MathLabs

Problem 2

Let a,b,n>1a,b,n>1 be integers and let a,ba,b be bases of two number systems. Using the same digits xn,…,x0x_n,\ldots,x_0, with xn≠0x_n\ne0 and xn−1≠0x_{n-1}\ne0, define An=xnxn−1⋯x0A_n=x_nx_{n-1}\cdots x_0 and An−1=xn−1⋯x0A_{n-1}=x_{n-1}\cdots x_0 in base aa, and Bn=xnxn−1⋯x0B_n=x_nx_{n-1}\cdots x_0 and Bn−1=xn−1⋯x0B_{n-1}=x_{n-1}\cdots x_0 in base bb. Prove that An−1An<Bn−1Bn\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n} if and only if a>ba>b.
Step 4 of 5: Recover the truncated ratios
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In plain words

A larger base gives the leading digit a larger share, leaving a smaller share for the remaining digits.

fracAn−1An=1−fracxnanAn,quadfracBn−1Bn=1−fracxnbnBn\\frac{A_{n-1}}{A_n}=1-\\frac{x_na^n}{A_n},\\quad\\frac{B_{n-1}}{B_n}=1-\\frac{x_nb^n}{B_n}
Detailed analysis

The decompositions are An=xnan+An−1A_n=x_na^n+A_{n-1} and Bn=xnbn+Bn−1B_n=x_nb^n+B_{n-1}. Multiply by xn>0x_n>0 and subtract from 11; the direction reverses and gives the target inequality.