# 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 \( \color{red}{\textbf{false}}\).

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 \(\infty\).

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**

Is \(-\infty < \infty\)? [preparation]

**Cite as:**Is \( \infty + \infty > \infty \) ?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/is-infty-infty-infty/