Binary Hex Decimal Octal Converter
Solve binary hex decimal octal problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Binary Hex Decimal Octal Converter
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Formula: N = d(n)*b^n + d(n-1)*b^(n-1) + ... + d(1)*b^1 + d(0)*b^0
Worked example — Binary: 11111111 | Hex: FF | Octal: 377
Formula
N = d(n)*b^n + d(n-1)*b^(n-1) + ... + d(1)*b^1 + d(0)*b^0
Any number N in base b is the sum of each digit d multiplied by the base raised to its positional power. For example, hex 1A3 = 1*16^2 + 10*16^1 + 3*16^0 = 256 + 160 + 3 = 419 in decimal.
Worked Examples
Example 1: Convert Decimal 255 to All Bases
Problem:Convert the decimal number 255 to binary, hexadecimal, and octal.
Solution:Decimal 255: Binary: 255 / 2 repeatedly gives 11111111 (8 ones) Hexadecimal: 255 / 16 = 15 remainder 15, so FF Octal: 255 / 8 = 31 r7, 31/8 = 3 r7, 3/8 = 0 r3, so 377 BCD: 0010 0101 0101 Bits: 8 | Bytes: 1 | All bits set
Result:Binary: 11111111 | Hex: FF | Octal: 377
Example 2: Convert Binary 10110100 to Other Bases
Problem:Convert the binary number 10110100 to decimal, hexadecimal, and octal.
Solution:Binary: 10110100 Decimal: 1*128 + 0*64 + 1*32 + 1*16 + 0*8 + 1*4 + 0*2 + 0*1 = 180 Hex: group into nibbles: 1011 0100 = B4 Octal: group into 3-bit: 010 110 100 = 264 ASCII: not printable (180 > 126)
Result:Decimal: 180 | Hex: B4 | Octal: 264
Frequently Asked Questions
What are the different number base systems used in computing?
The four most common number base systems are binary (base-2), octal (base-8), decimal (base-10), and hexadecimal (base-16). Binary uses only 0 and 1, representing the on/off states of electronic circuits, making it the native language of computers. Octal uses digits 0-7 and was historically used in early computing systems. Decimal is our everyday base-10 system using digits 0-9. Hexadecimal uses digits 0-9 and letters A-F (representing 10-15), providing a compact way to represent binary data since each hex digit corresponds to exactly 4 binary bits. Understanding conversions between these systems is essential for programmers and computer engineers.
How do you convert binary to hexadecimal?
Converting binary to hexadecimal is straightforward because each hexadecimal digit represents exactly 4 binary digits (bits). Group the binary number into groups of 4 bits starting from the right, padding with leading zeros if needed. Then convert each group to its hex equivalent. For example, binary 11010110 becomes 1101 0110, which converts to D6 in hex (1101 = 13 = D, 0110 = 6). This direct mapping makes hex a preferred shorthand for binary in computing. An 8-bit byte is always represented by exactly 2 hex digits, a 16-bit word by 4 hex digits, and a 32-bit value by 8 hex digits. This is why memory addresses and color codes use hexadecimal notation.
What is the octal number system and where is it used?
The octal system (base-8) uses digits 0 through 7. Each octal digit represents exactly 3 binary bits, making octal-to-binary conversion simple. For example, octal 375 equals binary 011 111 101. Octal was widely used in early computing when word sizes were multiples of 3 bits (such as 12-bit, 24-bit, and 36-bit architectures). Today, octal is still used in Unix/Linux file permissions, where each permission set (read, write, execute) is represented by 3 bits or one octal digit. For instance, chmod 755 sets permissions to rwxr-xr-x. Octal also appears in some programming languages for representing character codes and escape sequences.
What is BCD (Binary Coded Decimal) and how does it work?
Binary Coded Decimal (BCD) represents each individual decimal digit with its own 4-bit binary code. Unlike pure binary, which represents the entire number as a single binary value, BCD keeps each decimal digit separate. For example, the number 42 in BCD is 0100 0010 (4 = 0100, 2 = 0010), whereas in pure binary it is 101010. BCD is less space-efficient than pure binary but simplifies decimal display and avoids rounding errors in decimal arithmetic. BCD is commonly used in financial calculators, digital clocks, electronic meters, and any application where exact decimal representation is critical. The 8421 BCD is the most common variant, named after the bit position weights.
How do hexadecimal color codes work in web development?
Web color codes use hexadecimal to represent RGB (Red, Green, Blue) values. A hex color like #FF8C00 breaks down into three pairs: FF (red = 255), 8C (green = 140), 00 (blue = 0), creating an orange color. Each channel ranges from 00 (0 in decimal, no intensity) to FF (255 in decimal, full intensity), giving 256 possible values per channel and 16,777,216 total colors. The shorthand notation #F80 expands to #FF8800. Modern CSS also supports #RRGGBBAA format where the last pair is the alpha (transparency) channel. Understanding hex is essential for web developers because hex notation is the most compact way to specify precise colors, and most design tools report colors in hexadecimal format.
What is two complement representation for signed numbers?
Two complement is the standard method computers use to represent signed (positive and negative) integers. In an n-bit two complement system, the most significant bit serves as the sign bit (0 for positive, 1 for negative). Positive numbers are represented normally. To get the negative of a number, invert all bits and add 1. For example, in 8-bit: +5 = 00000101, inverting gives 11111010, adding 1 gives 11111011 = -5. The range for 8-bit two complement is -128 to +127. For 16-bit, it is -32768 to +32767. The beauty of two complement is that addition and subtraction work identically for signed and unsigned numbers at the hardware level, simplifying processor design significantly.
How do you perform arithmetic in binary?
Binary arithmetic follows the same rules as decimal but with only two digits. For addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1). For example, 1011 + 0110 = 10001 (11 + 6 = 17). Subtraction uses borrowing: 10-1=1, 0-1 requires borrowing. Multiplication works like decimal long multiplication but simpler since you only multiply by 0 or 1. Binary division follows long division with the same simplification. Computers perform all arithmetic in binary at the hardware level using logic gates. Addition is implemented with full-adder circuits, while multiplication is often done through repeated addition and bit-shifting. Understanding binary arithmetic helps debug low-level software issues and optimize code.
Why do programmers prefer hexadecimal over binary?
Programmers prefer hexadecimal because it provides a compact, human-readable representation of binary data. A single hex digit replaces 4 binary digits, so an 8-bit byte becomes just 2 hex characters instead of 8 binary digits. A 32-bit address becomes 8 hex characters versus 32 binary characters. This compactness makes hex much easier to read, type, and remember. For example, the binary value 11111111101011001101111000010010 is hard to parse at a glance, but its hex equivalent FFACDe12 is manageable. Hex also makes bit manipulation more intuitive: setting the high nibble is simply changing one hex digit. Memory dumps, network packet captures, and debug logs all use hex for readability.
What is the significance of powers of 2 in computing?
Powers of 2 are fundamental in computing because binary representation means each bit position represents a power of 2. Common powers include: 2^8 = 256 (one byte of possible values), 2^10 = 1024 (approximately 1K, used in KB/MB/GB), 2^16 = 65536 (maximum ports, old address space), and 2^32 = 4294967296 (IPv4 address space, 32-bit integer limit). Memory sizes are always powers of 2 because address decoding hardware works in binary. Checking if a number is a power of 2 is a common programming task done efficiently with the bit trick: n > 0 AND (n AND (n-1)) equals 0. This works because powers of 2 have exactly one bit set in their binary representation.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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