Binary Calculator
Result
Binary Result
Decimal Result
Understanding Binary Numbers
Binary is a base-2 number system that uses only two digits: 0 and 1. It's the fundamental language of computers and digital systems. Our binary calculator helps you perform calculations and conversions between binary and decimal number systems.
Binary Number System Basics
In the binary system, each digit represents a power of 2, starting from the right (which is 2⁰):
Practical Example: Binary to Decimal
Convert binary 1101 to decimal:
1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13
So, 1101 (binary) = 13 (decimal)
Decimal to Binary Conversion
To convert a decimal number to binary, repeatedly divide by 2 and record the remainders:
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading remainders from bottom to top: 1101
Practical Example: IP Subnetting
Network engineers use binary to calculate IP subnets. Understanding binary is essential for determining network addresses, broadcast addresses, and host ranges in subnetting calculations.
Binary Arithmetic Operations
Binary arithmetic follows the same rules as decimal arithmetic but with only two digits:
Binary Addition
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (write 0, carry 1)
Binary Subtraction
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (with borrow from next column)
Practical Example: Binary Addition
Add binary 1011 and 1101:
1011
+ 1101
------
11000
(11 + 13 = 24 in decimal)
Applications of Binary Numbers
- Computer Systems: All digital electronics and computers use binary to represent and process data
- Data Storage: Files, images, and programs are stored as binary data
- Networking: IP addresses and subnet masks use binary for network calculations
- Cryptography: Many encryption algorithms operate on binary data
- Digital Communications: Binary encoding is used in all digital transmission systems
Binary Representation Methods
Signed Binary Numbers
Negative numbers can be represented in binary using several methods:
- Sign-Magnitude: Leftmost bit indicates sign (0=positive, 1=negative)
- One's Complement: Negative numbers are represented by inverting all bits
- Two's Complement: Most common method - invert bits and add 1
Binary Coded Decimal (BCD)
Each decimal digit is represented by its 4-bit binary equivalent:
Frequently Asked Questions
Q: Why do computers use binary?
A: Computers use binary because electronic components can reliably represent and process two distinct states (on/off, high/low voltage) more easily than multiple states. Binary simplifies hardware design and improves reliability.
Q: How do I convert a large decimal number to binary?
A: Use the division-by-2 method: Divide the number by 2, record the remainder, and continue dividing the quotient by 2 until you reach 0. The binary number is the sequence of remainders read from bottom to top.
Q: What is the binary equivalent of hexadecimal numbers?
A: Each hexadecimal digit (0-9, A-F) corresponds to a 4-bit binary sequence. For example, hexadecimal 'A' is 1010 in binary, and 'F' is 1111.
Q: How do I perform binary multiplication?
A: Binary multiplication follows the same principles as decimal multiplication but with simpler rules (0×0=0, 0×1=0, 1×0=0, 1×1=1). You create partial products and then add them together, similar to long multiplication in decimal.