A googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
Then there is the Googolplex. It is 1 followed by Googol zeros. I can’t even write down the number, because there is not enough matter in the universe to form all the zeros: 10,000,000,000,000,000,000,000,000,000,000, … (Googol number of Zeros)
And a Googolplexian is a 1 followed by Googolplex zeros. Wow.
The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of U.S. mathematician Edward Kasner. He may have been inspired by the contemporary comic strip character Barney Google.
Kasner popularized the concept in his 1940 book Mathematics and the Imagination. Other names for this quantity include ten duotrigintillion on the short scale, ten thousand sexdecillion on the long scale, or ten sexdecilliard on the Peletier long scale.
What is the last number in the world?
There is no such thing as the last number when it comes to the natural number system. By definition, every number has a number larger than it.
In the base-10 number system, every number has a number larger than itself. The concept of infinity relates to the idea that there is no last or largest number; there is always a bigger number than another number chosen. For example, if someone chooses any abstract number N, and then adds 1 to it, the resulting sum is larger than N. For any number N chosen, N is not the last number that is possible.
We know that Infinity is the idea of something that has no end. therefore, we can say that there is no last number in the world. And we can call it an Infinity. Infinity is not a real number.
Size of Googol Number
A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities such as the number of subatomic particles in the visible universe or the number of hypothetical possibilities in a chess game.
Kasner used it to illustrate the difference between an unimaginably large number and infinity, and in this role it is sometimes used in teaching mathematics.
To give a sense of how big a googol really is, the mass of an electron, just under 10−30 kg, can be compared to the mass of the visible universe, estimated at between 1050 and 1060 kg. It is a ratio in the order of about 1080 to 1090, or at most one ten-billionth of a googol (0.00000001% of a googol).
Carl Sagan pointed out that the total number of elementary particles in the universe is around 1080 (the Eddington number) and that if the whole universe were packed with neutrons so that there would be no empty space anywhere, there would be around 10128. He also noted the similarity of the second calculation to that of Archimedes in The Sand Reckoner.
By Archimedes’s calculation, the universe of Aristarchus (roughly 2 light years in diameter), if fully packed with sand, would contain 1063 grains. If the much larger observable universe of today were filled with sand, it would still only equal 1095 grains. Another 100,000 observable universes filled with sand would be necessary to make a googol.
The decay time for a supermassive black hole of roughly 1 galaxy-mass (1011 solar masses) due to Hawking radiation is on the order of 10100 years. Therefore, the heat death of an expanding universe is lower-bounded to occur at least one googol years in the future.
A googol is considerably smaller than a centillion.
Properties of Googol
A googol is approximately 70! (Factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., 1 googol = 2(100/log102) ≈ 2332.19280949. However, a googol is well within the maximum bounds of an IEEE 754 double-precision floating point type, but without full precision in the mantissa.
Using modular arithmetic, the series of residues (mod n) of one googol, starting with mod 1, is as follows:
0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 4, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 16, 10, 5, 0, 1, 4, 25, 28, 10, 28, 16, 0, 1, 4, 31, 12, 10, 36, 27, 16, 11, 0, … (sequence A066298 in the OEIS)
This sequence is the same as that of the residues (mod n) of a googolplex up until the 17th position.
Some Very Big, and Very Small Numbers
|Quadrillionth||0.000 000 000 000 001||femto||f|
|Quintillionth||0.000 000 000 000 000 001||atto||a|
|Sextillionth||0.000 000 000 000 000 000 001||zepto||z|
|Septillionth||0.000 000 000 000 000 000 000 001||yocto||y|
All Big Numbers We Know
|Name||As a Power of 10||As a Decimal|
|Vigintillion||1063||1 followed by 63 zeros!|
All Small Numbers We Know
|Name||As a Power of 10||As a Decimal|
|Billionths||10-9||0.000 000 001|