What is infinity + infinity?
Understand the Problem
The question is asking about the result of adding infinity to infinity. In mathematics, infinity is not a real number, but rather a concept representing a quantity without any bound. Therefore, the operation infinity + infinity needs to be understood within the context of limits and mathematical analysis.
Answer
$\infty + \infty = \infty$
Answer for screen readers
$\infty + \infty = \infty$
Steps to Solve
- Understanding Infinity
Infinity ($\infty$) is a concept representing something that is endless or without any limit. It is not a real number and cannot be treated as such in arithmetic operations.
- Considering Limits
When we talk about infinity in mathematical operations, we often do so in the context of limits. The expression $\infty + \infty$ must be rigorously defined using limits.
- Analyzing the Sum
Consider two sequences $a_n$ and $b_n$ such that $\lim_{n\to\infty} a_n = \infty$ and $\lim_{n\to\infty} b_n = \infty$. We want to find the limit of their sum: $\lim_{n\to\infty} (a_n + b_n)$.
- Evaluating the Limit
Since both $a_n$ and $b_n$ approach infinity as $n$ goes to infinity, their sum will also approach infinity. Therefore, $\lim_{n\to\infty} (a_n + b_n) = \infty$. In this sense, $\infty + \infty = \infty$.
$\infty + \infty = \infty$
More Information
Infinity is not a number, but a concept. Therefore, adding infinity to infinity isn't a standard arithmetic operation. However, in the context of limits, if we have two quantities both approaching infinity, then their sum also approaches infinity.
Tips
Treating infinity as a real number and performing arithmetic operations directly can lead to incorrect conclusions. It's essential to work with limits rigorously. For example, $\infty - \infty$ is an indeterminate form and does not necessarily equal zero.
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