The exact answer to the question "What is 0 modulus 0?" is that it is undefined or indeterminate.
Understanding Modulus and Division by Zero
The modulus operation (often written as a mod n) finds the remainder when one number (a, the dividend) is divided by another number (n, the divisor or modulus). For example, 10 mod 3 equals 1 because when 10 is divided by 3, the remainder is 1.
However, a core principle in mathematics is that division by zero is not allowed. It is impossible to define a unique result for dividing any number by zero.
As stated in the reference: "When we take any number modulo 0, the result is undefined or indeterminate. This is because the modulus operation involves dividing a number by another number, and division by zero is undefined in mathematics."
Why 0 Modulus 0 is Undefined
Since 0 is considered "any number," applying the modulus operation with 0 as the divisor (the modulus) leads to the issue of division by zero. Specifically, to calculate 0 mod 0, you would conceptually consider the remainder of dividing 0 by 0. This falls directly under the rule that modulus by zero is undefined.
Consider the relationship: a = q * n + r, where r is the remainder (a mod n), q is the quotient, and 0 ≤ r < |n|.
If we try 0 mod 0, we get 0 = q * 0 + r. This simplifies to 0 = r. However, the condition 0 ≤ r < |n| becomes 0 ≤ 0 < |0|, which is 0 ≤ 0 < 0. This condition 0 < 0 is false, meaning no valid remainder r can satisfy the definition within the required range.
Furthermore, the expression 0 / 0 from the division aspect is an indeterminate form in calculus, meaning its value cannot be determined without further analysis (like using limits). In the context of basic arithmetic and the standard modulus operation, any division by zero is simply treated as undefined.
Examples
Let's look at how the modulus operation behaves with non-zero and zero moduli:
| Operation | Result | Explanation |
|---|---|---|
10 mod 3 |
1 | 10 divided by 3 is 3 with remainder 1 |
7 mod 2 |
1 | 7 divided by 2 is 3 with remainder 1 |
0 mod 5 |
0 | 0 divided by 5 is 0 with remainder 0 |
5 mod 0 |
Undefined / Indeterminate | Involves division by zero |
**0 mod 0** |
Undefined / Indeterminate | Involves division by zero, as 0 is the modulus |
As the table and the reference confirm, 0 mod 0 is not assigned a specific value in standard arithmetic due to the underlying principle of division by zero.
