable is defined in a program, it is not accessible in every portion of it. It might be accessed in some portion or in all portion of it. Another name for **scope **is **visibility. **So scope of a variable is actually the part of a program where it is visible. For better understanding the concept, lets consider two cases.

void afunction()

{

int firstvariable=10; //valid

int secondvariable=11; //valid

thirdvariable=9; // not valid

}

In the above **definition of function**, the fisrvariable and secondvariable are visible to the “afunction()” but the third variable is unknown to it. Limiting the visibility in programming has many benefits. First of all it helps you to provide security. Secondly, it helps the programmer to avoid mixing the values of different variables by different functions. This is an important feature of **structured programming**. It is also an important feature of object oriented programming.

- Local Variable
- Global Variable
- Static Local Variable

Local variables are those that are local to the function definition. It means they are only accessible where they have been declared. They are not accessible to the other functions defined in the same program.

void somefunction()

{

int var1=2;

float var2=5;

}

void otherfunction()

{

var1=5; // not accessible as it is locally declared in somefunction().

}

Another name for the **global variables **is the **external variables. **Global variables are those that are declared externally i.e., outside the functions. They are declared in the start of the program. That’s why they are visible to all functions used within a program.

int age;

void getvalue();

void showvalue();

void main()

{

while(getvalue!=0)

{

getvalue();

showvalue();

}

void getcvalue()

{

cin>>age;

}

void showvalue()

{

cout<<age;

}

As it can be seen the variable “age” is declared at the top of the program and it is accessible to both functions. The first function sets the value of age and the second function displays its value.

The global variables belongs to the **static storage class** which means they exist throughout the program. They are not like the local variables that are created only at the time of a function call and then destroyed once the function is executed.

If we want our program to remember the value of local variable then we can declare the variable as static. For this **static **keyword is used before it. static local variables come into existence once the definition of the function begins where they are declared. Once the function is executed the program remembers its value. So the lifetime of static local variable is same as that of global variables. The following example illustrates the concept of static local variable.

float takeavg(float);

void main()

{

clrscr();

float d1=1; average;

while(d1!=0)

{

cout<<“enter the value”<<endl;

cin>>d1;

average=takeavg(d1);

cout<<“new average value is”<<average<<endl;

}

}

float takeavg(float newdata)

{

static float t1=0;

static int c1=0;

c1++;

t1+=newdata;

return t1/c1;

}

both static variables t1 and c1 retain their values after the function takeavg() returns, so they are available the next time if needed.

int x=10; //this is the global variable as it is outside the main function

void main()

{

int x=6; // here x is local variable as it hides global variable x=10

int y=::x; // y=10 here x is global because of :: operator

{

int z=x; // z=6 (local variable)

int x=38; // now value of x is hided again (x=6)

int t=::x; // t=10

t=x; // t=38 here x is treated as local variable

}

int z=x; z=6

}

From above discussion it is concluded that the scope of variable is chosen by the programmer as per the need. If we want to keep the value accessible throughout the program then it is preferable to declare them as global variables.

you can also read here:

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]]>What are number systems in computer? Usually we humans are more comfortable with** decimal numbers** (base 10). But in computer system we have other types that are used as per the requirement. Computer understands only the logic 0 and 1. This logical representation is called **binary representation. **In this form, the base is 2. Similarly, there is a variety of number systems that has different** applications in computers**. For example ASCII codes are used for representing alphanumeric data and special characters in computers. **Hex numbers **are used for representing addresses of memory locations as they are the compact form of binary numbers.

If we make a list of commonly known number systems then we have

The range is determined using formula 0 to n-1.

So, for base 10 the range is 0 to (10-1=9).

So, what are the number systems? There are different types and each type is discussed in detail here.

Binary numbers need only bit 1 and 0 for their representation. Any decimal number can be represented in binary. Thus if we want to convert a decimal number 23 into binary we will divide it by 2 as shown in the figure below

The binary equivalent of 23 is (10111). Here 23 is divided by 2 and the remainder is written after the dash. Then 11 is divided by 2 and again the remainder 1 is written and so on until the quotient is less than 2 is obtained. The left most bit is called the **MSB (Most Significant Bit) **and the right most bit is called **LSB( Least Significant Bit). **

This is basically the reverse procedure of the above one. Here we add the binary weights of each bit and then the resultant is the decimal number. Lets do an example for understanding this.

Start multiplying the LSB with 2 raise to power 0 and MSB with 2 raise to power m-1, where m is the total number of bits in the binary expression.

Octal Numbers have base 8 and their range is from 0 to 7. 3 bits are needed for representing each octal digit into binary. The binary code for octal numbers is shown below.

If we want to convert any decimal number into octal number then we divide it with 8 just like we did in case of binary numbers.

So 98 in decimal is equal to 142 in octal. Here each digit is less than 8 (1<8, 4<8, 2<8).

The same procedure is followed while converting an octal number into decimal. But here the base is 8 is multiplied with each digit of octal representation.

The hexadecimal numbers have base 16. Their range is from 0 to 15 (F). Four bits are required for representing each hex digit. The following table shows the binary equivalent of each hex digit.

As we have mentioned already that the hex representation is the compact form of binary numbers so if we have some binary expression as

0000 1110 1101 0101 0010 1010 0101 0010

Then its hex can be written by combining 4-bits from Right side.

0010=2

0101=5

1010=A

0101=5

1101=D

1110=E

0000=0

So its Hex representation is

0ED5A52H.

**Notice that we write H at the end of hex representation usually. **

Convert 900 into Hex.

900=384 in hex.

In this representation, binary codes are assigned to decimal numbers. Each decimal digit needs four bits for its binary representation as shown in the following table.

- If we want to convert a decimal number into any base ‘n’ we divide that number with base n and note down the remainders at each step.
- If we want to convert any number represented in base ‘n’ we multiply each digit of that number with weights of that base ‘n’.

There are two ways to convert an octal number into hex

The first one is to convert that octal number into decimal, and then decimal to hex.

The second approach is by combining bits of its binary representation.

Convert 352 (in octal) into hex?

3=011

5=101

2=010

so, 352=011 101 010

As we know that each hex digit needs 4 bits for its representation so we will start combining 4 bits from right most and then append zeros for completing the number of bits on left side.

0000 1110 1010

0000=0

1110=E

1010=A

so 352 in octal is equal to 0EA in Hex.

https://www.youtube.com/watch?v=z5YfUNYwJSg

The complement of a number has a significant role in performing the subtraction of a number. In order to find the negative of a number we take complement in that particular base. There are two types of a complement in base ‘r’.

- Radix complement
- Diminishes Radix complement

For any integer N in base ‘r’ having ‘n’ no. of digits, the radix complement is given as

r^{n}-N

This is also called the r’s complement of a number. In case of a decimal number N=1234 the r’s complement is 10^{4}-1234=8766. Another way to find the 10’s complement of a decimal is to subtract the right most digit (if it is not zero, otherwise look for the next non zero digit) from 10 while all other digits are subtracted from 9.

999910

23456 –

__________

76544

__________

Similarly

99910

44150-

________

55850

________

This form of complement is also knows as the (r-1)’s complement. If An integer N has n bits in base r, then (r-1)’s complement is found by using the following formula

(r^{n}-1)-N

The following lecture explains multiple examples of complement and its use while subtracting two numbers.

https://www.youtube.com/watch?v=z26DA3IAx-Q

https://www.youtube.com/watch?v=pcxqOHq4yA4

Also read here

https://eevibes.com/computing/introduction-to-computing/what-is-binary-arithmetic/

What is binary arithmetic? How to add, subtract, multiply and divide two binary numbers?

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]]>How to calculate eigen values and eigen vectors? Eigen Values and Eigen Vectors has a great significance in many engineering problems specially case of** linear systems** where we want to study the “change”. In order to find the eigen values and eigen vectors a **vector equation** needs to be solved that is:

Ax= λx.

where A is called the** coefficient matrix of a linear system**, x is the eigen vector corresponding to the value λ. One thing to be noted is that eigen values are always found of a **square matrix**. If the order of a matrix is n×n then at most ‘n’ distinct eigen values exist or at least one distinct eigen value can be found. It can be seen clearly that x=0 is the solution of the above equation. But this solution does not give us the required information. So we are interested in some kind of non-zero eigen vectors so that we can interpret the obtained result.

The set of all eigen values of the system A is called the spectrum of A. While the maximum of absolute of eigen values is called the spectral radius of A. This concept will be shown in the later examples.

For finding the eigen values and eigen vectors of a system the following steps are followed.

Ax= λx

Ax -λx=0

(A- λI)x=0

det(A- λI)=0 gives a characteristic polynomial. The roots of this polynomial are called **eigen values (they can be distinct or repeated).** Then the next step is to find eigen vectors corresponding to each eigen value. After finding the eigen values, the next step is to determine eigen vectors corresponding to them. This can be done by solving the equation (A- λI)x=0. For solving such system of equations, **Gauss Elimination method** can be used. So the answer is given How to calculate eigen values and eigen vectors of a matrix?

The order of an eigen value as a root of characteristics polynomial is called “algebraic multiplicity”. It mean how my many times an eigen value is repeated.

It is defined as the number of linearly independent eigen vectors corresponding to λ. The difference between algebraic multiplicity and geometric multiplicity is called the defect of λ. The following figures depict the example which shows how eigen values and eigen vectors are determined. Also the defect has been calculated.

Here is the link of complete lecture

Though eigen values have large applications in engineering and in many other fields. I have chosen a very interesting application that will elaborate more how their concept can be applied in real life.

Generally speaking eigen values tell about the amount of change and the eigen vectors show the direction of change or it can be said they depict in which the change is introduced.

An elastic membrane in x_1 x_2 plane with boundary circle x_1^2+x_2^2=1 is stretched so that a point p (x1,x2) goes over into the point Q(y1,y2) given by y1=5×1+3×2

y2=3×1+5×2

Find the principal directions, that is the direction of the **position vector** x of P for which the direction of the position vector y of Q is the same or exactly opposite. What shape does the boundary circle take under this deformation?

There are also some other applications of eigen values and eigen vectors. For example the growth of a particular community either increasing or decreasing can be decided by them. For this, all we need is to determine the eigen values of that community matrix. If eigen values are positive then their population is increasing otherwise its decreasing.

Similarly the NETFLIX uses singular values decomposition procedure of eigen values and eigen vectors for ranking the movies on its site.

https://www.youtube.com/watch?v=KQAQ76hv3dw&t=18s

The power method is an iterative method for finding the largest eigen value of a system. With little modifications, it can also be used for finding the intermediate and the smallest eigen values. The plus point of this method is that we obtain the corresponding eigen vector as well in this method.

There are some cases where the power method will not converge to the largest value but to the second largest value.

In order to determine the smallest eigen value of a system we just need to find the inverse of that system. Then we can apply the same power method as applied in the above example. This will determine the largest of 1/λ. Then λ will be smallest eigen value. The following example illustrates this concept.

Here 0.964 is the smallest eigen value of the above matrix A.

Watch here

Also read here

https://eevibes.com/system-of-linear-equations-in-linear-algebra/

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]]>The post how parallel architecture is used for accelerating BL-SOM using dedicated hardware appeared first on EE-Vibes.

]]>how parallel architecture is used for accelerating BL-SOM using dedicated hardware. This is a review article and all rights are reserved by Ryota Miyauchi, Akira Kojima, Hideyuki Kawabata and Tetsuo Hironaka Graduate School of Information Sciences, Hiroshima City University. Lets just first learn about what is BL-SOM? SOM is an unsupervised method used for data analysis. In this method larger number of vectors of high dimension input vectors are converted into 2- dimension plane that represents the relationship between the input vectors. This learning process is independent of order of input vectors.

The problem with SOM is that its learned results changes when the order of the input vectors changes. The reason is it actually extracts the degree o similarity between each input vector.

**What is BL-SOM procedure?**

BL-SOM has two layer structure:

- an input layer
- an output layer

Three steps are involved in BL-SOM procedure

- Competitive Process
- Cooperation Process
- Adaptation Process

**Competitive Process **

This step is also called initialization of weight vectors. In this step, distance between all the input vectors and the reference vectors

of all nodes present on map is calculated. After calculating distance, the node which has smallest distance between reference letter and each input is determined. That is why it is called competitive process.

how parallel architecture is used for accelerating BL-SOM using dedicated hardware

**Cooperation Process **

After completing the competitive process, we determine the distance between all nodes on the map and the best matching nodes that were determined during competition process. Euclidean distance formula could be used for calculating the distance. But since it involves multiplier which increases the size of hardware. That is why Manhattan distance formula is used. This formula first subtract the reference vectors and input vectors, then finds their absolute value and finally adds the resulting values. It is noted that the distance calculation is independent for each node so the Manhattan distance calculation can be parallelized. Also computation of dimension can also be parallelized. The amount of learning is distributed to all best matching nodes on the map.

Neighborhood function is used so that learning nodes that are allocated to the nodes closer to the best matching nodes.

**Adaptation Process**

Based on the learning process, the reference letters of all the nodes are updated.

All these three processes are repeated number of times during learning. It can be observed that there is no transfer of data from competition process to cooperation process so both these methods can run in parallel and their computation process can be pipelined.

**BL-SOM Accelerator **

Here introduction to BL-SOM accelerator is given which consists of a controller, an input vector memory, individual units for all three stages.

Also read here:

https://ieeexplore.ieee.org/document/8793430

https://ieeexplore.ieee.org/document/8793430

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]]>The post What is binary arithmetic? How to add, subtract, multiply and divide two binary numbers? appeared first on EE-Vibes.

]]>What is binary arithmetic? Binary arithmetic is very interesting and easy to understand once you are familiar with its rules. In computer we need to perform addition, subtraction, multiplication, and division many times. Sometimes while **calculating the addresses** or sometimes while determining the **value of counters** and timers we need to perform addition and subtraction. Similarly **analog to digital conversion** demands to perform the division. Lets study about what is binary arithmetic in detail with examples.

The addition of binary addition follows the following rules:

- 0+0=0
- 0+1=1
- 1+1=0, carry=1

If we follow these simple rules, we can add any numbers of binary numbers easily.

Lets do some examples for understanding how binary numbers are added.

Here the addition of two binary numbers is shown. We can add two or more binary numbers using the same method. Whenever there is a carry out of MSB then the resultant is greater than the range of given bits available.

The following rules are used while performing the binary subtraction

- 0-0=0
- 1-0=0
- 1-1=0
- 0-1=1 borrow 1

Lets have a more closer look of these two examples. Lets first consider the subtraction of 10001111 from 00001111. If we say A=00001111 and B=00001111. We have performed A-B. As both numbers are 8 bits we can treat them as signed integers. The decimal value of A= 15 (since MSB=0 so it is a positive number). As the MSB of B=1 , so it is a negative number. For Finding its decimal equivalent, we need to take its **2’s complement**.

https://www.youtube.com/watch?v=pcxqOHq4yA4

1 1 1 1 1 1 1 2

1 0 0 0 1 1 1 1 __

____________________

0 1 1 1 0 0 0 1

____________________

Now convert it into decimals (0 1 1 1 0 0 0 1=113)

Now, append negative sign before it for making it a negative number. B=-113

A-B= (15)- (-113)

15+113=128

which is equal to the decimal equivalent of the resultant (1000 0000).

The same approach can be taken for interpreting the resultant of second example. One more example is shown below:

C=1 0 0 0 0 1 1 1

D=1 1 1 1 1 0 0 1 –

_________________

1 0 0 0 1 1 1 0 (C-D) =(-114) verify it by using 2’s complement !!!!

__________________

C=135, D=249, C-D=135-249=-114

C=10000111 -ve number

1 1 1 1 1 1 1 2

1 1 1 1 1 0 0 1

_______________

0 0 0 0 0 0 1 1 1 (-D)

_______________

add this -ve D into C

0 0 0 0 0 1 1 1

1 0 0 0 0 1 1 1 +

__________

1 0 0 0 1 1 1 0 =(-114)

__________

C=-121

D=-7

so, C-D=(-121)-(-7)

-121+7=-114

Lets verify the resultant.

1 0 1 0 1 1 1 0 (-ve number)

1 1 1 1 1 1 2

1 0 0 0 1 1 1 0 –

________________

0 1 1 1 0 0 1 0

________________

1 0 1 0 (10) 1 1 0 1 0 0 (52)

1 1 1- (7) 0 1 1 00 1 (25)-

____________ _______________

0 0 1 1 (3) 0 1 1 0 1 1 (27)

____________ ________________

The multiplication of binary numbers is very easy.

- 0X0=0
- 0X1=0
- 1X1=1

1 1 0 0 (12)

0 1 1 (3) X

___________________

1 1 0 0

1 1 0 0 X

0 0 0 0 X X

___________________

1 0 0 1 0 0 (36)

___________________

Dividing two binary numbers involves subtraction as major part. The following rules are adopted while performing the division of binary numbers.

- 0/0=0
- 0/1=0
- 1/1=1
- 1/0 =infinity

One easy trick for binary numbers division is: First convert both the dividend and the divisor into decimal. Since decimal numbers division is easy to do as compared to binary. After performing the decimal division, find its equivalent in binary.

The binary division is a lot simpler than the decimal division when you recall the accompanying division rules. The primary guidelines of the binary division include:

1÷1 = 1

1÷0 = Meaningless

0÷1 = 0

0÷0 = Meaningless

Like the decimal number framework, the binary division is comparable, which follows the four-venture measure:

Gap

Increase

Take away

Cut down

Significant Note: Binary division follows the long division strategy to locate the resultant in a simple manner.

Question: Solve 01111100 ÷ 0010

Arrangement:

Given

01111100 ÷ 0010

Here the profit is 01111100, and the divisor is 0010

Eliminate the zero’s in the Most Significant Bit in both the profit and divisor, that doesn’t change the estimation of the number.

So the profit gets 1111100, and the divisor gets 10.

Presently, utilize the long division technique.

Now, on the off chance that we need a whole number (entire number) answer in remainder furthermore, leftover portion structure, we’ll compose it as 101 r 1001. That is 5 leftover portion 9 in decimal. We can likewise compose that as 101 1001/1100 (5 9/12 decimal) by putting the rest of the divisor. That portion can be decreased to 101 11/100 (5 3/4 decimal) by isolating the numerator and denominator by 3. In the event that we need a “radix” answer – what could be compared to 5.75 – we can proceed with the long division measure by adding a radix point (the binary likeness a decimal point) and some following zeros on the profit. We’ll put a radix point in the remainder straightforwardly over the divisor’s too:

Presently we can proceed with the long division measure. We convey down the following digit (the first of our following zeros) and check: Can 1100 go into 10010? Note that for the deductions, we’ll overlook the radix point. Once more, the appropriate response is indeed, so we record a 1 (this one is to one side of the radix point in the remainder) and take away:

We’ll convey down the following digit once again and move over another place. Can 1100 go into 1100? Once more, the appropriate response is truly, so we record another 1 and take away. Since the consequence of this deduction is zero, we’re done and we have an accurate answer:

Similarly likewise with decimal, a few qualities won’t partition uniformly, and we’ll get a rehashing partial part. We can stop whenever, yet understand that we’ve just discovered a guess, not an accurate worth. One last note: If the divisor (the 1100 in this model) has a radix point in it, move the radix focuses in BOTH the profit and divisor to the privilege an equivalent number of spots adequate to eliminate it from the divisor. For instance, to isolate 11001.1 by 11.001, first move both radix focuses right 3 spots (you’ll need to add zeros to the profit.) Then separation 11001100 by 11001.

Also read here:

https://eevibes.com/computing/introduction-to-computing/what-is-binary-arithmetic/

What is binary arithmetic? How to add, subtract, multiply and divide two binary numbers?

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]]>