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The -yllion system is a fabricated system invented by Donald Knuth for naming large numbers.[1]

## Usage

Numbers up to 999 are named as they are normally.

Numbers from 1000 to 9999 are written without a comma and pronounced "a-ty b hundred c-ty d" for the number abcd. For example, 7283 is pronounced "seventy-two hundred eighty-three." The word thousand is not used in this system.

$$10^4$$ is called the myriad. All numbers from the myriad up to 9999,9999 are written with a comma between the ten thousands and thousands place. For example, 54325 would be written 5,4325.

Such numbers are pronounced like so:

• 45,7839 is pronounced "forty-five myriad seventy-eight hundred thirty-nine."
• 2423,3000 is pronounced "twenty-four hundred twenty-three myriad thirty hundred."
• 9999,9999 is pronounced "ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine."

$$10^8$$ is called the myllion. It is written 1;0000,0000. Note that a new punctuation mark, a semicolon, was used to represent the myllions place. Numbers up to 9999,9999;9999,9999 are named in the same way as the above numbers are pronounced: each group of four digits is pronounced as written above, and the commas are pronounced "myriad." The only difference is that the semicolon (;) is pronounced "myllion."

$$10^{16}$$ is called the byllion. It is written 1:0000,0000;0000,0000. Note that a new punctuation mark, a colon, was used to represent the byllions place. Numbers up to 9999,9999;9999,9999:9999,9999;9999,9999 are pronounced as above, but the colon (:) is pronounced "byllion."

Continuing in this method, we have n-yllion $$= 10^{2^{n+2}}$$, allowing the tryllion, quadryllion, quintyllion, etc.

## In googological notations

Notation Upper bound Condition
Fast-growing hierarchy $$f_2^2(n)$$ $$n>13$$
Hardy hierarchy $$H_{\omega^2 2}(n)$$ $$n>13$$
Slow-growing hierarchy $$g_{\omega^{\omega^\omega}}(n)$$ $$n>3$$

Knuth's system wouldn't be implemented well in Polish due to some numerals having -ylion suffix in basic forms due to rule of Polish language, which changes syllables -ti-, -ri-, -ci- into -ty-, -ry-, -cy- in adapted loanwoards, present in all thousands powers from trillion upwards, e.g. trylion as trillion, kwadrylion as quadrillion, kwintylion as quintillion etc. (nonilion as nonnillion is only exception, but also not always[2]), which creates system from 1032 upwards invalid.