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Binary Calculator: Addition, Subtraction, Multiplication & Division

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How to Use a Binary Calculator for Maths Problems

A binary calculator is a tool that is used to estimate binary digits or numbers. The binary calculator is used for the addition of binary numbers, for the subtraction of binary numbers, multiplication, and we can also divide binary numbers very easily. A binary calculator contains a total of eleven operations which is capable of performing the logic operations on the given numbers or digits. A binary calculator provides the results in binary, decimal, and hex numbers.

We know that according to the concepts of digital electronics and mathematics, a binary number is a number expressed in terms of the base-2 number system which is also known as the binary number system. The binary number system comprises of only two digits that include: “0” (Zero or off state) and “1” (one or on-state). The binary number system (or the base-2 number system) is said to be a positional notation with a radix of 2. Furthermore, every digit is known as a bit of a binary digit.

When we say a binary number, we need to refer to each digit, for example, the binary number “1112” is simply expressed as “one one one two”. Once we exactly know about the binary term, we don’t get confused with the decimal number. Every individual binary digit (such as “0” or “1”) will be called a “bit”. For example, 1110101 is seven bits long.


Binary Addition Calculator

The binary number system is one of the numerical systems that works identical to the decimal number system that which is most widely used by everyone. The decimal number system is explained by the number 10 i.e., the decimal number system includes digits 0 to 9 for counting. In other words, a decimal system is a base 10 number system, whereas the binary number system is a base 2 number system.

In the binary number systems we use only 0 and 1, these are known as “bits” or “binary digits. All the algebraic operations such as addition, multiplication, subtraction, and division, apart from these variations, are measured according to the same principles as the decimal system.


The addition of binary numbers obeys the same rule as in the decimal addition, but it generally carries 1 instead of 10. The binary calculator executes the adding rules for the addition of binary digits. 

Electronic equipment that performs the binary addition is popularly known as a binary adder in digital electronics & communications and in general binary addition calculators. Any arithmetic and algebraic operation in digital circuits takes place in the binary form, thus, the Binary addition is one of the most fundamental & vital arithmetic operations to process the instruction.

Let us have a look at an example for binary addition:

\[\Rightarrow\] \[1 + 0\] = \[1\]

\[\Rightarrow\] \[0 + 1\] = \[0\], and carry 1 to the next more significant bit


Binary Subtraction Calculator With Steps

Similar to the binary addition, the binary subtraction of binary numbers is almost the same, excluding those emerging from the use of only numbers 0 and 1. If the amount withdrawn is greater than the number withdrawn, the number will always be borrowed.

In binary subtraction, borrowing only makes sense if 1 is subtracted from 0. In this case, 0 in the "Borrow" column becomes 2, and  1 in the "Borrow" column decreases. If the next column is also  0, then each column must be subsequently borrowed to reduce the columns with values ​​1 to 0.

Let us now have a look at the binary subtraction examples:

\[\Rightarrow\] \[1 - 0\] = \[1\]

\[\Rightarrow\] \[0 - 1\] = \[1\] ( with a borrow of 1)

Electronic equipment that performs binary subtraction is popularly known as a binary subtractor in digital electronics & communications. In general, binary subtraction calculators are used to performing the calculations. 


Binary Multiplication

Like binary addition and subtraction, binary multiplication is not as complicated as it looks. The only values ​​used are 0 and 1, so the number you add is the same as the first word or 0. Note that you need to insert placeholder 0 in each subsequent section and shift the value to the left, similar to 10 base multiplication. 

For example:

\[\Rightarrow\] \[1 \times 0\] = \[0\]

\[\Rightarrow\] \[1 \times 1\] = \[1\] (No borrow or carry method is applicable here)

Binary multiplication may seem a bit tricky, as it repeats binary additions, but it's not that hard. The binary process is the same as decimal multiplication, as shown in the example. Notice that the placeholder on the second line is 0. The zero placeholders for decimal multiplication is usually not physically visible. We can do perform all these multiplication operations on a binary multiplication calculator for quick calculations.

If 0 is not specified, you can make the mistake of excluding 0 when applying the above binary values. Remember again that in a binary system, the 0 to the right of the 1 is important, but the 0 to the left of the last 1 is not important in that sense.


Binary Division

The binary division process is similar to the tedious decimal system division. The dividends are always evenly separated by the divisor. The only major difference is the use of binary numbers instead of decimal subtraction. It is important to understand the binary subtraction of binary division. Let's use an example to understand binary division.

\[\Rightarrow\] \[1 \div 1\] = \[1\]

\[\Rightarrow\] \[0 \div 1\] = \[0\]

\[\Rightarrow\] \[1 \div 0\] = \[0 \div 0\] = meaningless

Further, a binary division calculator can be used for performing seamless calculations.


Conversion of Binary Numbers to Decimal

The conversion of binary to decimal conversion can be done in the easiest way by adding the products of each binary digit with its weight (It will be in the form of binary digit × 2 raised to a power of the position of the digit) starting from the right-most digit which has a weight of  \[2^{0}\].

Binary to decimal conversion is usually done by using two methods as listed below: 

  • The positional notation method

  • Doubling method. 

Most widely used is the positional notation method for converting binary to decimal.

In binary, 8 is considered as 1000. If we read from right to left, the 1 represents 23, the first 0 represents 22, the second 0 represents 21, and the 0 at the left represents 20. The only difference between decimal and binary systems is having a base of 2 instead of 10, rest is just similar to the decimal system. 

Let us have a look at the following conversion example;

\[(11001)_{2}\] =  \[(\textrm{1}\] \[\times\] \[2^{4})\] + \[(\textrm{1}\] \[\times\] \[2^{3})\] + \[(\textrm{0}\] \[\times\] \[2^{2})\] + \[(\textrm{0}\] \[\times\] \[2^{1})\] + \[(\textrm{1}\] \[\times\] \[2^{0})\]   = \[\textrm{16 + 1 + 0 + 0 + 1}\] = \[(18)_{10}\]


Binary to Decimal Conversion

Similarly, we can do decimal to binary conversion just like binary to decimal conversion. Now let us have a look at an example for the conversion.

For converting a decimal number to binary we divide the given decimal number by 2 till we get 0 as quotient. The first remainder obtained by dividing the given number by 2 will least significant digit and it will be placed on the rightmost. The last remainder obtained will be the most significant digit and this will be placed on the leftmost. Let us convert 21 to binary digits:


Division by 2

Quotient

Remainder

212

10

1 (LSB)

102

5

0

52

2

1

22

1

0

12

0

1 (MSB)


Therefore from the above calculation, we found that (21)10 decimal number binary equivalent is (10101)2

It is easier to convert from binary to decimal. Determine all positions of 1 and calculate the sum of the values.

These are all the types of a binary arithmetic calculator, we can even find methods for binary addition converter with steps on this page.


Did You Know?

  • We can easily convert the binary number to a decimal number (upto a count of 15) by using the “8421” method. 

  • For example, consider the binary number 1101, now let us use “8421” method to convert it to a decimal number. Place the binary number as it is without changing the positions of 8421 and then find the sum of 1’s. 


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  • By position method: \[(1101)_{2}\] = \[\textrm{1}\] \[\times\] \[2^{3}\] + \[\textrm{1}\] \[\times\] \[2^{2}\] + \[\textrm{0}\] \[\times\] \[2^{2}\] + \[\textrm{1}\] \[\times\] \[2^{0}\] = \[\textrm{8 + 4 + 0 + 1}\] = \[(13)_{10}\]

FAQs on Binary Calculator: Addition, Subtraction, Multiplication & Division

1. What is a binary calculator and what are its primary functions?

A binary calculator is a specialised tool designed to perform arithmetic operations using the binary number system (base-2), which only uses two digits: 0 and 1. Its primary functions include performing binary addition, subtraction, multiplication, and division. Many binary calculators can also convert numbers between the binary system and other systems like decimal, octal, and hexadecimal.

2. What are the four fundamental rules for performing binary addition?

Binary addition follows four simple rules, which are the basis for all calculations in digital electronics. These rules are:

  • 0 + 0 = 0

  • 0 + 1 = 1

  • 1 + 0 = 1

  • 1 + 1 = 0 (with a carry-over of 1 to the next column)

3. How is binary subtraction performed, especially when borrowing is required?

Binary subtraction is similar to decimal subtraction and follows specific rules. The most important concept is borrowing. The basic rules are 1 - 1 = 0 and 1 - 0 = 1. When you need to calculate 0 - 1, you must borrow from the next more significant bit (the column to the left). This changes the 0 to '10' (which is 2 in decimal), and the calculation becomes 10 - 1 = 1.

4. Why is the binary number system so important for digital computers and calculators?

The binary system is fundamental to all digital computing because computer hardware is built from electronic components, primarily transistors, which operate in two distinct states: on or off. These two states can be perfectly represented by the binary digits 1 (on) and 0 (off). This simple, reliable system allows computers to represent, store, and process all types of complex data and instructions efficiently.

5. What is the process for multiplying two binary numbers?

Multiplying binary numbers is very similar to decimal multiplication. You multiply the first number (the multiplicand) by each digit of the second number (the multiplier) to create partial products. The rules for multiplication are simpler than decimal: 1 × 1 = 1, and any multiplication involving a 0 results in 0. Finally, you add all the partial products together using binary addition rules to get the final result.

6. How does a binary calculator convert a binary number into its decimal equivalent?

To convert a binary number to a decimal number, a calculator uses the positional value of each digit. Starting from the rightmost digit, each binary digit is multiplied by 2 raised to the power of its position (starting from 0). For example, the binary number 1101 is converted as (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰), which equals 8 + 4 + 0 + 1, giving the decimal number 13.

7. What happens if a calculation on a binary calculator results in an overflow error?

An overflow error occurs when the result of an arithmetic operation is larger than the maximum value that can be stored in the available number of bits (e.g., an 8-bit system). When this happens, the calculator cannot represent the number correctly, leading to a wraparound effect and an incorrect result. Processors typically handle this by setting an 'overflow flag', which is a special bit that signals an error has occurred.

8. Can you explain the basic steps of binary division?

Binary division follows the same logic as decimal long division. The steps are:

  • Compare: Compare the divisor with the dividend, starting from the left. If the dividend part is greater than or equal to the divisor, you can divide.

  • Quotient: Place a '1' in the quotient.

  • Subtract: Perform binary subtraction of the divisor from that part of the dividend.

  • Bring Down: Bring down the next bit from the dividend and repeat the process until no bits are left.