Different languages name big numbers in different ways. Korean gives a new name every 10,000× (10⁴) — man, eok, jo, gyeong. English gives a new name every 1,000× (10³) — thousand, million, billion. So even for the exact same number, the two languages' "sense of scale" often lines up differently. But even these naming systems eventually hit a wall. Once a number gets big enough, naming it at all starts to lose meaning.
A googol is a number with a 1 followed by 100 zeros (10¹⁰⁰). It's said the name came about when the mathematician Edward Kasner asked his 9-year-old nephew to make up a name for an enormously large number. For reference, the total number of atoms in the observable universe is estimated at around 10⁸⁰ — and a googol is far larger than even that.
A googolplex is 10 multiplied by itself a googol times — that is, 10^(10¹⁰⁰). It has a googol zeros, so large that even using every single atom in the universe to write one zero each, you still couldn't write them all. It's a number so large you couldn't even finish writing it out on paper. In the face of a number this big, even "trying to imagine how large it is" starts to lose all meaning.
But whether it's a googol or a googolplex, no matter how large, it's still ultimately a finite number. Keep counting the natural numbers endlessly — 1, 2, 3... — and eventually you'll pass even a googolplex. Mathematicians call this endless, never-ending count of all the natural numbers "infinity," and the 19th-century mathematician Georg Cantor took this one step further, proving that even infinity comes in different sizes.
First, natural numbers, integers, and even rational numbers (fractions) are all the same size of infinity, in the sense that they can all be "paired up one-to-one" with the natural numbers — an infinity like this is called countable (or countably infinite). The story of Hilbert's Hotel illustrates this well. Even a hotel that's already completely full, because it has infinitely many rooms, can make room for a new guest just by shifting every existing guest over by one room. But try to pair up every real number between 0 and 1 with the natural numbers, and a method called the "diagonal argument" proves you can always construct a new real number that isn't anywhere on the list. In other words, real numbers form a "larger" infinity than natural numbers — an uncountable infinity.
This concept of comparing infinities by size is called "cardinality." It feels strange to our intuition, but by the strict standard of "can it be paired up perfectly, one-to-one," the infinity of natural numbers and the infinity of real numbers are clearly different sizes. On the activity page, you can move a slider through powers of 10 to check big-number names, compare sizes from Avogadro's number all the way up to googolplex, and directly see Hilbert's Hotel and the diagonal argument in action with your own eyes.