Great names for great numbers

Chemists commonly speak of Avogadro's number, the term for the number of particles in a mole of matter: it is written 6.02214076 × 10²³. That is certainly a "large" number, but mathematicians, often faced with squares, cubes, various n^th powers, and exponentials, which very quickly become "immense", have had to invent notational tricks to write them.

A suitable notation --------------------

With the advent of computers, the notation m $\uparrow \backslash uparrow$ n was introduced; today, it is often written simply m^n (using only the head of the vertical arrow), which evokes *mⁿ, while avoiding exponents. Just as multiplication is repeated addition (m × n = m + m + … + m, n times), exponentiation is repeated multiplication: mⁿ = m × m × … × m, with n copies of the factor m*, as the arrow notation also shows.

The notation extends to m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ n = m $\uparrow \backslash uparrow$ m $\uparrow \backslash uparrow$ … $\uparrow \backslash uparrow$ m, with n copies of m, then to m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ n = m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ … $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ m, again with m occurring n times, then to m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ n = m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ m $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ … $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ m, once more with m occurring n times. This notation was introduced in 1976 by Donald Knuth, an American mathematician born in 1938 and a pioneer of programming, who called the double upward arrow tetration because it represents iterated exponentiation. Thus 3 $\uparrow \backslash uparrow$ 3 = 3³ = 27, 3 $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ 3 = 3 $\uparrow \backslash uparrow$ (3 $\uparrow \backslash uparrow$ 3) = 3 $\uparrow \backslash uparrow$ 27 = 7,625,597,484,987… and we soon see why this notation is useful: it lets us write very large numbers compactly. Without it, it is hard even to write the number 3 $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ 3 = 3 $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ (3 $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ 3) = 3 $\uparrow \backslash uparrow$ $\uparrow \backslash uparrow$ (3²⁷) = 3 $\uparrow \backslash uparrow$ (3²⁷ $\uparrow \backslash uparrow$ 3²⁷), which we can scarcely imagine written out in full.

Great numbers that made a name for themselves --------------------------------------------