# Emulating digits

In gen­eral, given enough $$n$$-digits, you can em­u­late an $$m$$-digit, for any $$m, n \in$$ $$\mathbb N$$. If $$m < n,$$ you can em­u­late an $$m$$-digit us­ing just one $$n$$-digit — in other words, you can use a digit wheel like a $$7$$-digit if you want to, by just ig­nor­ing three of the pos­si­ble ways to set the digit wheel. If $$m > n,$$ things are a bit more difficult, but only slightly.

Ba­si­cally, with 2 $$n$$-digits, you can em­u­late a $$n^2$$-digit, as fol­lows. Us­ing your two $$n$$-digits, en­code a num­ber $$(x, y)$$ where $$0 \le x < n$$ and $$0 \le y < n$$. In­ter­pret $$(x, y)$$ as $$xn + y.$$ You have now en­coded a num­ber be­tween 0 (if $$x = y = 0$$) and $$n^2 - 1$$ (if $$x = y = n-1$$). Con­grat­u­la­tions, you just used two $$n$$-digits to make an $$n^2$$ digit!

You can use the same strat­egy to em­u­late $$n^3$$-digits (in­ter­pret $$(x, y, z)$$ as $$xn^2 + yn + z$$), $$n^4$$-digits (you get the pic­ture), and so on. Now, to em­u­late an $$m$$-digit, just pick an ex­po­nent $$a$$ such that $$n^a > m,$$ col­lect $$a$$ copies of an $$n$$-digit, and you’re done.

This isn’t nec­es­sar­ily the most effi­cient way to use $$n$$-digits to en­code $$m$$-digits. For ex­am­ple, if $$m$$ is 1,000,001 and $$n$$ is 10, then you need seven 10-digits. Seven 10-digits are enough to em­u­late a 10-mil­lion-digit, whereas $$m$$ is a mere mil­lion-and-one-digit — pay­ing for a 10-mil­lion-digit when all you needed was an $$m$$-digit seems a bit ex­ces­sive. For some differ­ent meth­ods you can use to re­cover your losses when en­cod­ing one type of digit us­ing an­other type of digit, see Frac­tional digits and Frac­tional bits. (Th­ese tech­niques are fairly use­ful in prac­tice, given that mod­ern com­put­ers en­code ev­ery­thing us­ing bits, i.e. 2-digits, and so it’s use­ful to know how to effi­ciently en­code $$m$$-mes­sages us­ing bits when $$m$$ is pretty far from the near­est power of 2.)

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.