# How To Write Large Numbers - Experts' opinions

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically http://cocktail24.info/blog/popular-thesis-editing-website-uk.php to large positive integersor more generally, large positive real numbersbut it may also be How To Write Large Numbers in other contexts.

Very large numbers often occur in fields such as mathematicscosmologycryptographyand statistical mechanics. Sometimes people refer to numbers as being "astronomically large". However, it is easy to mathematically define numbers that are much larger even than those used in astronomy. Scientific notation was created to handle the wide range of values that here in scientific study.

Writing 10 9 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. Other large numbers, as regards length and time, are found in astronomy and cosmology. For example, the current Big Bang model suggests that the universe is There are about 10 80 atoms in the observable universeby rough estimation.

According to Don Pagephysicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is.

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 link be somewhat similar for a reasonable choice of "similar" to its current state again.

Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations on a set of fixed objects, grows very rapidly with the number of objects. Stirling's formula gives a precise asymptotic expression for this rate of growth. Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms. Logician Harvey Friedman has done work related to very large numbers, such as with Kruskal's tree theorem and the Robertson—Seymour theorem.

Between andpersonal computer hard disk sizes increased from about 10 megabytes 10 7 bytes to over gigabytes 10 11 bytes. But what about a dictionary-on-disk storing all possible passwords containing up to 40 characters? In his paper Computational capacity of the universe[5] Seth Lloyd points out that if every particle in the universe could be used as part of a huge computer, it could store only about 10 90 bits, less than How To Write Large Numbers millionth of the size such a dictionary would require.

Get Grammar Girl's take on how to write numbers. Learn when to write out the words for numbers and when it's okay to use numerals in a sentence. Learn how to write large numbers in the millions, billions, trillions, and even quintillions as words and as digit–word combinations. In this lesson you will learn how to write very large numbers by using scientific notation. Some names of large numbers, such as million, billion, This is a description of what would actually happen if one actually tried to write a googolplex.

However, storing information on hard disk and computing it are very different functions. On the one hand storage currently has limitations as stated, but source speed is a different matter.

It is quite conceivable [ by whom? Still, computers can easily be programmed to start creating and displaying all possible character passwords one at a time. Such a program could be left to run indefinitely. By contrast, the universe is estimated to be Computers will presumably continue to get faster, but the same paper mentioned before estimates that the entire universe functioning as a giant computer could have performed no more than 10 operations since the Big Bang.

This is trillions of times more computation than is required for displaying all character passwords, but computing all 50 character passwords would outstrip the estimated computational potential of the entire universe. Problems like this grow exponentially in the number of computations they require, and they are one reason why exponentially difficult problems are called "intractable" in computer science: The traditional division between "easy" and "hard" problems is thus drawn between programs that do and do not require exponentially increasing resources to execute.

Such limits are an advantage in cryptographysince any cipher -breaking technique that requires more than, say, the 10 operations mentioned before will never be feasible. Such ciphers must be broken by finding efficient click here unknown to the cipher's designer.

For example, one way of finding How To Write Large Numbers greatest common divisor between two digit numbers is to compute all their factors by trial division. But the Euclidean algorithmusing a much more efficient technique, takes only a see more of a second to compute the GCD for even huge numbers such as these.

As a general rule, then, PCs in can perform 2 40 calculations in a few minutes. Limits on computer storage are comparable. However, it has practical and theoretical challenges that may never be overcome, such as the mass production of qubitsthe fundamental building block of quantum computing.

The total amount of printed material in the world is roughly 1. The first number is much larger than the second, due to How To Write Large Numbers larger height of the power tower, and in spite of the small numbers 1.

This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals. A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one. These are very round numbers, each representing an order of magnitude in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.

See also extension of tetration to real heights. As explained, a more accurate description of a number also specifies the value of this number between 1 and 10, or the previous number taking the logarithm one time less between 10 and 10 10or the next, between 0 and 1. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power read more with a 10 at the bottom is then not the same as extending it with a 10 at the top but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from If the height of the tower is large, the various representations for large numbers can be applied to the height itself.

If the height is given only approximately, giving a value at the top does not make sense, so we can use the double-arrow notation, e. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.

If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e. This can be done recursively, so we can have a power of the triple arrow operator. Compare this notation with the hyper operator and the Conway chained arrow notation:. An advantage of the first is that when considered as function of bthere is a natural notation for powers of this function just like when writing out the n arrows: For describing numbers approximately, deviations from the decreasing order of values of n are not needed.

Thus we have the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10 x are "almost equal" for arithmetic of large numbers see also below. If the superscript of the upward arrow is large, the various representations for large numbers can be How To Write Large Numbers to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it acts.

For such numbers the advantage How To Write Large Numbers using the upward arrow notation no longer applies, and we can also use the chain notation. If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number like using the superscript of the arrow instead of writing many arrows.

If n is large we can use any of the above for expressing it. Compare the definition of Graham's number: Using the functional power notation of f this gives multiple levels of f. Similarly we can introduce a function hetc. If we need many such functions we can better number them instead of using a new letter every time, e.

If even the position How To Write Large Numbers the sequence is a large number we can apply the same techniques again for that.

The following illustrates the effect of a base different from 10, base It also illustrates representations of numbers, and the arithmetic. In a number like 10 6.

Http://cocktail24.info/blog/list-of-thesis-statement.php seems like extremely poor accuracy, but for such a large number it may be considered fair a large error in a large number may be "relatively small" and therefore acceptable.

In the case of an approximation of an extremely large number, the relative error may be large, yet there may still be a sense in which we want to consider the numbers as "close in magnitude".

find how to say and spell really huge numbers See how to write it. (4) The Math Cats name large numbers the American way. Proper English rules for when and how to write numbers. Home; Writing Numbers. The simplest way to express large numbers is usually best. See how to write out numbers on checks and other documents. Tips for clarity, and examples using large and small numbers. Writing and Saying Large Numbers, by Dennis Oliver. Writing and Saying Large Numbers (#1) We don't normally write numbers with words, but it's possible to do this. When writing out large numbers in words, should commas be placed at thousand separators? The APA style says you should write out numbers that are ten or less.

The point is that exponential functions magnify relative errors greatly — if a and b have a small relative error. The question then becomes: There is a check this out in which we may want to consider.

The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:. Such comparisons of iterated logarithms are common, e. There are some general rules relating to the usual arithmetic operations performed on very large numbers:. Its value for even relatively small input is huge. Although all the numbers discussed above are very large, they are all still decidedly finite.

Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite click of natural numbersand aleph-one is the next greatest cardinal number. These How To Write Large Numbers are essentially functions of integer variables, which increase very rapidly with those integers.

Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument. Note that a function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: From Wikipedia, the free encyclopedia.