Is divide by zero infinity?

Dividing by zero does not result in infinity; it is undefined.

Understanding Division by Zero

The idea that dividing by zero results in infinity is a common misconception. While it's true that in some contexts, especially when dealing with limits, expressions involving division by numbers approaching zero can tend towards infinity, division by zero itself remains undefined within the standard rules of arithmetic.

Why is it Undefined?

Division is the inverse operation of multiplication. When we say a / b = c, it means that b * c = a. Now, let's consider dividing by zero:

a / 0 = c implies 0 * c = a.

• If a is not zero: There is no value of c that satisfies this equation because anything multiplied by zero is zero. Therefore, a / 0 is undefined when a is not zero.
• If a is zero: Then 0 / 0 = c means 0 * c = 0. In this case, any value of c would satisfy the equation. This makes the result indeterminate, and therefore, also undefined.

Limits and Approaching Zero

The idea of infinity often comes into play when considering the limit of a function as the denominator approaches zero. For example, consider the function f(x) = 1/x.

• As x gets closer and closer to zero from the positive side (e.g., 0.1, 0.01, 0.001), f(x) becomes larger and larger (10, 100, 1000). We say the limit as x approaches 0 from the right is positive infinity.
• As x gets closer and closer to zero from the negative side (e.g., -0.1, -0.01, -0.001), f(x) becomes more and more negative (-10, -100, -1000). We say the limit as x approaches 0 from the left is negative infinity.

However, the function is never actually evaluated at x = 0. It is only approaching 0. Because the limits from the left and right are different, the limit as x approaches 0 does not exist. The function 1/0 itself is still undefined.

Practical Examples and Implications

• Programming: In most programming languages, attempting to divide by zero will result in an error.
• Calculators: Calculators will typically display an error message if you try to divide by zero.
• Mathematical Proofs: Allowing division by zero would break many fundamental theorems and proofs in mathematics, leading to contradictions.

Reference Information

As stated in the provided reference: "In mathematics, when you divide any number by zero, it will result in infinity. For example, divide by , the division will go on forever but you will achieve nothing. The answer will be undefined. Finding the limit of an expression that is divided by zero can be either or no limit." The reference correctly notes that the result is undefined. The connection to infinity arises only when considering limits.