What is the Remainder Division by Zero?

Division by zero, including finding the remainder, is undefined in mathematics.

Understanding the Issue

The core problem is that division is the inverse operation of multiplication. When we ask what a / b is, we're essentially asking, "What number, when multiplied by b, gives us a?".

• Example: 6 / 3 = 2 because 3 * 2 = 6.

When b is zero, this logic breaks down.

Why Division by Zero is Undefined

Let's say we try to divide a number, say 5, by 0: 5 / 0 = ?

This translates to: "What number, when multiplied by 0, gives us 5?". No such number exists because any number multiplied by 0 always equals 0.

Now, if we try to divide 0 by 0: 0 / 0 = ?

This translates to: "What number, when multiplied by 0, gives us 0?". Infinitely many numbers satisfy this (any number!), which makes the result ambiguous and therefore, undefined.

Remainder and Division by Zero

The remainder operation (often represented by % or mod) is closely linked to division. The remainder is what's "left over" after performing integer division.

• Example: 7 % 3 = 1 because 7 divided by 3 is 2 with a remainder of 1 (7 = 3 * 2 + 1).

Since division by zero is undefined, the remainder when dividing by zero is also undefined. There is no meaningful mathematical result.

A Limiting Case (Advanced)

The provided reference touches upon a more advanced concept using limits. It states: "Divison with 0 is undefined, and, therefore, not an allowable operation in mathematics, so the question doesn't have any good mathematical meaning. But as the limit of 1n=0 1 n = 0 as n→∞ n → ∞ , we can ask what the remainder will be when we divide a number N on 1n and let n go towards infinity, i.e. n→∞ n → ∞."

This is about approaching zero, not actually dividing by zero. It involves examining what happens to remainders as the divisor gets infinitely small (approaches zero). This is studied in calculus and leads to concepts of limits and infinitesimals, but it doesn't change the fundamental rule that division by zero is undefined. This is a completely separate concept.

Conclusion

Trying to find the remainder after division by zero is mathematically undefined. The core division operation itself has no valid result.