Science & Technology

# These Are the Biggest Numbers in Mathematics

Counting to three is so easy, a salamander can do it. Seriously. Lab experiments have shown that captive salamanders are able to distinguish between piles of two fruit flies and piles of three. If you're not impressed, we understand. A human being who'd never taken a single math class would have no trouble doing the same thing. Some single-digit numbers like one, two and three are so small that our minds can recognize their value without even needing to count. Put a tray of three cookies in front of your average adult and he or she will immediately (and intuitively) know how many there are. No fingers or calculators required!

Yet as numbers grow bigger, our ability to comprehend their values starts to break down. The word "billion" gets tossed around a lot by economists and politicians, but it's hard to appreciate just how large that sum is. One billion seconds is equal to 31 years, 251 days, 13 hours, 34 minutes, and 54.7843 seconds (not counting leap days and leap seconds)

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By the commonly accepted definition we use today, one billion is equal to a thousand millions. Numerically, it is expressed as 1,000,000,000. One trillion is understood to be a million millions, or 1,000,000,000,000. To put that in perspective, let's say you've pulled an H.G. Wells and built a functional time machine. If you ordered it to take you one trillion seconds back in time, you'd get to hang out with mammoths and saber-toothed cats because one trillion seconds is the equivalent of 31,546 years.

Note that a trillion is written as a one followed by twelve zeroes. The next order of magnitude is a quadrillion, which contains fifteen zeroes. (You may be interested to know that a supercomputer that was recently unveiled at the Oak Ridge National Laboratory in Tennessee can make up to 200 quadrillion calculations per second. It's roughly a million times faster than the average laptop.)

Now take a pen, grab some paper, and write down a nice, tidy row of 100 individual zeroes. Then put a "1" in front of them. The massive figure you'll see before you is 10100. Mathematician Edward Kasner took a fancy to this number in 1923. His nine-year-old nephew Milton Sioratta came up with a name for it, calling the super-large sum a "Googol." Many years later, a misspelling of this term would be used as the name of the internet's top search engine — and a brand worth \$132.1 billion. Don't know what we're talking about? Google it.

## And Then Your Mind Blows...

The number's size will blow your mind. Remember when we said that a Googol is 10100? Well get this: Astronomers estimate that there are only 1,078 to 1,082 atoms in the known, observable universe — an area which encompasses 93 billion light years.

Enormous as a Googol is, at least you can write it down numerically. By this, we mean to say that you could — if you felt so inclined — write a 1 followed by 100 zeroes. The same cannot be said of a Googolplex. That, dear reader, is a one followed by a Googol's worth of zeroes. No matter how tiny your handwriting is, you'll never be able to jot down all those 0s; there are more zeroes in a Googolplex than there are atoms in the observable universe. The only way to commit this figure to paper is by using exponential notation. Written out that way, a Googolplex is:

And if you think a Googolplex is big, get a load of Skewes' number, which looks like so:

This one derives its name from Stanley Skewes, a South African mathematician with an interest in prime numbers. You probably know that a prime is any number that can only be divided by itself and by the number one. Therefore, three is a prime, but four is not because it's divisible by two. To make a long story short, Skewes was studying a mathematical function that's been used to give rough estimates of how many primes there are between zero and any number you might care to name (eg: 1,000).

Skewes introduced his eponymous number to the world in a 1933 paper on said function. In the words of one colleague, this was — at the time, at least — the "largest number which has ever served any definite purpose in mathematics."

It's since lost that distinction to still-bigger sums like Graham's number and the monstrous TREE(3). Both of these are way too vast for the human mind to grasp. Yet each is finite and mathematically useful in its own way.

Before wrapping up this discussion, let's take a step back to acknowledge a smaller figure. In January 2018, math enthusiast Jonathan Pace identified what is, to date, the biggest known prime number. Named M77232917, it contains more than 23 million digits — 23,249,425 of them to be exact. As such, it is 910,807 digits larger than the previous record-holder. To be sure, M77232917 isn't in the same league as the Googol, the Googolplex, or Skewes' number. But if you wrote the newfound number out in its entirety at a rate of five digits per inch, the whole thing would exceed 73 miles (118 kilometers) in length. Sounds like a surefire way to get finger cramps.

Correction 3/3/2019: A previous version of this article defined a Googol as 10100. That was a formatting error; a Googol is 10100. The article has been corrected.

For more mathematical explorations, check out "The Prime Number Conspiracy: The Biggest Ideas in Math from Quanta." We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

Mark Mancini for HowStuffWorks