Hey there! Great question—let's unpack what you're looking for and how to frame this mathematically.

First, let's clarify your requirements to make sure we're on the same page:

• For a finite decimal like x = 45, you want f(x) = 0.54 (reverse the digits, shifting the decimal point so the original integer part becomes the new decimal part, and vice versa)
• For x = 376.59, f(x) = 95.673 (same rule: reverse all digits including the decimal's position)
• For irrational numbers (like π or e, which have infinite non-repeating decimal expansions), you want f(x) = ∞ since you can't fully reverse an infinite non-repeating sequence.

Breaking down the finite decimal case #

Let's start with finite decimals, since these are well-behaved. Any positive finite decimal can be written as:

x = N / 10^k
where N is a positive integer (the digits of x without the decimal point) and k is a non-negative integer (the number of digits after the decimal point in x).

For example:

• 45 = 45 / 10^0 (so N=45, k=0)
• 376.59 = 37659 / 10^2 (so N=37659, k=2)

Next, we need a way to reverse the digits of N. Let's call this reversed integer rev(N):

• rev(45) = 54
• rev(37659) = 95673

Mathematically, we can define rev(N) using a summation. If N is an m-digit number (so m = floor(log₁₀N) + 1), then each digit d_i of N (from left to right) can be extracted with d_i = floor(N / 10^{m-1-i}) mod 10. Then:

rev(N) = Σ_{i=0}^{m-1} d_i * 10^i

Now, to get the final reversed number matching your examples:
We need to divide rev(N) by 10^{m - k}. Let's test this with your examples:

• For x=45: m=2 (digits in N=45), k=0 → 10^{2-0}=100 → 54 / 100 = 0.54 (perfect!)
• For x=376.59: m=5 (digits in N=37659), k=2 → 10^{5-2}=1000 → 95673 / 1000 = 95.673 (exactly what you wanted!)

Handling irrational numbers #

Irrational numbers have infinite non-repeating decimal expansions—you can never write them as N / 10^k (since that's a finite decimal, a type of rational number). Since you can't fully reverse an infinite non-repeating sequence, defining f(x) = ∞ for irrational x makes intuitive sense. We can formalize this by saying:

• If x is a rational number that can be written as N / 10^k (finite decimal), use the formula above.
• If x is irrational, f(x) = ∞.

A quick note on edge cases #

• What about x=0? Well, reversing 0 is just 0, so f(0)=0.
• Negative numbers: If you want to handle negatives, you can just apply the same rule to the absolute value and keep the negative sign, e.g., f(-45) = -0.54.

备注：内容来源于stack exchange，提问作者G Beck