The theory of numbers, by Robert D. Carmichael ...

22

THEORY OF NUMBERS

~ II. SCALES OF NOTATION
I. If mn and n are positive integers and n>I, then m can be
represented in terms of n in one and in only one way in the form
m=aonh+alnh-l+.. +ahn+aa,
where
a0o, o a, oan, I, 2,.., h.
That such a representation of m exists is readily proved by
means of the fundamental theorem of Euclid. For we have
no=non+ah-, o < an- <n,
nl = 21+ah- 2, O < (h-2 <n,
nh-3 =nh-2n+a2, o0 a2<n,
nnh2 =nh-1n+al, o i< c,<,
nh- = ao, o < ao<n.
If the value of nh- given in the last of these equations is substituted in the second last we have
nn-2 =aon+al.
This with the preceding gives
h- 3 = aon2+aln+a2.
Substituting from this in the preceding and continuing the
process we have finally
m =aonh +alnh-l-...ah_ +ah,
a representation of m in the form specified in the theorem.
To prove that this representation is unique, we shall suppose
that n has the representation
m=bonk +bln-l+.. +bkn+bk,
where
bono, o<b<n, i=o, I, 2,..., k,