How do you convert binary to decimal?

Binary to decimal: multiply each bit by its positional value (power of 2) and sum the results. Positions are counted from right starting at 0. Formula: decimal = sum of (bit × 2^position) Example: 1010 in binary - Position 3: 1 × 2^3 = 1 × 8 = 8 - Position 2: 0 × 2^2 = 0 × 4 = 0 - Position 1: 1 × 2^1 = 1 × 2 = 2 - Position 0: 0 × 2^0 = 0 × 1 = 0 - Total: 8 + 0 + 2 + 0 = 10 So 1010₂ = 10₁₀ Another example: 11111111₂ = 128+64+32+16+8+4+2+1 = 255₁₀

How do you convert between binary and hexadecimal?

Binary and hex convert cleanly because 16 = 2^4. Each hex digit represents exactly 4 binary bits (a nibble). Binary to hex: group bits in 4 from the right, convert each group. 1010 1111₂ → A F₁₆ → AF₁₆ Hex to binary: replace each hex digit with its 4-bit binary equivalent. FF₁₆ → F=1111, F=1111 → 11111111₂ Hex digit table: 0=0000, 1=0001, 2=0010, 3=0011 4=0100, 5=0101, 6=0110, 7=0111 8=1000, 9=1001, A=1010, B=1011 C=1100, D=1101, E=1110, F=1111 This is why hex is used in programming: it is compact (2 hex digits = 1 byte = 8 bits) and converts directly to binary without division.

Why do computers use binary?

Computers use binary (base 2) because electronic circuits have two stable states: on (1) and off (0). These correspond directly to high voltage and low voltage in transistors. Key reasons: - Reliability: distinguishing between 2 states (on/off) is much more reliable than 10 states - Simplicity: Boolean logic (AND, OR, NOT) maps perfectly to binary arithmetic - Physics: transistors are essentially binary switches All computer data — text, images, audio, video — is ultimately stored as sequences of 0s and 1s. Common binary groupings: - 1 bit: 0 or 1 - 4 bits (nibble): 0-15 - 8 bits (byte): 0-255 - 16 bits (word): 0-65,535 - 32 bits (dword): 0-4,294,967,295 - 64 bits (qword): 0-18,446,744,073,709,551,615 Hexadecimal is used by programmers as a compact way to write binary — 2 hex digits represent exactly 1 byte.

What is hexadecimal and why is it used in programming?

Hexadecimal (hex) is base 16, using digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15). Why programmers use hex: - Compact: 2 hex digits represent 8 bits (1 byte) exactly - Binary-friendly: each hex digit = 4 bits, so conversion is trivial - Readable: FF is much easier to read than 11111111 Common uses: - Colour codes: #FF5733 = RGB(255, 87, 51) - Memory addresses: 0x7FFE1234 - ASCII character codes: 0x41 = 'A' (decimal 65) - Network addresses: MAC address like 00:1A:2B:3C:4D:5E - File formats: hex editors show raw bytes as hex pairs Conversion shortcuts: - 0xFF = 255 (max byte value) - 0x100 = 256 - 0x1000 = 4096 - 0xFFFF = 65535 (max 16-bit value)

Binary Converter — Binary, Octal, Decimal, Hex

The Binary Converter converts any number between binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16) instantly. Select your input base, type the number, and all four conversions appear simultaneously. Step-by-step working shows the repeated division method for decimal-to-bi...

NUMBER BASE CONVERTER

CONVERT FROM

Enter Decimal number(digits: 0-9)

Binary (base 2)

101010

Octal (base 8)

Decimal (base 10)

input

Hexadecimal (base 16)

Bit length

Bytes

Hex prefix

0x2A

STEP-BY-STEP

Decimal 42 → Binary by repeated division:

42 ÷ 2 = 21 remainder 0

21 ÷ 2 = 10 remainder 1

10 ÷ 2 = 5 remainder 0

5 ÷ 2 = 2 remainder 1

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Read remainders bottom to top: 101010

QUICK EXAMPLES

BIT VISUALIZER

BIT PATTERN

Byte 1

POWERS OF 2 REFERENCE

2ⁿ	n	Decimal	Hex
1	0	1	1
2	1	2	2
4	2	4	4
8	3	8	8
16	4	16	10
32	5	32	20
64	6	64	40
128	7	128	80
256	8	256	100
512	9	512	200
1024	10	1024	400
32768	15	32768	8000
65536	16	65536	10000

Created with❤️byeaglecalculator.com

FORMULAS

NUMBER BASE CONVERSION REFERENCE

NAME	FORMULA	DESCRIPTION
Binary to decimal	Sum of (bit × 2^position) for each bit	1010 = 1×8 + 0×4 + 1×2 + 0×1 = 10
Decimal to binary	Repeated division by 2, read remainders bottom-up	10 ÷ 2 = 5 r0, 5 ÷ 2 = 2 r1, 2 ÷ 2 = 1 r0, 1 ÷ 2 = 0 r1 → 1010
Hex to decimal	Sum of (digit × 16^position) for each digit	FF = 15×16 + 15×1 = 255
Binary to hex	Group bits in 4 from right, convert each group	1010 1111 → A F → AF
Octal to decimal	Sum of (digit × 8^position) for each digit	77 = 7×8 + 7×1 = 63
Bit length	floor(log2(n)) + 1 bits	Number of bits needed to represent n
Byte count	ceil(bit_length / 8) bytes	Number of bytes needed
Hex prefix	0x prefix: 0xFF = 255	Convention: 0b for binary, 0o for octal, 0x for hex

HOW TO USE

1
Select the base of your input number: Binary (base 2, digits 0-1), Octal (base 8, digits 0-7), Decimal (base 10, digits 0-9), or Hexadecimal (base 16, digits 0-9 and A-F).
2
Type your number in the input box. The converter validates as you type — the border turns red if you enter an invalid digit for the selected base. All four conversions appear instantly.
3
The result table shows your number in all four bases simultaneously. Click any Copy button to copy that representation to clipboard.
4
The bit visualiser on the right shows the binary representation as individual bit cells, colour-coded black for 1 and grey for 0. Bit position numbers are shown below each cell.
5
The step-by-step panel shows the conversion process — for decimal input it shows repeated division by 2, for binary it shows the positional expansion (each bit × its power of 2).

WORKED EXAMPLE

42 decimal to binary: 42÷2=21 r0, 21÷2=10 r1, 10÷2=5 r0, 5÷2=2 r1, 2÷2=1 r0, 1÷2=0 r1. Read bottom-up: 101010. 42 = 101010 binary = 2A hex = 52 octal. Verification: 32+0+8+0+2+0=42 ✓. 255 = 11111111 binary = FF hex = 377 octal.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.