Binary Calculator

Decimal to Binary

Binary to Decimal

Binary result Decimal result Hex result Calculation steps

The ultimate Binary Calculator is a fast, free, and reliable tool that simplifies binary math like never before. Whether you’re a student learning number systems, a programmer working with binary logic, or someone in digital electronics, this calculator is designed to save your time and effort. From basic binary arithmetic operations, such as addition, subtraction, multiplication, and division, to advanced calculations, everything is handled with just a few clicks, eliminating the need for manual conversion that takes a lot of time and energy.

In addition to binary calculations, you can easily convert values between binary and decimal, binary and hexadecimal, or even view your results in octal format. The tool provides instant answers in multiple formats, allowing you to understand the logic clearly and avoid calculation mistakes. Bitwise operations like AND, OR, NOT, XOR, left, and right shifts are also supported, making it a versatile calculator for low-level computing tasks or circuit design.

What is a Binary Calculator?

A binary calculator is a specialized digital tool designed to perform arithmetic and logical operations using binary numbers, the base-2 numeral system composed only of 0s and 1s. Unlike standard calculators that work with decimal numbers (base-10), binary calculators help users input, manipulate, and compute values in binary format, which is fundamental in computer science, digital electronics, and programming.

By providing immediate and accurate results, binary calculators serve as an essential learning aid for students, developers, and engineers who work closely with computer systems and digital logic.

Supported Operations with the Binary Calculator:

By providing immediate and accurate results, binary calculators serve as an essential learning aid for students, developers, and engineers who work closely with computer systems and digital logic.

Supported Operations with the Binary Calculator:

Quick Binary Practice Examples

Add 1001₂ + 1110₂ = 10111₂
Subtract 1010₂ from 1100₂ = 10₂
Multiply 101₂ × 10₂ = 1010₂
Divide 11010₂ by 11₂ = Quotient: 101₂, Remainder: 1₂

Binary Addition:

Binary addition works similarly to decimal addition but only uses two digits, 0 and 1. The rules are:

Operand 1

Operand 2

Result

carry

Binary Addition Example

Example: Add 1011₂ and 1101₂

  1011
+ 1101
-------
 11000

Step-by-step:

1 + 1 = 0 (carry 1)
1 + 0 + carry 1 = 0 (carry 1)
0 + 1 + carry 1 = 0 (carry 1)
1 + 1 + carry 1 = 1 (carry 1 leftmost)

Final result: 11000₂

Binary Subtraction:

Binary subtraction works like decimal subtraction, with borrowing when subtracting 1 from 0.Example:
Subtract 1010₂ from 1101₂

Markdown

Operand 1

Operand 2

Result

Carry-Back

Binary Subtraction Example

Example: Subtract 1011₂ from 1101₂

  1101
- 1010
-------
  0011

Step-by-step:

1 - 0 = 1
0 - 1 (borrow 1) = 1
1 - 0 = 1
1 - 1 = 0

Final result: 0011₂

Binary Multiplication:

Binary multiplication is similar to decimal multiplication but simpler since it only involves multiplying by 0 or 1.

Binary Multiplication Example

Example: Multiply 101₂ by 11₂

   101
 ×  11
-------
   101     (101 × 1)
+ 1010     (101 × 1 shifted left)
-------
  1111

Step-by-step:

Multiply 101 by the rightmost 1 → 101
Multiply 101 by the left 1 and shift left → 1010
Add the results: 101 + 1010 = 1111

Final result: 1111₂

Binary Division:

Binary multiplication is similar to decimal multiplication but simpler since it only involves multiplying by 0 or 1.

Binary Division Example

Example: Divide 1101₂ by 10₂

1101 ÷ 10 = 110 remainder 1

Step-by-step:

10 fits into 11 once → quotient bit 1
Subtract 10 from 11 → remainder 1
Bring down next bit 0 → 10 fits once → quotient bit 1
Subtract 10 → remainder 0
Bring down next bit 1 → 10 does not fit → quotient bit 0

Final quotient: 110₂ (decimal 6)

Remainder: 1

Binary Table:

Understanding the binary and decimal number correlation is essential. Here’s a basic reference table:

Decimal

Binary

100

101

110

111

1000

1001

Potential Uses of a Binary Calculator

A binary calculator is an authentic calculator that is very useful in the fields of computer science, circuit design, networking, electronics, and cryptography. Some of them are discussed below:

Programming:

This is very helpful for programming. Binary arithmetic in the algorithm is used for low-level programming, data processing, and logic gates. It also plays an important role in bitwise operations, which is useful for memory management and optimizing code.

Data Communication and Networking:

It is valuable for IP address calculations. It plays an important role in error detection and correction of supporting data encoding methods like CRC( Cyclic Redundancy Check).

Circuit designs and Digital electronics:

It aids in microprocessor and calculating systems. It also helps engineers to create and test digital circuits based on binary logic.

In the Educational System:

It is a useful tool for the student to learn about the binary system, Boolean algebra, Digital systems, and binary arithmetic. It also gives us a way to learn about machine-level operations, how the computer works, and store data.

What is a binary number?

A binary number is a number expressed using only two digits: 0 and 1. This system is known as the base-2 numeral system, in contrast to the decimal system (base-10), which uses digits from 0 to 9.

In binary, each digit (called a bit) represents an increasing power of 2, starting from the right.

Binary to Decimal Conversion

Binary: 1011₂
Powers of 2: 2³, 2², 2¹, 2⁰
Calculation: 8 + 0 + 2 + 1
Decimal: 11₁₀

What is a binary number?

In binary, each digit (called a bit) represents an increasing power of 2, starting from the right.

Binary to Decimal Example

Binary: 1011₂
Powers of 2: 2³, 2², 2¹, 2⁰
Decimal Conversion: 8 + 0 + 2 + 1 = 11

Why is Binary Important?

Binary numbers are fundamental to computing because digital systems operate using two electrical states: ON (1) and OFF (0). Everything from storage to processing, including internet data and media files, relies on binary.

A binary calculator makes it easy to convert between binary, decimal, hexadecimal, and octal systems, which is essential for programmers, engineers, and students.

What is a Decimal Number System?

Understanding the Decimal Number System

The decimal number system, also known as the base-10 numeral system, is the most commonly used numbering system in everyday life. It consists of ten digits ranging from 0 to 9.

This system is the foundation of most human counting and mathematical operations because it aligns naturally with our ten fingers.

Each position in a decimal number represents a power of 10, starting from the rightmost digit:

Rightmost digit → 10⁰ = 1 (ones place)
Next digit to the left → 10¹ = 10 (tens place)
Next digit → 10² = 100 (hundreds place)
And so on...

Example:
543 = (5 × 10²) + (4 × 10¹) + (3 × 10⁰)
= 500 + 40 + 3

Because of its simplicity and universality, the decimal system is used globally in commerce, science, education, and daily calculations.

It also serves as the foundation for converting to other number systems like binary and hexadecimal, especially in computing.

Understanding the decimal system is essential before moving on to more specialized numbering systems, such as binary, which computers use at their most basic level.

Converting To and From Binary Numbers:

Converting between binary and decimal numbers is a fundamental concept in computer science. A binary calculator is a helpful tool that can quickly and accurately perform these conversions, but understanding the underlying process helps build a solid foundation.

Binary to Decimal Conversion:

To convert a binary number to its decimal equivalent, multiply each digit by 2 raised to the power of its position (starting from 0 at the rightmost digit), and then sum the results.

Binary to Decimal Conversion Formula

To convert a binary number to decimal, we use the following formula:

Decimal = (b_n × 2ⁿ) + (b_n−1 × 2ⁿ⁻¹) + … + (b₁ × 2¹) + (b₀ × 2⁰)

b = binary digit (0 or 1)
n = position from the right, starting at 0

This is the same logic used by a binary calculator for precise conversion.

Binary to Decimal Conversion Examples

Let us look at some binary-to-decimal examples to understand how the formula works:

Example 1: Binary = 1010
= (1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰)
= 8 + 0 + 2 + 0 = 10

Example 2: Binary = 1101
= (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
= 8 + 4 + 0 + 1 = 13

Example 3: Binary = 1001
= (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)
= 8 + 0 + 0 + 1 = 9

You can also use an online binary calculator to perform these calculations instantly and accurately.

Binary to Decimal Conversion Table:

The following table shows the binary equivalents of the first 20 decimal numbers:

Decimal

Binary

100

101

110

111

1000

1001

1010

1011

1100

1101

1110

11111

10000

10001

10010

10011

10100

Decimal to Binary Conversion:

Converting decimal numbers to binary involves dividing the number by 2 repeatedly and recording the remainders. A binary calculator can help automate this, but understanding the manual process helps solidify the concept.

Decimal to Binary Conversion Formula:

To convert a decimal number to binary, there is a direct mathematical formula, like binary to decimal. Instead, the process follows a repeated division by 2, and you collect the remainders.

However, it can be described algorithmically as:

Step-by-step formula:

Divide the decimal number by 2.
Record the remainder (0 or 1).
Use the quotient as the new number.
Repeat steps 1–3 until the quotient becomes 0.
The binary number is the reversed sequence of remainders.

Keep dividing the decimal number by 2 and collecting the remainders, and reverse them at the end.

Decimal to Binary Conversion Examples

Let’s see a few step-by-step examples:

Example 1: Decimal = 10

10 ÷ 2 = 5, remainder = 0
5 ÷ 2 = 2, remainder = 1
2 ÷ 2 = 1, remainder = 0
1 ÷ 2 = 0, remainder = 1

Binary: 1010₂

Example 2: Decimal = 13

13 ÷ 2 = 6, remainder = 1
6 ÷ 2 = 3, remainder = 0
3 ÷ 2 = 1, remainder = 1
1 ÷ 2 = 0, remainder = 1

Binary: 1101₂

Example 3: Decimal = 7

7 ÷ 2 = 3, remainder = 1
3 ÷ 2 = 1, remainder = 1
1 ÷ 2 = 0, remainder = 1

Binary: 111₂

You can double-check your result with a binary calculator to ensure accuracy.

Decimal to Binary Conversion Table:

Here’s a reference table for the first 20 decimal numbers and their binary equivalents:

Decimal

Binary

100

101

110

111

1000

1001

1010

1011

1100

1101

1110

1111

10000

10001

10010

10011

10100

Positional Notation Method:

The value of a digit in a binary number is determined by its position is known as the positional notation method, weighted by powers of 2. You should follow these steps:

Multiply each binary digit, starting from the rightmost digit, by powers of 2. Begin with 20 and increase the exponent by 1 for each digit as you move to the left.
Add the results of all these products. The total represents the decimal equivalent of the binary number.

Conclusion:

A binary calculator is more than just an easy tool, it’s an important tool for anyone working with binary and decimal number systems. In fields like computer science, electronics, programming, and data encoding, binary numbers form the backbone of how machines process and store information. Manually converting between binary and decimal using formulas is the best way to learn, but it can be time-consuming and error-prone, especially with large values.

It allows you to instantly convert numbers between systems and reduces the potential for mistakes. If you’re a student checking homework, a developer fixing low-level code, having access to a binary calculator saves time and boosts productivity. Many modern calculators provide advanced features like signed/unsigned conversions, bitwise operations, and support for hexadecimal and octal systems, making them ideal for professionals who need more than just basic number translation.

Frequently Asked Questions (FAQs)

What is the difference between binary and decimal?

Binary is a base-2 system using digits 0 and 1, while decimal is a base-10 system using digits 0 through 9.

What is a binary calculator?

A binary calculator is a digital tool used to perform binary arithmetic operations and conversions between binary, decimal, octal, and hexadecimal number systems.

Can a binary calculator perform bitwise operations?

Some advanced binary calculators support bitwise operations such as AND, OR, XOR, and bit shifts.

Why is the binary number system important?

Binary is the foundation of digital computing systems, as it represents data using only two states: on (1) and off (0).

What is the fastest way to convert a decimal to binary?

Use a binary calculator for instant conversion or divide the decimal number by 2 repeatedly, recording remainders.

How to convert 11011 binary to decimal?

1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 16 + 8 + 0 + 2 + 1 = 27

Can I use a binary calculator on my phone?

Yes, many binary calculator apps are available for Android and iOS, and most websites offering them are mobile-friendly.