Binary Converter
Convert between binary and decimal
Decimal to Binary
Binary System:
Binary uses only 0 and 1. Each digit represents a power of 2.
Binary to Decimal
Pro Tip
Binary is the foundation of all computer systems. Every piece of data is stored as 0s and 1s!
Privacy & Security
Your conversions are completely private. All binary calculations are performed locally in your browser - no data is transmitted, stored, or tracked. Use the converter freely with complete confidentiality.
What is a Binary Converter?
A binary converter is an essential tool that translates between the decimal number system we use daily (base 10) and the binary number system (base 2) that forms the foundation of all digital computing. Binary represents numbers using only two digits: 0 and 1, called bits (binary digits). Each position in a binary number represents a power of 2, just as each position in decimal represents a power of 10. For example, the decimal number 13 converts to binary 1101, which breaks down as (1×8) + (1×4) + (0×2) + (1×1) = 8+4+0+1 = 13. Understanding binary is fundamental to computer science, programming, digital electronics, and data storage because computers process all information as sequences of binary digits at the hardware level. Every character you type, image you view, song you hear, and program you run exists inside computers as patterns of 1s and 0s. Binary's simplicity makes it ideal for electronic circuits where transistors have two states: on (1) or off (0). The binary converter serves multiple purposes: helping students learn fundamental computer science concepts, enabling programmers to understand low-level data representation, assisting in digital circuit design and debugging, supporting network engineers working with IP addresses and subnet masks, aiding in understanding file sizes and memory addresses, and solving problems in mathematics and logic. While humans naturally think in decimal, learning to read and convert binary provides insight into how computers process information. A byte contains 8 bits and can represent 256 different values (2^8), from binary 00000000 to 11111111 (decimal 0 to 255). This 8-bit structure forms the basis of character encoding, color representation (RGB values), and countless other digital systems. Understanding binary also clarifies concepts like megabytes (roughly 1 million bytes) and gigabytes (roughly 1 billion bytes), though technically defined as powers of 2 (1 mebibyte = 2^20 = 1,048,576 bytes). The converter demonstrates that the same quantity can be represented in different number systems - the value doesn't change, only its notation. This concept extends to hexadecimal (base 16) and octal (base 8) systems also used in computing.
Key Features
Bidirectional Conversion
Convert from decimal to binary and from binary to decimal with equal ease
Step-by-Step Breakdown
See detailed conversion process showing how each binary digit contributes to the final value
Large Number Support
Handle conversions for numbers ranging from 0 to billions with accurate results
Negative Number Conversion
Convert negative numbers using two's complement representation used in computers
Bit Count Display
Shows how many bits are required to represent each number
Binary Arithmetic Examples
Learn how computers add, subtract, and manipulate binary numbers
Real-Time Validation
Instant error detection for invalid binary input (digits other than 0 and 1)
Educational Explanations
Understand binary number system fundamentals with clear, accessible explanations
How to Use the Binary Converter
Select Conversion Direction
Choose whether you want to convert from decimal to binary or from binary to decimal. The input field adjusts accordingly to accept the appropriate number format.
Enter Your Number
Type the decimal number (using digits 0-9) or binary number (using only 0s and 1s). The converter validates your input in real-time and shows errors for invalid entries.
View Instant Results
See the converted result immediately. For decimal to binary, you'll see the binary representation. For binary to decimal, you'll see the numerical value.
Read the Breakdown
Review the step-by-step explanation showing exactly how the conversion works, including positional values and calculations for educational understanding.
Check Bit Information
See additional information like the number of bits used, the maximum value representable with that many bits, and other relevant binary details.
Copy or Convert More
Copy the result for use in programming or documentation, or convert additional numbers to learn patterns and build binary fluency.
Binary Conversion Tips
- Memorize Powers of 2: Learn powers of 2 from 2^0 to 2^16: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536 for faster conversions.
- Practice Small Numbers: Master converting numbers 0-31 until they become automatic. These 5-bit patterns appear frequently and build intuition for larger numbers.
- Use the Rightmost Bit Trick: The rightmost bit instantly tells you if a number is odd (1) or even (0) without full conversion.
- Recognize Common Patterns: All 1s means one less than the next power of 2 (1111 = 15 = 16-1). A single 1 followed by 0s is a power of 2 (10000 = 16).
- Check Your Work: After converting decimal to binary, convert back to decimal to verify accuracy. This catches mistakes and builds confidence.
- Understand Context Matters: The same binary pattern means different things in different contexts: 11111111 could be 255 (unsigned), -1 (two's complement), or 'ÿ' (character).
Frequently Asked Questions
What is binary and why do computers use it?
Binary is a base-2 number system using only two digits (0 and 1) to represent all numbers, in contrast to the decimal (base-10) system that uses ten digits (0-9). Computers use binary because of the physical nature of digital electronics. Computer processors and memory are built from billions of transistors - tiny electronic switches that have two stable states: on or off, conducting or non-conducting, high voltage or low voltage. These two states perfectly map to binary's two digits, making binary the natural language of digital electronics. Representing 0 as off/low voltage and 1 as on/high voltage allows computers to reliably store and process information using simple electronic components. Attempting to use decimal in electronic circuits would require distinguishing between ten different voltage levels, which is unreliable due to electrical noise, component variation, and signal degradation. Binary's two-state system is robust and error-resistant - a transistor is clearly either on or off with little ambiguity. This reliability is crucial when billions of operations occur every second. Additionally, binary makes logical operations straightforward using Boolean algebra, where AND, OR, NOT, and other logical functions operate on binary values, enabling complex decision-making and computation. All data in computers - numbers, text, images, sounds, videos, programs - ultimately reduces to binary patterns. A character like 'A' is stored as binary 01000001 (decimal 65 in ASCII encoding), a color like pure red is 11111111 00000000 00000000 (RGB 255, 0, 0), and program instructions are binary codes the processor interprets. While programmers rarely work directly in binary (using higher-level languages instead), understanding binary is fundamental to computer science because it reveals how computers actually represent and manipulate information at the hardware level. The simplicity and reliability of binary make it perfect for the physical implementation of computation.
How do you convert decimal numbers to binary?
Converting decimal to binary involves repeatedly dividing the decimal number by 2 and tracking the remainders, which become the binary digits. The process works as follows: divide the decimal number by 2 and note the remainder (either 0 or 1) and the quotient. Continue dividing each successive quotient by 2, recording remainders, until the quotient reaches 0. The binary result is the sequence of remainders read in reverse order (bottom to top). For example, to convert decimal 13 to binary: 13÷2 = 6 remainder 1 (rightmost binary digit); 6÷2 = 3 remainder 0; 3÷2 = 1 remainder 1; 1÷2 = 0 remainder 1 (leftmost binary digit). Reading remainders bottom to top gives 1101, which is binary for 13. This method works because it essentially decomposes the number into powers of 2. Alternatively, you can use the subtraction method: find the largest power of 2 that doesn't exceed the decimal number, subtract it, mark a 1 in that position, and repeat with the remainder until you reach 0. For 13: largest power of 2 ≤ 13 is 8 (2^3), so mark 1 in the 8s position (1000); remainder is 5, largest power of 2 ≤ 5 is 4 (2^2), mark 1 in the 4s position (1100); remainder is 1, which is 2^0, mark 1 in the 1s position (1101). Both methods produce the same result. For programmers, most languages provide built-in functions for conversion - in Python, bin(13) returns '0b1101', in JavaScript, (13).toString(2) returns '1101'. Understanding the manual process, however, builds intuition about how binary representation works. Each binary digit (bit) represents a power of 2: the rightmost bit is 2^0 (1), next is 2^1 (2), then 2^2 (4), 2^3 (8), 2^4 (16), and so on. A number's binary representation shows which powers of 2 sum to that number - 1101 means 8+4+0+1. With practice, small numbers become recognizable: 1111 is always 15, 10000 is always 16, 11111111 is always 255 (8 bits all set).
How do you convert binary numbers to decimal?
Converting binary to decimal is more straightforward than the reverse process - you multiply each binary digit by its positional value (a power of 2) and sum the results. Binary digits are numbered from right to left starting at position 0. Position 0 (rightmost) represents 2^0 (1), position 1 represents 2^1 (2), position 2 represents 2^2 (4), position 3 represents 2^3 (8), and so on, doubling each position. For example, to convert binary 1101 to decimal: start from the right: (1 × 2^0) + (0 × 2^1) + (1 × 2^2) + (1 × 2^3) = (1×1) + (0×2) + (1×4) + (1×8) = 1 + 0 + 4 + 8 = 13. Any binary digit that's 0 contributes nothing to the sum, while digits that are 1 contribute their positional value. For binary 10110: (0×1) + (1×2) + (1×4) + (0×8) + (1×16) = 0+2+4+0+16 = 22. A shortcut is to only add the positions where the bit is 1, ignoring positions with 0. For 10110, that's positions 1, 2, and 4, giving values 2, 4, and 16, which sum to 22. This conversion reveals that binary is just another way of writing numbers using powers of 2 instead of powers of 10. In decimal, the number 3,245 means (3×1000) + (2×100) + (4×10) + (5×1) using powers of 10 (10^3, 10^2, 10^1, 10^0). Binary works identically but with powers of 2. After practicing conversions, patterns emerge: any binary number ending in 1 is odd (since the 2^0 position is 1), while ending in 0 is even. A binary number with all 1s equals 2^n - 1 where n is the number of bits (1111 = 15 = 2^4-1). Leading zeros don't change the value (01101 = 1101 = 13), just like in decimal (0025 = 25). Common binary numbers become familiar: 1010 is 10, 1100 is 12, 11111111 is 255. Understanding this conversion process is essential for programming because it clarifies how computers interpret binary data in memory and why numbers like 255, 256, 1024, and 65535 appear frequently in computing (they're powers of 2 or related values).
What is a bit and what is a byte?
A bit (binary digit) is the smallest unit of data in computing, representing a single binary value of either 0 or 1. Bits are the atomic unit of digital information - everything in computers ultimately reduces to patterns of bits. A single bit can represent two states: yes/no, true/false, on/off, or any two-option choice. However, single bits have limited expressive power, so they're grouped into larger units. A byte is a group of 8 bits, representing the standard unit for measuring data and memory in computers. With 8 bits, a byte can represent 256 different values (2^8 = 256), from 00000000 to 11111111 in binary, or 0 to 255 in decimal. Bytes are ubiquitous in computing: each character in a text file typically occupies one byte (in ASCII encoding), computer memory addresses are byte-addressable, file sizes are measured in bytes (kilobytes, megabytes, gigabytes), and data transmission rates often use bytes per second. The 256 values in a byte are perfect for representing characters (ASCII uses values 0-127, extended ASCII uses 0-255), small integers, or individual components of data like red, green, and blue color values (each 0-255) in RGB images. Larger numbers require multiple bytes: a 16-bit integer uses 2 bytes (values 0-65,535 unsigned), a 32-bit integer uses 4 bytes (values 0-4,294,967,295 unsigned or approximately -2 billion to +2 billion signed), and a 64-bit integer uses 8 bytes. Understanding bits and bytes clarifies why certain numbers are significant in computing: 255 is the maximum value in one byte, 256 is the number of values representable in one byte, 1024 is 2^10 (often approximated as '1,000' leading to kilobyte definitions), 65,536 is 2^16 (two bytes), and so on. Terminology for larger units follows: kilobyte (KB) is roughly 1,000 bytes (technically 1,024), megabyte (MB) is roughly 1 million bytes (technically 1,024 KB), gigabyte (GB) is roughly 1 billion bytes (technically 1,024 MB), and terabyte (TB) is roughly 1 trillion bytes (technically 1,024 GB). The discrepancy between decimal approximations (1,000) and binary reality (1,024) causes confusion - a '1 GB' drive might show as 0.93 GB because manufacturers use decimal definitions while computers use binary.
How do computers represent negative numbers in binary?
Computers represent negative numbers using a system called two's complement, which enables both positive and negative integers to coexist in binary while allowing normal binary arithmetic to work correctly for both. In two's complement, the leftmost bit (called the sign bit) indicates whether the number is positive (0) or negative (1), but the representation isn't simply flipping the sign bit. For an n-bit two's complement number, positive values range from 0 to 2^(n-1)-1, while negative values range from -1 to -2^(n-1). In 8-bit two's complement, values range from -128 to +127. To convert a positive number to its negative equivalent in two's complement: first, write the positive number in binary; second, invert all the bits (change 0s to 1s and 1s to 0s) - this is called the one's complement; third, add 1 to the result. For example, to represent -13 in 8-bit two's complement: +13 is 00001101; invert to get 11110010; add 1 to get 11110011 = -13. This system has several advantages: there's only one representation for zero (all 0s), whereas older signed magnitude and one's complement systems had both +0 and -0. More importantly, two's complement allows subtraction to be performed as addition: to compute 5-3, the computer instead computes 5+(-3), converting 3 to -3 via two's complement and adding. Normal binary addition rules work correctly, with overflow bits discarded, producing the right answer. This eliminates the need for separate subtraction circuitry in processors. The sign bit effectively has a negative weight: in 8-bit two's complement, bits represent (from left to right) -128, 64, 32, 16, 8, 4, 2, 1. So 11110011 = -128+64+32+16+2+1 = -13. All negative numbers have the leftmost bit set to 1, while all non-negative numbers have it set to 0, making sign determination instant. When expanding a two's complement number to more bits (like 8-bit to 16-bit), you perform sign extension: copy the sign bit into all new leftmost positions. For 11110011 (-13), sign-extending to 16 bits gives 1111111111110011, still representing -13. Two's complement is virtually universal in modern computers for representing signed integers.
What are binary operations and how are they used?
Binary operations are mathematical and logical operations performed directly on binary numbers, forming the foundation of all computer processing. The most fundamental are bitwise operations that work on individual bits. The AND operation compares corresponding bits from two numbers and returns 1 only when both bits are 1, otherwise 0 - for example, 1101 AND 1011 = 1001. The OR operation returns 1 when either bit or both bits are 1, and 0 only when both are 0 - 1101 OR 1011 = 1111. The XOR (exclusive OR) operation returns 1 when bits differ and 0 when they match - 1101 XOR 1011 = 0110. The NOT operation inverts all bits, changing 0s to 1s and 1s to 0s - NOT 1101 = 0010 (in 4-bit representation). Shift operations move all bits left or right: left shift multiplies by 2 for each position (1101 << 1 = 11010, effectively 13×2=26), while right shift divides by 2 (1101 >> 1 = 0110, effectively 13÷2=6 with truncation). These operations have crucial applications: AND with a bitmask extracts specific bits (checking if a number is even uses AND with 1, checking the rightmost bit). OR sets specific bits to 1 (used in setting flags). XOR toggles specific bits and is used in encryption, checksums, and detecting changes. NOT inverts bit patterns. Shifts implement efficient multiplication and division by powers of 2, far faster than general multiplication. Programmers use these operations extensively for: manipulating individual bits in flags and settings, implementing efficient algorithms, working with hardware registers, compressing data, implementing cryptographic algorithms, and optimizing performance-critical code. For example, checking if a number is a power of 2 uses (n AND (n-1)) == 0. Swapping two variables without a temporary variable uses XOR: a = a XOR b; b = a XOR b; a = a XOR b. Computer graphics, network protocols, embedded systems, and operating systems rely heavily on bitwise operations for efficiency and precise control over data at the bit level. Understanding these operations reveals how computers perform complex tasks through combinations of simple binary manipulations.
Why do computers use powers of 2 for memory and storage sizes?
Computer memory and storage sizes are powers of 2 (2^10=1024, 2^20≈1 million, 2^30≈1 billion) because of binary addressing and the fundamental architecture of digital systems. Memory is organized as an array of locations, each identified by a unique address. With n address bits, you can uniquely identify 2^n different memory locations. For example, 10 address bits can identify 1,024 locations (2^10), 20 bits can address about 1 million locations (2^20 = 1,048,576), and 32 bits can address about 4 billion locations (2^32 = 4,294,967,296). Using power-of-2 sizes allows efficient binary addressing where each additional address bit doubles the addressable space. Computer architectures are designed around binary: registers are sized in powers of 2 (8-bit, 16-bit, 32-bit, 64-bit processors), data buses transfer data in power-of-2 widths, and memory chips are manufactured with power-of-2 capacities because the internal circuitry uses binary addressing. This creates a natural preference for power-of-2 quantities throughout computing. The common term 'kilobyte' originally meant 1,024 bytes (2^10), not 1,000, because this aligned with binary addressing. Similarly, 'megabyte' meant 1,048,576 bytes (2^20), 'gigabyte' meant 1,073,741,824 bytes (2^30), and so forth. However, hard drive manufacturers adopted decimal definitions (1 KB = 1,000 bytes) for marketing reasons, creating confusion. To resolve this, international standards now define 'kibibyte' (KiB) = 1,024 bytes, 'mebibyte' (MiB) = 1,048,576 bytes, and 'gibibyte' (GiB) = 1,073,741,824 bytes for binary-based units, reserving KB, MB, GB for decimal-based units, though adoption varies. This explains why a '500 GB' hard drive shows as only 465 GiB in operating systems - different definitions. The prevalence of power-of-2 sizes throughout computing means numbers like 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, and 65536 appear constantly in technical specifications, memory configurations, buffer sizes, and maximum values. Understanding this binary foundation clarifies why these specific numbers are ubiquitous in computing and why memory comes in sizes like 8 GB, 16 GB, or 32 GB rather than 10 GB, 20 GB, or 30 GB.
How is binary used in IP addresses and networking?
Binary is fundamental to IP addressing and networking because network protocols and routing decisions operate on binary representations of addresses. An IPv4 address like 192.168.1.1 appears in decimal for human readability, but computers process it as a 32-bit binary number: 11000000.10101000.00000001.00000001. Each of the four decimal sections (called octets) represents 8 bits (one byte), so values range from 0-255. Understanding binary is crucial for subnet masks and CIDR notation. A subnet mask like 255.255.255.0 in binary is 11111111.11111111.11111111.00000000, where consecutive 1s represent the network portion of an address and 0s represent the host portion. Performing a bitwise AND operation between an IP address and its subnet mask extracts the network address. For example, 192.168.1.50 AND 255.255.255.0 gives 192.168.1.0 (the network address). CIDR notation like 192.168.1.0/24 indicates that the first 24 bits are the network portion (the '/24' means 24 consecutive 1s in the subnet mask). This binary understanding enables calculating how many hosts a subnet can contain: a /24 network has 8 bits for hosts (32 total minus 24 network bits), so 2^8 = 256 total addresses, minus 2 reserved addresses (network and broadcast) = 254 usable host addresses. Network engineers regularly work with binary to design subnets, troubleshoot routing issues, and understand access control lists (ACLs). IPv6 addresses are 128 bits (written in hexadecimal for compactness), and binary understanding remains essential for subnetting. Binary operations also appear in network protocols: TCP flags are individual bits indicating packet properties (SYN, ACK, FIN, etc.), accessed via bitmask operations. Understanding binary makes networking concepts like address allocation, subnetting, supernetting, and VLSM (Variable Length Subnet Masking) much clearer. Many networking certification exams test binary-to-decimal conversion specifically for subnet calculation. The binary foundation of networking explains why certain address ranges are special: 127.0.0.0/8 (loopback), 10.0.0.0/8, 172.16.0.0/12, and 192.168.0.0/16 (private addresses) - these ranges were chosen for specific binary patterns in their network portions.
Why Use Our Binary Converter?
Understanding binary is fundamental to computer science, programming, and digital electronics. Our binary converter makes learning and using binary effortless, whether you're a student mastering number systems, a programmer debugging low-level code, a network engineer calculating subnets, or a curious learner exploring how computers work. With instant bidirectional conversion, step-by-step explanations, and educational content, you'll build binary fluency quickly. No registration required - just enter your number and discover the binary representation that powers all digital technology.