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1Repair Windows errors before they cause bigger problems2Scan for outdated or missing drivers - takes under a minute3Clear out junk files and repair common Windows errorsA decimal-to-hexadecimal converter rewrites a number from base 10 into base 16. Hexadecimal uses 0–9 and A–F, and each hexadecimal digit represents four binary bits. For an integer, repeatedly divide by 16 and read the remainders from last to first. For a fractional part, repeatedly multiply by 16 and record each integer part.
That ordinary base conversion is different from displaying a floating-point value’s stored bits in hexadecimal. A bit pattern requires a specified format, such as binary32 or binary64, width, and rounding behavior.
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What hexadecimal means
Hexadecimal, or base 16, uses sixteen symbols: 0 through 9 and A through F (where A=10 and F=15). The relationship with binary is convenient because one hex digit maps exactly to four bits. IEEE 754 describes hexadecimal character conversions for floating-point values as well as the rounding required when a conversion changes radix or precision: IEEE Std 754-2008.
A suffix such as ₁₀ or ₁₆ identifies the base. Thus, FE₁₆ is not a different quantity from 254₁₀; it is a different written representation.
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Convert a decimal integer to hexadecimal
- Divide the decimal integer by 16.
- Keep the remainder. Remainders 0–9 stay as digits; 10–15 become A–F.
- Divide the quotient by 16 again.
- Continue until the quotient is zero.
- Read the remainders upward, from the last one produced to the first.
This repeated-division and remainder method is documented for radix conversion in Microchip’s application note AN575: IEEE 754 Compliant Floating Point Routines.
Worked example: 254
| Operation | Quotient | Remainder |
|---|---|---|
| 254 ÷ 16 | 15 | 14 = E |
| 15 ÷ 16 | 0 | 15 = F |
Reading the remainders backward gives 254₁₀ = FE₁₆.
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Zero and negative inputs
Zero converts to 0. For an ordinary signed value, keep the minus sign and convert the magnitude, so −254 is −FE₁₆. A machine representation using two’s-complement bits is a separate question: it cannot be determined without choosing a bit width (for example, 8, 16, or 32 bits).
Convert a decimal fraction
Separate the number at the radix point. Convert the integer part by division, then handle the fractional part independently:
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- Multiply the remaining fraction by 16.
- Record the integer part of the result as the next hex digit.
- Keep the new fractional remainder and multiply it by 16 again.
- Stop when the remainder is exactly zero or when the required number of digits has been reached.
If the expansion does not terminate, a converter must state how many digits it emits and whether the final digit is rounded or simply truncated. The multiplication procedure and the need to specify precision follow the conversion algorithms in Microchip AN575; IEEE’s rounding requirements are described in IEEE 754-2008.
Worked example: 10.625
The integer part, 10, is A. For the fraction:
| Step | Calculation | Digit | Remainder |
|---|---|---|---|
| 1 | 0.625 × 16 = 10.000 | A | 0 |
Therefore, 10.625₁₀ = A.A₁₆. Because the fractional remainder reaches zero, no further digits are needed.
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Choose the conversion mode
| Question | Ordinary base conversion | Floating-point representation |
|---|---|---|
| What is converted? | The numeric value written in base 16 | A value in a selected floating-point format or its stored bits |
| What must be specified? | Input value and desired fractional precision | Format and width, such as binary32 or binary64, plus rounding behavior |
| Typical output | FE or A.A |
A hexadecimal value string or a raw hexadecimal bit pattern |
| Can the answer vary? | Only with a different precision or rounding choice for an inexact fraction | Yes; precision, format, and rounding can change the result |
Do not call a raw bit pattern “the hexadecimal version” of a decimal number without naming the format. IEEE 754 requires rounding when conversion goes to a different radix or to a narrower precision in the same radix.
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Programming languages add syntax that is not the same as writing an ordinary hexadecimal integer. In C, a hexadecimal floating constant uses a binary exponent marked with p or P; GNU documents the syntax and decimal-to-binary rounding caveat in Exact Floating Constants. Java hexadecimal floating-point literals likewise require a p or P exponent, as described in the OpenJDK specification page Update to IEEE 754 Terminology.
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For example, a language may write a hexadecimal floating literal in a form like 0x1.Ap2. The p2 is a power-of-two exponent; it is not a hexadecimal exponent. Exact accepted syntax depends on the language version.
How to specify a converter result
- State whether the input is an integer, a real decimal value, or a floating-point object.
- For fractions, give the number of hexadecimal places requested.
- Say whether omitted digits are rounded or truncated.
- For bit-level output, name the format and width, such as binary32 or binary64.
- Distinguish a hexadecimal value string from the raw bits stored in memory.
With those choices explicit, 254 has the unambiguous ordinary conversion FE, while a floating-point answer is only well-defined after its representation format and precision are selected.
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