# 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.

परस्परसंवादी आवृत्ती: https://www.calcopenly.com/mr/programming/binary-hex-converter
विषय: प्रोग्रामिंग आणि तंत्रज्ञानाचे कॅल्क्युलेटर

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.

## इनपुट

- **Number**: Use a point for fractions. In auto mode 0x, 0b and 0o prefixes pick the base. Spaces and underscores are ignored.
- **From base** (पर्याय: Auto (prefix, else decimal), Binary (2), Octal (8), Decimal (10), Hex (16), Other base)
- **Source base**
- **To base** (पर्याय: Binary (2), Octal (8), Decimal (10), Hex (16), Other base)
- **Target base**
- **Two's-complement width** (पर्याय: 8-bit, 16-bit, 32-bit, 64-bit, 128-bit)
- **अंकांचे गट**: Applies to every base except decimal; binary in 4s shows nibbles, in 8s shows bytes. (पर्याय: None, Groups of 4, Groups of 8)
- **Fraction digits to show**

## निकाल

- निकाल — मुख्य निकाल
- Binary
- Octal
- Decimal
- Hex
- Fraction in the target base
- Exact form
- Repeating block length
- Two's-complement bits
- Two's complement in hex
- Bits read as unsigned
- Bits read as signed

## सूत्र

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

## सोडवलेली उदाहरणे

### 255 to hex

- Number: 255
- From base: Decimal (10)
- To base: Hex (16)
- Two's-complement width: 16-bit
- अंकांचे गट: Groups of 4
- **निकाल: FF**
- **Binary: 1111 1111**
- **Octal: 377**
- **Fraction in the target base: Whole number**
- पडताळणीचा स्रोत: 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
- अंकांचे गट: Groups of 4
- **निकाल: -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**
- पडताळणीचा स्रोत: 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
- अंकांचे गट: None
- Fraction digits to show: 12
- **निकाल: 0.000110011001…**
- **Exact form: 0.0(0011)**
- **Fraction in the target base: Repeating**
- **Repeating block length: 4**
- पडताळणीचा स्रोत: 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
- अंकांचे गट: Groups of 4
- **निकाल: 1.5**
- **Fraction in the target base: Terminating**
- **Binary: 1.1**
- पडताळणीचा स्रोत: Python 3.8: float.fromhex('0x1.8') = 1.5

### 128 overflows signed 8-bit

- Number: 128
- From base: Decimal (10)
- To base: Binary (2)
- Two's-complement width: 8-bit
- अंकांचे गट: None
- **निकाल: 10000000**
- **Two's-complement bits: 10000000**
- **Bits read as unsigned: 128**
- **Bits read as signed: -128**
- पडताळणीचा स्रोत: Python 3.8: int.from_bytes(bytes([128]), 'big', signed=True) = -128

### Base 36 ZZ

- Number: ZZ
- From base: Other base
- Source base: 36
- To base: Decimal (10)
- Two's-complement width: 16-bit
- अंकांचे गट: Groups of 4
- **निकाल: 1295**
- **Hex: 50F**
- पडताळणीचा स्रोत: Python 3.8: int('ZZ', 36) = 1295, format(1295, 'X') = '50F'

## प्रश्न

### 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.

### “Binary, hex and decimal converter” किती अचूक आहे?

अचूकता तुमच्या इनपुटवर आणि पद्धतीच्या गृहीतकांवर अवलंबून असते. दशांश गणना 50 सार्थ अंक वापरते, पण अंदाज, संख्यात्मक पद्धती आणि मूळ डेटा कमी अचूक असू शकतात; दाखवलेल्या मूल्यांचे पूर्णांकन केल्याने या मर्यादा दूर होत नाहीत. स्वतंत्र स्रोतांतील सोडवलेल्या उदाहरणांशी पडताळणी: ७. उदाहरणार्थ, “255 to hex” ची पडताळणी Python 3.8: format(255, 'X'), format(255, 'b'), format(255, 'o') याच्याशी केली आहे.

### या पद्धतीचा स्रोत कोणता?

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).

## स्रोत

- 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)](https://www.open-std.org/jtc1/sc22/wg14/www/docs/n3096.pdf)