## Monday, 5 December 2011

### Googolplex

A googolplex is the number 10googol, i.e. $10^{(10^{100})}$.

In 1938, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition "because different people get tired at different times and it would never do to have Carnera be a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer". It thus became standardized to $10^{(10^{100})}$.

In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe provides.

An average book of 60 cubic inches can be printed with 5×105 zeroes (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3×103 zeros per cubic inch. The observable (i.e. past light cone) universe contains 6×1083 cubic inches (4/3 × π × (14×109 light years in inches)3). This math implies that if the universe is stuffed with paper printed with 0's, it could contain only 5.3×1087 zeros—far short of a googol of zeros. In fact there are only about 2.5×1089 elementary particles in the observable universe so even if one were to use an elementary particle to represent each digit, one would run out of particles well before reaching a googol of digits.

Consider printing the digits of a googolplex in unreadable, one-point font (0.353 mm per digit). It would take about 3.5×1096 metres to write a googolplex in one-point font. The observable universe is estimated to be 8.80×1026 meters, or 93 billion light-years, in diameter,so the distance required to write the necessary zeroes is 4.0×1069 times as long as the estimated universe.

The time it would take to write such a number also renders the task implausible: if a person can write two digits per second, it would take around about 1.51×1092 years, which is about 1.1×1082times the age of the universe, to write a googolplex.