Can a number divided by 0 equal infinity?

No, a number divided by 0 does not equal infinity. Division by zero is undefined.

While the limit of a number divided by a value approaching zero can tend towards infinity, it's crucial to understand the distinction between a limit and an actual mathematical operation.

Here's a breakdown:

• Division by Zero is Undefined: In standard mathematics, division is defined as the inverse operation of multiplication. For example, 6 / 3 = 2 because 2 * 3 = 6. If we tried to divide by zero, say 6 / 0 = ?, we'd be asking "What number, when multiplied by 0, equals 6?". There is no such number. Anything multiplied by zero is zero. Therefore, division by zero is undefined.

• Limits and Approaching Zero: The concept of a limit deals with the behavior of a function as its input approaches a certain value. Consider the function f(x) = 1/x. As x gets closer and closer to zero (from the positive side), the value of f(x) gets larger and larger, approaching positive infinity. We write this as lim (x→0+) 1/x = +∞. Similarly, as x approaches zero from the negative side, f(x) approaches negative infinity: lim (x→0-) 1/x = -∞. However, this doesn't mean 1/0 equals infinity; it means the limit of 1/x as x approaches zero is infinity.

• Why the Distinction Matters: Treating division by zero as equal to infinity can lead to mathematical contradictions and nonsensical results. For example, if we incorrectly assume that 1/0 = ∞, we could "prove" that 1 = 2.

• Computer Handling: As the reference pointed out, computers usually don't "consider infinity." In programming, attempting to divide by zero typically results in an error (e.g., ZeroDivisionError in Python) or a special value like NaN (Not a Number) to indicate an undefined result. This is because computers cannot accurately represent infinity.

In summary, while the limit of a number divided by a value approaching zero can be infinite, division by zero itself remains undefined. Don't confuse the limit with the operation.