Hex Calculator

Arithmetic in base 16

Hex arithmetic follows exactly the same rules as decimal. Only the point at which a column overflows changes: sixteen instead of ten.

That single substitution covers everything. If you can add decimal by hand you can add hex by hand, once you stop expecting a carry at ten.

Addition

Worked through in more depth, with the carry rule and fixed-width overflow, on the hex addition page.

  2F
+ 1A
----

  ones:  F + A = 15 + 10 = 25 = 16 + 9  → write 9, carry 1
  16s:   2 + 1 + 1 (carry) = 4          → write 4

  result → 49   (47 + 26 = 73 in decimal)

Subtraction

Borrowing has its own page, with two's complement wrapping explained: hex subtraction.

  40
- 1B
----

  ones:  0 - B  — borrow 16 from the next column
         16 - 11 = 5                    → write 5
  16s:   4 - 1 - 1 (borrowed) = 2       → write 2

  result → 25   (64 - 27 = 37 in decimal)

Multiplication

The shift shortcuts for powers of sixteen are covered on the hex multiplication page.

  1F × 3

  F × 3 = 15 × 3 = 45 = 2×16 + 13  → write D, carry 2
  1 × 3 = 3, plus carry 2 = 5        → write 5

  result → 5D   (31 × 3 = 93 in decimal)

Division

There is a fuller treatment, including the digit-dropping shortcut, on the hex division page. The calculator does integer division and reports the quotient and remainder separately, which is what you almost always want in hex. There is no fractional part, because a hex fraction is rarely what anyone means.

  FF ÷ 10

  255 ÷ 16 = 15 remainder 15

  quotient  → F
  remainder → F

Powers of two, and why they matter here

Multiplying or dividing by a power of two shifts bits, and when the power is a multiple of four it shifts whole hex digits. That turns a lot of arithmetic into moving characters.

Overflow, and what this calculator does not do

Real hardware works at a fixed width. An 8-bit register holding FF and adding 1 gives 00, not 100 — the carry has nowhere to go.

This calculator has no width limit. It uses arbitrary-precision integers, so FF + 1 gives 100 and stays exact no matter how large the operands grow. That is the right answer mathematically, and the wrong answer if you are simulating a specific register. To model overflow, take the last n digits: two for 8-bit, four for 16-bit, eight for 32-bit.

Negative results

Subtracting a larger value from a smaller one gives a negative result, shown here with a minus sign. Hardware would instead wrap to the two's complement representation at whatever width the register is: 0 − 1 becomes FF at 8 bits and FFFFFFFF at 32. The hex to decimal converter reads those values back as signed numbers.

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