Binary, hex and decimal converter

Convert numbers between binary, decimal, hex, octal and any base 2–36, including fractions, repeating digits and two's-complement bits for negatives.

Обновлено 28 сентября 2026 г.Проверенные примеры: 7

Результат

-2A

Binary: -10 1010
Octal: -52
Decimal: -42
Fraction in the target base: Whole number
Two's-complement bits: 1111 1111 1101 0110
Two's complement in hex: FFD6
Bits read as unsigned: 65494
Bits read as signed: -42

-42 in decimal is -2A in hex. A 16-bit register stores it as FFD6 in hex, which reads as 65494 if the same bits are treated as unsigned.

16-bit pattern

16-bit two's complement

Как выполняется расчёт S

Read the decimal digits
$\texttt{42}_{10} = 4 \times 10^{1} + 2 \times 10^{0} = 42$
The minus sign is carried through unchanged.
Integer part: divide by 16 repeatedly
$\begin{aligned}42 \div 16 &= 2 \text{ remainder } 10 \;(A) \\ 2 \div 16 &= 0 \text{ remainder } 2 \;(2)\end{aligned}$
Reading the remainders from last to first gives the digits.
Store in 16 bits (two's complement)
$2^{16} - 42 = 65494 = \texttt{1111\,1111\,1101\,0110}_2$
Read as unsigned the bits are 65494; read as signed the top bit counts −2¹⁵, giving -42.

О калькуляторе: Binary, hex and decimal converter

A number written in base b is a sum of digits times powers of b, so 2A in hex is 2 × 16 + 10 = 42. To write a whole number in base b, the converter divides by b repeatedly and reads the remainders from last to first; for a fraction it multiplies by b repeatedly and takes each whole part as the next digit. It works with exact fractions, so it can tell whether the digits after the point end or repeat.

Programmers use it for memory addresses, color codes, file permissions and bit masks. The default, −42, is −2A in hex and 1111 1111 1101 0110 in 16-bit two's complement, which reads as 65,494 when the same bits are treated as unsigned. Decimal 0.1 never ends in binary: it is 0.0(0011), with the block 0011 repeating.

Two's-complement bits are shown for whole numbers at widths from 8 to 128 bits; a value outside the chosen width wraps to its low bits, with a warning.

Примеры с решением

255 to hex

Number: 255
From base: Decimal (10)
To base: Hex (16)
Two's-complement width: 16-bit
Группировка цифр: Groups of 4

Результат: FF
Binary: 1111 1111
Octal: 377
Fraction in the target base: Whole number

Источник проверки: Python 3.8: format(255, 'X'), format(255, 'b'), format(255, 'o')

−42 in 16-bit two's complement

Number: -42
From base: Decimal (10)
To base: Hex (16)
Two's-complement width: 16-bit
Группировка цифр: Groups of 4

Результат: -2A
Two's-complement bits: 1111 1111 1101 0110
Two's complement in hex: FFD6
Bits read as unsigned: 65494
Bits read as signed: -42

Источник проверки: Python 3.8: format(-42 & 0xFFFF, '016b') = '1111111111010110', format(-42 & 0xFFFF, 'X') = 'FFD6', 2**16 − 42 = 65494

0.1 decimal is repeating in binary

Number: 0.1
From base: Decimal (10)
To base: Binary (2)
Two's-complement width: 16-bit
Группировка цифр: None
Fraction digits to show: 12

Результат: 0.000110011001…
Exact form: 0.0(0011)
Fraction in the target base: Repeating
Repeating block length: 4

Источник проверки: Python 3.8: format(int(Fraction(1, 10) * 2**12), '012b') = '000110011001'; a separate long-division script finds the remainder cycle of length 4 starting after the first digit

Hex fraction 0x1.8

Number: 0x1.8
From base: Auto (prefix, else decimal)
To base: Decimal (10)
Two's-complement width: 16-bit
Группировка цифр: Groups of 4

Результат: 1.5
Fraction in the target base: Terminating
Binary: 1.1

Источник проверки: Python 3.8: float.fromhex('0x1.8') = 1.5

Вопросы

How do you convert binary to decimal?

Multiply each binary digit by 2 raised to its position, counting from 0 at the right, and add the results. 1111 1111 is 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255, and 101010 is 32 + 8 + 2 = 42. Digits after a binary point use negative powers, so 1.1 is 1 + 1/2 = 1.5.

How do you convert decimal to hexadecimal?

Divide by 16 repeatedly and write the remainders from last to first, using A–F for 10–15. For 255, 255 ÷ 16 = 15 remainder 15, then 15 ÷ 16 = 0 remainder 15, giving FF; for 1295 the remainders 15, 0 and 5 give 50F. Each hex digit stands for exactly 4 bits, which is why hex is used to write bytes: FF is 1111 1111.

What is two's complement?

Two's complement is how CPUs store signed integers: a negative number −x in w bits is stored as 2^w − x, so the top bit carries a weight of −2^(w−1). In 16 bits, −42 is 65,536 − 42 = 65,494, or FFD6 in hex. An 8-bit value runs from −128 to 127, so 128 wraps to −128. C23 (ISO/IEC 9899:2024) requires two's complement for signed integers.

Why is 0.1 not exact in binary?

A fraction ends in base b only if its denominator, in lowest terms, has no prime factors other than those of b. 0.1 is 1/10, and 10 = 2 × 5, so in binary it repeats forever: 0.000110011…, written 0.0(0011). That is why 0.1 + 0.2 gives 0.30000000000000004 in IEEE 754 double precision, the number type behind JavaScript numbers and Python floats.

What is octal used for?

Octal (base 8) groups binary digits in threes, and its main use today is Unix file permissions: each digit adds read (4), write (2) and execute (1) for the owner, group and others, so chmod 755 means rwxr-xr-x. Decimal 255 is 377 in octal. Python 3 and JavaScript write octal literals as 0o755; C writes them with a leading zero, 0755.

Насколько точен «Binary, hex and decimal converter»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 7. Например, «255 to hex» проверяется по источнику Python 3.8: format(255, 'X'), format(255, 'b'), format(255, 'o').

Откуда взята методика?

Knuth, The Art of Computer Programming, Vol. 2, §4.1 Positional number systems and §4.4 Radix conversion; ISO/IEC 9899:2024 (C23) §6.2.6.2 — signed integers use two's complement (draft N3096).

Об этом калькуляторе

\begin{aligned} (d_{k-1}\cdots d_1 d_0.d_{-1}d_{-2}\cdots)_b &= \sum_i d_i\, b^{i} \\ -x \text{ in } w \text{ bits} &= 2^w - x\end{aligned}

Источники

Knuth, The Art of Computer Programming, Vol. 2, §4.1 Positional number systems and §4.4 Radix conversion
ISO/IEC 9899:2024 (C23) §6.2.6.2 — signed integers use two's complement (draft N3096)

Проверено по источникам

В калькулятор включены решённые примеры с ответами из независимых источников. Их количество: 7. Они входят в набор тестов, и вы также можете запустить их здесь.