Problem 2

Let

a,b,n>1

be integers and let

a,b

be bases of two number systems. Using the same digits

x_n,\ldots,x_0

, with

x_n\ne0

and

x_{n-1}\ne0

, define

A_n=x_nx_{n-1}\cdots x_0

and

A_{n-1}=x_{n-1}\cdots x_0

in base

a

, and

B_n=x_nx_{n-1}\cdots x_0

and

B_{n-1}=x_{n-1}\cdots x_0

in base

b

. Prove that

\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n}

if and only if

a>b

Step 1 of 5: Compare each digit weight

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In plain words

A larger base gives a larger weight to every earlier digit.

a>b\\Longrightarrow a^nb^k\\ge b^na^k\\quad(0\\le k\\le n)

Detailed analysis

If $a>b$ , then $a^{n-k}\ge b^{n-k}$ for every $k\le n$ . Multiplying by the positive factor $a^kb^k$ gives the displayed inequality, strict when $k<n$ .