## FANDOM

10,385 Pages

Although we value [your] donations, we were somewhat surprised to note that none of them ended in "-illion."
—Stephen Colbert[src]

The -illion system is a suffix used to represent the powers of ten,[1] the only large number system that is widely in use except for systems in eastern Asia which are based on a myriad. It has two distinct forms, the long scale and the short scale. English uses the short scale, while other languages such as French and German use the long scale. The long scale was also used in UK English until 1974, when the British usage and the American usage became identical.

The short scale goes as follows:

The long scale goes as follows:

## In googological notations

For the short scale:

Notation Expression
Fast-growing hierarchy between $$f_2(f_1^3(n))$$ and $$f_2(f_1^4(n))$$
Hardy hierarchy between $$H_{\omega^2+\omega 3}(n)$$ and $$H_{\omega^2+\omega 4}(n)$$
Slow-growing hierarchy $$g_{\omega^{n+1}}(1,000)$$; just below $$g_{\omega^\omega}(n)$$

For the long scale:

Notation Expression
Fast-growing hierarchy between $$f_2(f_1^4(n))$$ and $$f_2(f_1^5(n))$$
Hardy hierarchy between $$H_{\omega^2+\omega 4}(n)$$ and $$H_{\omega^2+\omega 5}(n)$$
Slow-growing hierarchy $$g_{\omega^{2n}}(1,000)$$; just below $$g_{\omega^\omega}(n)$$

For -illiard:

Notation Expression
Fast-growing hierarchy between $$f_2(f_1^4(n))$$ and $$f_2(f_1^5(n))$$
Hardy hierarchy between $$H_{\omega^2+\omega 4}(n)$$ and $$H_{\omega^2+\omega 5}(n)$$
Slow-growing hierarchy $$g_{\omega^{2n+1}}(1,000)$$; just below $$g_{\omega^\omega}(n)$$

## References

1. Billion from Wolfram Mathworld