Binary Calculator

- April 01, 2025

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Convert and calculate between binary, decimal, hexadecimal, and octal number systems. This calculator performs basic arithmetic operations on binary numbers and provides step-by-step conversion details.

Number Input

Enter Number:

Input Base:

Operation (Optional)

Select Operation:

Display Options

Output Base:

Show conversion steps

Format binary in groups of 4

Number System Basics

Different number systems use different bases to represent numerical values. The most common are binary (base 2), decimal (base 10), hexadecimal (base 16), and octal (base 8).

Binary (Base 2)

Uses only 0 and 1. Each position represents a power of 2. For example, binary 1010 = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10 in decimal.

Decimal (Base 10)

Uses digits 0-9. Most familiar to humans. Each position represents a power of 10.

Hexadecimal (Base 16)

Uses digits 0-9 and letters A-F (where A=10, B=11, ..., F=15). Each position represents a power of 16. For example, hex A = 10 in decimal, and FF = 15×16¹ + 15×16⁰ = 255 in decimal.

Octal (Base 8)

Uses digits 0-7. Each position represents a power of 8. For example, octal 12 = 1×8¹ + 2×8⁰ = 8 + 2 = 10 in decimal.

Logical Operations

Operation	Description	Example
AND (&)	1 if both bits are 1, otherwise 0	1010 & 1100 = 1000
OR (\|)	1 if either bit is 1, otherwise 0	1010 \| 1100 = 1110
XOR (^)	1 if bits are different, otherwise 0	1010 ^ 1100 = 0110
NOT (~)	Inverts all bits	~1010 = 0101 (simplified)
Left Shift (<<)	Shift bits left, fill with 0	1010 << 1 = 10100
Right Shift (>>)	Shift bits right, discard overflow	1010 >> 1 = 101

What Is the Binary Calculator?

The Binary Calculator is a simple, interactive tool that allows users to convert and perform arithmetic and logical operations between binary, decimal, hexadecimal, and octal number systems. Whether you're a student learning about number systems, an engineer working with binary data, or just curious about conversions, this calculator is a quick way to get precise results with helpful explanations.

Formula Used in Binary Conversion

To convert a binary number to decimal:

Decimal Value = \( b_n \times 2^n + b_{n-1} \times 2^{n-1} + \dots + b_0 \times 2^0 \)

How to Use the Binary Calculator

This calculator is user-friendly and does not require any advanced knowledge. Here’s how you can use it effectively:

Step 1: Enter your number in the input field.
Step 2: Select the base of your input (Binary, Decimal, Hexadecimal, or Octal).
Step 3 (Optional): Choose an operation like addition, subtraction, or logical operations such as AND, OR, or NOT.
Step 4: If using an operation that needs two inputs, enter the second number and its base.
Step 5: Choose the output base or select "All Bases" to view the result in binary, decimal, hex, and octal.
Step 6: Click "Calculate" to see results and step-by-step conversion details.

Features of the Calculator

Works with Binary, Decimal, Hexadecimal, and Octal numbers
Supports arithmetic operations: +, −, ×, ÷
Includes logical operations: AND, OR, XOR, NOT
Step-by-step explanations for each conversion and operation
Option to display output in multiple number systems
Formats binary in groups of 4 for better readability

Why Use This Tool?

This binary arithmetic helper is ideal for learning and verifying binary number operations. It’s especially useful for:

Students exploring binary conversion tools and logical operations
Technicians and developers needing quick number system transformations
Anyone wanting a practical alternative to manual calculations in base-2, base-8, base-10, or base-16

It also complements tools like the Percent Error Calculator (great for evaluating accuracy), the Matrix Calculator (for Linear Algebra tasks), and the Scientific Calculator (handling advanced calculations including trigonometry and exponents).

Frequently Asked Questions (FAQ)

What number bases does the calculator support?

The calculator supports Binary (Base 2), Decimal (Base 10), Hexadecimal (Base 16), and Octal (Base 8).

Can I perform logical operations?

Yes. Logical operations like AND, OR, XOR, NOT, as well as bitwise shifts, are supported and explained.

What is the benefit of seeing all base results?

It helps users understand how a number appears in different systems, useful in fields like programming and electronics.

Is this calculator suitable for learning?

Absolutely. With step-by-step breakdowns, it works like a learning tool and a calculator in one.

How does this compare with a Scientific Calculator or a Fraction Calculator?

While a Scientific Calculator focuses on functions like trigonometry or exponentials, and a Fraction Calculator helps simplify or compare fractions, this Binary Calculator focuses on number system conversions and binary operations.

Conclusion

Whether you want to convert a binary number, understand how base conversions work, or perform logical bitwise operations, this calculator makes it easy. It’s quick, informative, and helpful across educational and professional tasks.