Integer Calculator

Enter two integers and choose an operation to add, subtract, multiply, or divide them — the calculator shows the result along with its sign, absolute value, and (for division) the integer quotient and remainder using Euclidean division.

Quick Facts

Different-sign addition
Subtract |smaller| from |larger|, keep sign of larger
Example: 12 + (−5) = 7, while −12 + 5 = −7.
Multiplication & division sign rule
Same signs → positive, different signs → negative
Example: (−4) × (−3) = 12 and (−12) ÷ 4 = −3.
Euclidean division
a = b·q + r, with 0 ≤ r < |b|
For any integers a and nonzero b, this equation defines a unique quotient q and remainder r.

Your Results

Calculated
Exact Quotient (a ÷ b)
-
Result of the selected operation
Absolute Value
-
|Result| of the operation
Sign
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Positive, negative, or zero
Integer Quotient & Remainder
-
a = b·q + r (Euclidean division)

Ready

Enter two integers and an operation, then press Calculate.

How the Integer Calculator Works

Integers are whole numbers — positive, negative, or zero — with no fractional or decimal part: …, −3, −2, −1, 0, 1, 2, 3, …. This calculator takes two integers, a and b, applies the operation you choose (addition, subtraction, multiplication, or division), and reports the result along with its sign, its absolute value, and — for division — the integer quotient and remainder found through Euclidean division.

The sign rules for addition, subtraction, multiplication, and division

Adding two integers with the same sign adds their absolute values and keeps that sign: (−4) + (−3) = −7. Adding integers with different signs means subtracting the smaller absolute value from the larger and keeping the sign of the number with the larger absolute value: 12 + (−5) = 7, but −12 + 5 = −7. Subtraction is addition of the opposite, a − b = a + (−b), so it follows the same rules once you flip the sign of b. For multiplication and division, two integers with the same sign (both positive or both negative) always produce a positive result, and two integers with different signs always produce a negative result: (−4) × (−3) = 12, but (−4) × 3 = −12.

Quotient and remainder: Euclidean division

Integers are not closed under division — 17 ÷ 5 is not itself an integer. When you select Division, the calculator reports the exact result of a ÷ b (which may be a decimal) as well as the integer quotient q and remainder r, defined so that a = b·q + r, with the remainder always satisfying 0 ≤ r < |b|. This is the Euclidean division convention taught in most math courses: for example, 17 ÷ 5 has quotient q = 3 and remainder r = 2, because 17 = 5×3 + 2. The same rule extends to negative dividends: −17 = 5×(−4) + 3, so −17 ÷ 5 has quotient −4 and remainder 3 (never a negative remainder).

Common mistakes

  • Treating subtraction as its own sign rule: a − (−b) is the same as a + b — two negatives next to each other become a plus.
  • Assuming remainders can be negative: under the Euclidean convention used here, the remainder is always between 0 and |b| − 1, even when the dividend is negative.
  • Entering decimals as "integers": the calculator requires whole numbers for both a and b; a value like 4.5 is not an integer and will be rejected.
  • Dividing by zero: division by zero is undefined for every number, including integers, and has no quotient or remainder.

Real-world applications

  • Splitting a quantity into equal groups with leftovers — e.g., packing 17 items into boxes of 5 gives 3 full boxes and 2 items left over.
  • Modular arithmetic used in clocks, calendars, checksums, and cryptography relies directly on the Euclidean remainder.
  • Tracking gains and losses, temperature changes, or elevation changes uses signed-integer addition and subtraction.
  • Programming and computer science use integer division and remainder (quotient/modulo) constantly for indexing, loops, and hashing.

Frequently Asked Questions

How do I add and subtract integers with different signs?
To add two integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value: 12 + (−5) = 7, while −12 + 5 = −7. Subtracting an integer is the same as adding its opposite, so a − b = a + (−b).
What is the sign rule for multiplying and dividing integers?
Two integers with the same sign (both positive or both negative) multiply or divide to a positive result, and two integers with different signs multiply or divide to a negative result. For example, (−4) × (−3) = 12 and (−12) ÷ 4 = −3.
What are the quotient and remainder in integer division?
When one integer does not divide another evenly, division produces a quotient q and a remainder r such that a = b·q + r, with the remainder always between 0 (inclusive) and the absolute value of the divisor (exclusive). This is Euclidean division. For example, 17 ÷ 5 gives a quotient of 3 and a remainder of 2, since 17 = 5×3 + 2.
Is zero an integer, and can you divide by zero?
Yes, zero is an integer — it is neither positive nor negative. Division by zero is undefined for all numbers, including integers, so this calculator shows an error if the second integer is 0 and division is selected.

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