Never running out
What Is Infinity?
Infinity is not a huge number. It is the fact that the counting never has to stop.
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Step 1 of 5
Name the biggest number you can.
A million. A billion. A billion billion. Whatever you say, adding one gives something bigger — so your number was not the biggest.
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Step 2 of 5
So the counting numbers never end.
The rule for making a new number never fails. That is what never ending means. There is no last one. Not because we have not found it, but because the rule for making a new one always works. That endlessness is what infinity names.
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Step 3 of 5
Infinity is not a number you can do sums with.
Asking what infinity minus one comes to is like asking what colour Tuesday is. You cannot write infinity minus one and get something sensible. It describes a property of a collection — how it behaves, not how big it is.
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Step 4 of 5
Endless things can fit in small spaces.
Between 0 and 1 there are infinitely many fractions: a half, a third, a quarter, and on forever. Infinity does not need infinite room.
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Step 5 of 5
And some infinities are bigger than others.
Cantor proved in 1874 that the endless list of fractions is smaller than the endless list of all decimals. Mathematicians thought he had lost his mind.
The ∞ symbol was invented by John Wallis in 1655. Nobody is sure why he chose that shape.
The short version
Infinity is not a number you can do sums with; it names a property of a collection — how it behaves, not how big it is. Infinitely many fractions fit between 0 and 1, and in 1874 Cantor proved that some infinities are bigger than others.
Try it yourself
Ask a friend for the biggest number they can think of, then say theirs plus one. You can keep winning forever.