Is $\infty + \infty > \infty$ ?

This is part of a series on common misconceptions.

Is this true or false?

$$\infty + \infty > \infty$$

Why some people say it's true: We have $2 = 1 + 1 > 1$ and $2x = x + x > x$ with all positive real values of $x$. Therefore $\infty + \infty > \infty$

Why some people say it's false: Based on the $\infty$'s property, we know that $\infty + \infty = \infty$. Therefore $\infty + \infty$ cannot be larger than $\infty$

The statement is $\color{#D61F06}{\textbf{false}}$.

Proof: You may say that $\infty + \infty = \infty$, but that means we have $$\lim_{x\to\infty}{f(x)} = \infty$$ and $$\lim_{x\to\infty}{g(x)} = \infty$$ so $$\lim_{x\to\infty}{f(x)} + \lim_{x\to\infty}{g(x)} = \infty$$

We cannot do ANY arithmetic operator to infinity. It is not legal to do so since $\infty$ is a concept, not a number. It is also uncountable. Therefore we cannot compare $\infty + \infty$ and $\infty$

Rebuttal: Then what is $\infty + \infty$ in the school?

Reply: That is only when you evaluate sum of the limits of $f(x)$ and $g(x)$ altogether. If both of them are infinity, then the sum will end up as $\infty$.

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Rebuttal: But if you cannot do any arithmetic operator with limit, how comes $\infty \times (- \infty) = - \infty$?

Reply: Let's put it in this way: $\infty$ > 0 and $- \infty$ < 0. A number greater than 0 multiplied by a number less than 0 will result in a negative number; therefore, a really large positive number multiplied by a really large negative number will end up with a really large negative number.

See Also

• List of Common Misconceptions

• Is $-\infty < \infty$? [preparation]

• Infinity