Is \( \infty + 1 > \infty \)?

This is part of a series on common misconceptions.

Is this true or false?

$$\infty + 1 > \infty$$

Why some people say it's true: Adding one to any number always makes it bigger.

Why some people say it's false: Infinity is not a real number, so this arithmetic is not valid. Also, according to the infinity rules, $x + \infty = \infty$, with $x$ is any real number. Hence, $\infty + 1 = \infty$

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

Proof:

We cannot do ANY ARITHMETIC OPERATION with infinity, since it is a concept, not a number.

But let's take a look at the limit to prove it is wrong.

Let $$\lim_{x\to\infty}{f(x)} = \infty$$ and $$\lim_{x\to\infty}{g(x)} = 1$$

We have: $$\lim_{x\to\infty}{f(x)} + \lim_{x\to\infty}{g(x)} = \lim_{x\to\infty}{(f(x)+g(x))} = \infty + 1 = \infty$$

Hence, the statement is $\color{#D61F06}{\textbf{false}}$.

Rebuttal: But what if you have a very large number, say, $x = 3141592653589\dots$, will it be larger than infinity?

Reply: You may want to see Is $\infty + \infty > \infty$ ?

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Rebuttal: If you cannot do any arithmetic operation with infinity, what does $\infty + 1 = \infty$ mean in your proof?

Reply:

That means you use the one of the properties of infinity to evaluate the limit of a function.

See Also

• List of Common Misconceptions

• Infinity