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Concept
Large numbers
Summary
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts.
Googology is the study of nomenclature and properties of large numbers.
Scientific notationLogarithmic scaleOrders of magnitude and Names of large numbers
Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, or a 1 followed by nine zeros: 1 000 000 000. The reciprocal, 1.0 × 10−9, means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. In addition to scientific (powers of 10) notation, the following examples include (short scale) systematic nomenclature of large numbers.
Examples of large numbers describing everyday real-world objects include:
The number of cells in the human body (estimated at 3.72 × 1013), or 37.2 trillion
The number of bits on a computer hard disk (, typically about 1013, 1–2 TB), or 10 trillion
The number of neuronal connections in the human brain (estimated at 1014), or 100 trillion
The Avogadro constant is the number of “elementary entities” (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 - approximately 6.022e23, or 602.2 sextillion.
The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3 ± 3.6) × 1037, or 53±36 undecillion
The mass of Earth consists of about 4 × 1051, or 4 sexdecillion, nucleons
The estimated number of atoms in the observable universe (1080), or 100 quinvigintillion
The lower bound on the game-tree complexity of chess, also known as the “Shannon number” (estimated at around 10120), or 1 novemtrigintillion
Note that this value of the Shannon number is for Standard Chess.
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Names of large numbers
Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America. These naming procedures are based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion.
Hyperoperation
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity.
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.
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