The answer is ∞ (infinity).
When dealing with infinity, it's important to remember it's not a number in the traditional sense, but rather a concept representing something without any bound or limit. Adding infinity to itself doesn't suddenly create a larger infinity; it simply remains unbounded.
Understanding Infinity
Infinity is often used to describe:
- Limits: The value that a function approaches as the input gets arbitrarily large.
- Sets: The size of a set that contains an unlimited number of elements.
Why ∞ + ∞ = ∞
Imagine you have an infinitely large collection of objects. If you then combine it with another infinitely large collection of objects, you still have an infinitely large collection. You haven't changed the fundamental nature of being unbounded.
This can be further illustrated with examples:
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Integers: The set of all positive integers is infinite. The set of all negative integers is also infinite. Combining them (along with zero) still results in an infinite set (all integers).
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A Number Line: A number line extends infinitely in both positive and negative directions. Taking the positive part (0 to infinity) and adding an equivalent positive number line to it still results in a positive number line that extends to infinity.
Caution: Indeterminate Forms
It's crucial to distinguish ∞ + ∞ from other expressions involving infinity, which can be indeterminate. Examples include:
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∞ - ∞: This is indeterminate because the result depends on how the two infinities are approaching infinity. One infinity might be growing much faster than the other.
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0 * ∞: This is also indeterminate.
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∞ / ∞: This is another indeterminate form.
However, ∞ + ∞ definitively equals ∞.
