Numbers - Hierarchy

Jai Ganesh · 2022-10-10 03:39:59

Names of large numbers - Prefixes

This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.

Two naming scales 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

(short scale) or (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion.

Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as 10 with a numeric superscript.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores. English also has many words, such as "zillion", used informally to mean large but unspecified amounts.

Standard dictionary numbers

Jai Ganesh · 2022-10-10 13:24:26

Large Numbers

Large numbers are numbers above one million that are usually represented either with the use of an exponent such as

or by terms such as billion or thousand millions that frequently differ from system to system. The American system of numeration for denominations above one million was modeled on a French system, but in 1948 the French system was changed to correspond to the German and British systems. In the American system each of the denominations above 1,000 millions (the American billion) is 1,000 times the preceding one (one trillion = 1,000 billions; one quadrillion = 1,000 trillions). In the British system each of the denominations is 1,000,000 times the preceding one (one trillion = 1,000,000 billions) with the sole exception of milliard, which is sometimes used for 1,000 millions. In recent years British usage has reflected widespread and increasing use of the American system.

Have you ever wondered what number comes after a trillion? Or how many zeros there are in a vigintillion? Some day you might need to know this for a science or math class, or if you happen to enter one of several mathematical or scientific fields.

Numbers Bigger Than a Trillion

The digit zero plays an important role as you count very large numbers. It helps to track these multiples of 10 because the larger the number is, the more zeros are needed.

Metric Numbers.

Jai Ganesh · 2022-10-10 13:39:10

Have a look at these Ginormous Numbers:

.

Googol

A googol is the large number

. 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.

Etymology

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.

Size

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.

Jai Ganesh · 2022-10-14 22:06:47

Centillion

Celtillion a cardinal number represented in the U.S. by 1 followed by 303 zeros, and in Great Britain by 1 followed by 600 zeros.

Jai Ganesh · 2022-10-15 02:58:39

Skewes's number

Part - I

In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number

for which

,
where is the prime-counting function and li is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing near It is not known whether it is the smallest.

Jai Ganesh · 2022-10-24 22:15:47

Skewes's number

Part - II

Skewes's numbers
John Edensor Littlewood, who was Skewes's research supervisor, had proved in Littlewood (1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference

changes infinitely many times. All numerical evidence then available seemed to suggest that was always less than Littlewood's proof did not, however, exhibit a concrete such number x.

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating

below

In Skewes (1955), without assuming the Riemann hypothesis, Skewes proved that there must exist a value of x below

Skewes's task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to Georg Kreisel, this was at the time not considered obvious even in principle.

Jai Ganesh · 2023-01-02 13:33:31

Googolplex

A googolplex is the number

, or equivalently, or . Written out in ordinary decimal notation, it is 1 followed by zeroes; that is, a 1 followed by a googol of zeroes.

History

In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is

, and 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 , due to the right-associativity of exponentiation.

Size

A typical book can be printed with

zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires such books to print all the zeros of a googolplex (that is, printing a googol zeros). If each book had a mass of 100 grams, all of them would have a total mass of kilograms. In comparison, Earth's mass is kilograms, the mass of the Milky Way galaxy is estimated at kilograms, and the total mass of all the stars in the observable universe is estimated at .

To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the masses of the Milky Way and the Andromeda galaxies combined (by a factor of roughly

), and greater than the mass of the observable universe by a factor of roughly .

In pure mathematics

In pure mathematics, there are several notational methods for representing large numbers by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation, or Conway chained arrow notation.

Jai Ganesh · 2023-11-23 14:23:13

Tetration

Introduction

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as

a'=a+1, is considered to be the zeroth operation.

Addition

n copies of 1 added to a combined by succession.

Multiplication

n copies of a combined by addition.

Exponentiation

n copies of a combined by multiplication.

Tetration

n copies of a combined by exponentiation, right-to-left.

Note that nested exponents are conventionally interpreted from the top down:

means

and not

Europe2048 · 2024-04-01 09:14:57

The study of big numbers is also called "googology".

Jai Ganesh · 2024-04-01 18:32:21

I checked up.

Googology: Googology is the study of nomenclature and properties of large numbers.

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