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This is going to sound really elementary but I had this question about (this is a problem in Number Theory)

'find all four digit numbers such that, when written in decimal,...'

Now the question itself is not that bad but what does it mean precisely when a 4 digit number is written in decimal? Does it just mean it is not written in binary form? If that is not the case, is $1000$ a 4 digit number written in decimal? Is $9.001$ a $4$ digit number written in decimal? Also what would be the last digit of $9.001$ is? Would it be $9$ or $1$?

UnsinkableSam
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    I interpret the phrase "All four-digit decimal numbers" to mean the list of nine-thousand natural numbers $1000,1001,1002,\dots,9997,9998,9999$. In particular, no decimal point occurs, the numbers are interpreted as being base10, the digits occurring are all members of the base10 list of digits $0,1,2,\dots,9$, and the leading digit is nonzero. $0027$ is considered a two-digit number, $27$. – JMoravitz Apr 30 '20 at 16:51
  • To me, it sounds like the four digit numbers are not in decimal at the start. Why else would one then 'write them in decimal'? The clip you posted is just too short to know what's going on. Not enough context. –  Apr 30 '20 at 17:08

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In everyday life, numbers are strongly associated with their decimal representations – so much so that most people who don’t usually deal with other representations of numbers identify the two and are somewhat surprised when they learn that decimal representations are just arbitrary representations and there are other, equally valid representations and numbers are an abstraction independent of any of these representations.

There are many ways to represent numbers. For instance, $\sqrt2$ is a number, which is written neither in decimal form, nor in binary form, nor in any other digit-based representation; the number is specified by stating its square, $2$ (and the convention that by $\sqrt2$ we mean the positive number whose square is $2$, not the negative one).

The author of the problem you quote was apparently also afflicted by the general confusion between numbers and their decimal representations: “four-digit numbers such that, when written in decimal” makes no sense, because the property of having four digits is a property of the (presumably decimal) representation, not of the number. Of course we often say “four-digit number” as a short form for “number with a four-digit decimal representation” – and rightly so, since otherwise we’d be using a lot more words than necessary. But in this sentence, an explicit distinction is being made between a number and its decimal form, and then it doesn’t make sense to apply the property “four-digit” to the number instead of the representation.

But to answer your question: No, being written in decimal form doesn’t just mean not being written in binary form; there are infinitely many other forms a number can be written in, including e.g. ternary, octal, hexadecimal. Writing it in decimal form means writing it out with digits from $0$ to $9$ like most people usually do. $1000$ has four digits when written in decimal form. One usually doesn’t refer to numbers like $9.001$ as four-digit numbers, perhaps because once you start writing digits after the decimal point, you could add any number of zeros, e.g. $9.0010$ is the same number, so for non-integers it’s not so obvious how many digits one should count. Usually, “four-digit numbers” refers to four-digit integers. But that’s just a linguistic convention; $9.001$ certainly does have four digits in its standard decimal representation without unnecessary zeros. The last digit of that representation is indeed $1$.

joriki
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