Confusion about normalised floating point number systems

Question

I was assigned the following question:

List all numbers that can be represented exactly in a normalised floating- point number system with base 10, two digits in the fraction, and an exponent 0 ≤ e ≤ 2. How many are there in total (including zero)?

The bit in particular that confuses me is the 'base 10' aspect. After doing some research I found that a normalised floating-point number system is supposed to have a '1' before the decimal point and the rest afterwards but that only works for binary. Also, if I am supposed to list the numbers and this is in base 10, it would seem like a really large amount to list.

So it seems I need to answer the question in binary, but I am really confused about why 'base 10' is being mentioned.

Thanks

Answer 1

Floating-point numbers in modern computers are almost exclusively one of a few types of IEEE-defined floating-point types that differ only in how many bits are used for each number.

But the concept of a floating-point representation is not limited to what the vast majority of actual applications are using. You're being asked a theoretical question about another kind of floating-point representation that could be used.

The "leading digit is always $1$" trick only works in binary, and therefore you cannot use it in base ten. There is a bit of ambiguity in the question regarding whether "two digits in the fraction" means they want numbers like $2.34 \times 10^1$ or like $0.34 \times 10^1.$ In the absence of any other indication, if I had to guess, I would guess they mean $0.34 \times 10^1$ just because the other way is ten times as many numbers to write. But if you can ask the person who assigned this question, I would recommend you do.

Even if we only use two fractional digits (like $0.34 \times 10^1$), there are still a few hundred possible numbers. If you really literally must list them all, perhaps you can get them printed out by computer. A simple spreadsheet can do the job. But I suspect that only only need to list enough of them in a pattern so that it is clear how you would fill in the missing numbers (and how many there would be), much the way that if I write $1,2,3,4,\ldots,59,60$ you can see I have in mind the first sixty positive integers. Again, that is something to ask the person who assigned this if you can.