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

## 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 $$g_{\omega^{\omega^\omega}}(n)$$ $$n>3$$

## Sources

1. The Knuth -yllion Notation - Large Numbers (page 2) at MROB