Cal11 calculator
Search calculator...

How to Calculate Remainder of Negative Number

Reviewed by Calculator Editorial Team

Calculating the remainder of negative numbers is a common operation in programming and mathematics. This guide explains how to perform these calculations correctly, including the rules for negative dividends and divisors.

What is a remainder?

The remainder is what's left after dividing one number by another. In the equation dividend ÷ divisor = quotient with remainder, the remainder is always less than the divisor and has the same sign as the dividend.

Remainder formula:
remainder = dividend - (divisor × quotient)

For example, in 10 ÷ 3 = 3 with remainder 1, the remainder is 1 because 10 - (3 × 3) = 1.

Remainder with negative numbers

When working with negative numbers, the remainder follows these rules:

  • The remainder always has the same sign as the dividend
  • If the dividend is negative, the remainder will be negative
  • If the dividend is positive, the remainder will be positive
  • The absolute value of the remainder is always less than the absolute value of the divisor

Important: In programming languages like Python and JavaScript, the remainder operator (%) follows these rules. However, in some languages like C, the behavior differs for negative numbers.

Calculation method

To calculate the remainder of negative numbers:

  1. Divide the dividend by the divisor to find the quotient
  2. Multiply the divisor by the quotient
  3. Subtract this product from the dividend to get the remainder

This method works for both positive and negative numbers.

Worked examples

Example 1: Positive dividend and divisor

Calculate the remainder of 10 ÷ 3:

  1. 10 ÷ 3 = 3 with remainder 1
  2. 3 × 3 = 9
  3. 10 - 9 = 1 (remainder)

Example 2: Negative dividend

Calculate the remainder of -10 ÷ 3:

  1. -10 ÷ 3 = -3 with remainder -1
  2. 3 × -3 = -9
  3. -10 - (-9) = -1 (remainder)

Example 3: Negative divisor

Calculate the remainder of 10 ÷ -3:

  1. 10 ÷ -3 = -3 with remainder 1
  2. -3 × -3 = 9
  3. 10 - 9 = 1 (remainder)

FAQ

Why does the remainder have the same sign as the dividend?
The remainder represents what's left after division, so it should logically have the same sign as what you started with.
What's the difference between remainder and modulo operations?
In mathematics, remainder and modulo operations are the same. However, in programming, some languages implement modulo differently for negative numbers.
How do I calculate remainder in programming?
In most programming languages, you can use the modulus operator (%). For example, in Python: remainder = 10 % 3.
What if both numbers are negative?
The remainder will be negative if the dividend is negative, regardless of the divisor's sign.
Is there a difference between remainder and floor division?
Yes. Floor division (// in Python) gives the largest integer less than or equal to the division result, while remainder gives what's left after division.