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Large numbers
Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics. Googology studies the naming conventions and properties of these immense numbers.
Since the customary decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (1012) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic scale.
A natural language numbering system represents large numbers using names rather than a series of digits. For example "billion" may be easier to comprehend for some readers than "1,000,000,000". Sometimes it is shortened by using a suffix, for example 2,340,000,000 = 2.34 B (B = billion). A numeric value can be lengthy when expressed in words, for example, "2,345,789" is "two million, three hundred forty five thousand, seven hundred and eighty nine".
Scientific notation was devised to represent the vast range of values encountered in scientific research in a format that is more compact than traditional formats yet allows for high precision when called for. A value is represented as a decimal fraction times a multiple power of 10. The factor is intended to make reading comprehension easier than a lengthy series of zeros. For example, 1.0×109 expresses one billion – 1 followed by nine zeros. The reciprocal, one billionth, is 1.0×10−9. Sometimes the letter e replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0×109.
Examples of large numbers describing real-world things:
In astronomy and cosmology large numbers for measures of length and time are encountered. For instance, according to the prevailing Big Bang model, the universe is approximately 13.8 billion years old (equivalent to 4.355×1017 seconds). The observable universe spans 93 billion light years (approximately 8.8×1026 meters) and hosts around 5×1022 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about 1080 atoms within the observable universe.
According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
(which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses), roughly 10^10^1.288*10^3.884 T This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
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Large numbers
Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics. Googology studies the naming conventions and properties of these immense numbers.
Since the customary decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (1012) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic scale.
A natural language numbering system represents large numbers using names rather than a series of digits. For example "billion" may be easier to comprehend for some readers than "1,000,000,000". Sometimes it is shortened by using a suffix, for example 2,340,000,000 = 2.34 B (B = billion). A numeric value can be lengthy when expressed in words, for example, "2,345,789" is "two million, three hundred forty five thousand, seven hundred and eighty nine".
Scientific notation was devised to represent the vast range of values encountered in scientific research in a format that is more compact than traditional formats yet allows for high precision when called for. A value is represented as a decimal fraction times a multiple power of 10. The factor is intended to make reading comprehension easier than a lengthy series of zeros. For example, 1.0×109 expresses one billion – 1 followed by nine zeros. The reciprocal, one billionth, is 1.0×10−9. Sometimes the letter e replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0×109.
Examples of large numbers describing real-world things:
In astronomy and cosmology large numbers for measures of length and time are encountered. For instance, according to the prevailing Big Bang model, the universe is approximately 13.8 billion years old (equivalent to 4.355×1017 seconds). The observable universe spans 93 billion light years (approximately 8.8×1026 meters) and hosts around 5×1022 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about 1080 atoms within the observable universe.
According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
(which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses), roughly 10^10^1.288*10^3.884 T This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.