Hex Calculator

Three calculators sit above this. The first adds, subtracts, multiplies or divides two hexadecimal values and reports the answer in both hex and decimal. The second turns a hex value into decimal and the third goes the other way. All three show the working, including the column-by-column carrying that makes hex arithmetic feel unfamiliar at first.

What hexadecimal is

Hexadecimal works exactly like decimal and binary, but on a base of 16 rather than 10 or 2. It needs sixteen digits, so after 0 to 9 it borrows the letters A to F for the values 10 to 15. Case does not matter: A and a are the same digit.

Its usefulness comes from the relationship with binary. Sixteen is two to the fourth power, so every hex digit stands for exactly four binary digits, a group known as a nibble. That makes conversion between the two mechanical rather than arithmetic: the binary value 1010101010 splits into nibbles as 10 1010 1010 and reads straight off as 2AA. A byte is always two hex digits, which is why hex is how computers show binary to people.

Hex, binary and decimal side by side

Hex	Binary	Decimal
0	0	0
1	1	1
2	10	2
3	11	3
4	100	4
5	101	5
6	110	6
7	111	7
8	1000	8
9	1001	9
A	1010	10
B	1011	11
C	1100	12
D	1101	13
E	1110	14
F	1111	15
14	10100	20
3F	111111	63

The first sixteen rows are worth knowing by sight, because every hex operation eventually comes down to them.

Place values

Each column in a hex number is a power of 16, counting from zero on the right. In 2AA the right-hand A is the ones place, 16⁰; the second A is 16¹, or sixteens; and the 2 is 16², or 256s. Remembering that A is 10 in decimal:

$$\text{2AA}=(2\times {16}^2)+(\text{A}\times {16}^1)+(\text{A}\times {16}^0)$$$$=(2\times 256)+(10\times 16)+(10\times 1)=512+160+10=682$$

Converting from hex to decimal is always this: multiply each digit by its place value and add the results. Letter digits change nothing about the method.

$$\text{1024}=(1\times {16}^3)+(0\times {16}^2)+(2\times {16}^1)+(4\times {16}^0)=4132$$

Decimal to hex

Going the other way takes a little more work. One approach is repeated division by 16, keeping the remainders and reading them upwards, which is what the third calculator above shows. The other is to work down through the place values:

1. Find the largest power of 16 that fits inside the number.
2. Work out how many times it goes in, and note that count.
3. Multiply the count by the power of 16 and subtract, leaving a remainder.
4. Repeat with the remainder until 16 is larger than what is left; whatever remains is the 16⁰ digit.
5. Write each count in its place value, converting 10 to 15 into A to F.

Take 1500. The largest useful power is 16² = 256, which goes in 5 times: 5 × 256 = 1280, leaving 220. Sixteen goes into 220 thirteen times: 13 × 16 = 208, leaving 12. Since 16 is larger than 12, the last digit is 12. So 1500 = (5 × 16²) + (13 × 16¹) + (12 × 16⁰), and converting 13 to D and 12 to C gives 5DC.

Hex addition

Hex addition follows the same rules as decimal addition, with one difference: a column carries when it reaches 16, not 10.

 1 1
   8 A B
 +  B 7 8
 = 1 4 2 3

Reading right to left: B + 8 is 11 + 8 = 19 in decimal, and 19 is 13 in hex, being one 16 with 3 left over. Write the 3 and carry the 1. The next column is 1 + A + 7 = 1 + 10 + 7 = 18 decimal, which is 12 hex, so write 2 and carry again. The last column is 1 + 8 + B = 1 + 8 + 11 = 20 decimal, or 14 hex. The result is 1423.

Keeping the A to F values in mind while adding is the whole skill. Everything else is ordinary column arithmetic.

Hex subtraction

Subtraction works the same way with one thing to watch: a borrowed 1 is worth 16, not 10, because the column you borrow from is sixteen times larger than the one receiving it.

   5 D ¹C
 -  3 A F
 =  2 2 D

In the right-hand column C is 12 and F is 15, so 12 is too small and the column has to borrow. That reduces the D to C and lends 16, giving 16 + 12 − 15 = 13 decimal, which is D. The remaining columns need no borrowing: C − A is 12 − 10 = 2, and 5 − 3 = 2. The answer is 22D.

When the number being subtracted is the larger of the two, swap them, subtract, and put a minus sign in front. Reversing the example above to 3AF − 5DC gives −22D.

Hex multiplication

Multiplication is the fiddly one, because the intermediate products are larger and each needs converting. A multiplication table helps, and one is below.

     F A        3 × A = 1E, carry 1 to F
 ×   C 3        3 × F = 2D, + 1 = 2E
   2 E E      C × A = 78, carry 7 to F
 + B B 8 0    C × F = B4, + 7 = BB
 = B E 6 E

Each partial product is built the same way as in decimal, then the partials are added using hex addition. The result of FA × C3 is BE6E, which is 250 × 195 = 48,750 in decimal.

Hex division

Long division in hex is identical to long division in decimal, except that the multiplying and subtracting inside it happen in hex. The alternative is to convert to decimal, divide there, and convert back, which is what the calculator above does internally.

        D E F      12 × D = EA
   12 ) F A C E    FA − EA = 10
        E A        12 × E = 10C
        1 0 C      10C − FC = 10
          F C      12 × F = 10E
          1 0 E
          1 0 E
                0

Here FACE divided by 12 gives DEF with nothing left over: 64,206 ÷ 18 = 3,567 in decimal. Remember that any borrowing inside the subtraction steps is worth 16.

Hexadecimal multiplication table

Negative values and two's complement

The calculator above accepts a minus sign and reports a negative answer plainly, which is what you want for ordinary arithmetic. Computers do it differently, and the difference catches people out when reading a memory dump.

Inside a fixed-width register there is no room for a sign character, so negative numbers are stored using two's complement: the top bit carries the sign, and a negative value is written as the positive one subtracted from 2 raised to the width. In a single signed byte that makes FF equal to −1, FE equal to −2, and 80 the most negative value at −128, while 7F is the largest positive at 127.

So the same two hex digits mean 255 or −1 depending entirely on whether the byte is being read as signed or unsigned, and nothing in the digits themselves tells you which. A value of FFFFFFFF in a 32-bit register is either about 4.29 billion or simply −1. When a debugger shows an implausibly enormous number, an unsigned reading of a negative value is usually the reason.

Notation you will meet

The same value gets written several ways depending on where it appears. 0x1A3 is the C convention and the one most languages adopted. #1A3 means a colour in CSS and HTML. 1A3h is the assembler style, with a trailing h. Some documentation uses a subscript, as in 1A3₁₆. None of these change the number, and the calculators above accept a plain value with or without the 0x prefix.

Two habits are worth keeping. Case is irrelevant to the value but not to consistency, and mixed case such as 1a3F is legal and horrible to read; pick upper or lower and stay there. Leading zeros are also free, and hex normally keeps them so a value fills its width, since 0A and 0x0000000A carry the same value as A but say something useful about the size of the field holding it.

Hexadecimal displaced octal, base 8, for the same reason it displaced raw binary. Octal groups bits in threes, which suited machines with 12-, 24- and 36-bit words, but once the 8-bit byte became standard the threes stopped aligning with anything. Four bits divide a byte evenly and eight bits divide a 32-bit word evenly, so hex fits modern hardware and octal does not. Octal survives mainly in Unix file permissions, where three bits per group is exactly what is wanted.

Where hex actually turns up

Web and design colours are three bytes of hex, written #RRGGBB, so #FF0000 is full red and #AABBCC is the same value the second calculator would read as 11,189,196. The shorthand #ABC expands to #AABBCC.

Memory addresses and machine code are shown in hex because a byte is exactly two digits, so a dump lines up in neat columns. MAC addresses are six bytes in hex, and IPv6 addresses are eight groups of four hex digits, which is why they look so different from IPv4. Unicode code points are written U+ followed by hex, so U+1F600 is the grinning face. Hashes such as SHA-256 are quoted as 64 hex characters, which is exactly 256 bits at four bits per character.

The habit even shows up in jokes: words made only from A to F, such as DEADBEEF, CAFEBABE and FACE, are valid hex numbers, and programmers use them as recognisable marker values in memory.

Reading your result

Check that your input only uses 0-9 and A-F. A stray O for zero or an I for one is the usual cause of a rejected value, and the calculator will not guess which you meant.

The first calculator gives both forms of the answer, and the decimal line is the one to check against a spreadsheet or another tool. Division reports a quotient and a remainder rather than a fraction, since hex division here is whole-number division.

All three run in exact whole-number arithmetic, so a sixteen-digit hex value such as a memory address or a hash fragment converts without losing its last digits, which is not true of tools that convert through an ordinary decimal number.