# 0.999 = 1

1. Feb 18, 2009

### jobyts

... can we convert this equation to binary notation?

Also another one, would an irrational number be irrational in any number format?

2. Feb 18, 2009

### statdad

$$0.999 = 1$$

Use the standard algorithms to convert $$0.\overline{000}$$ to decimal form (do the same for $$1$$, but that isn't as exciting). The results will look different, but if two symbols represent the same quantity in one number system, the corresponding symbols represent the same quantity in any number system.

If you convert (say) $$\sqrt 2$$ to another base, you won't get a decimal form, since decimal refers to base 10 alone. You will get a representation, in that other base, of a number that we refer to as irrational in base 10.

3. Feb 18, 2009

### HallsofIvy

Staff Emeritus
In binary it would be $0.\overline{1111}= 1$ and, yes, it is true.

statdad, while incorrect, there just isn't any good term to replace "decimal fraction" in another base so I would cut jobytx some slack on that.

jobytx, the distinction between "rational" and "irrational" is a property of numbers not numerals and has nothing to do with whether it is represented in base 2 or base 10 or even Roman numerals (although I confess I don't know how one would represent a non-integer in Roman numerals!).

4. Feb 18, 2009

### statdad

HallsofIvy;
I agree (I think, unless you are saying my comment is incorrect) with you - language is awkward with this stuff. Here's my reason for the comment.
I'm currently teaching an applied course for folk majoring in computer areas (programming, mostly) and we discuss number systems for our CIS colleagues. They have (for their own reasons) specifically asked us not to refer to "decimal" when using other number systems, whatever the base: they prefer we refer to them as "non-integers", or "representations of decimals".
Think too long and hard like that and it seeps outside the classroom.

If it seems I was being harsh to the OP, I do apologize - that was not my intent.

5. Feb 18, 2009

### HallsofIvy

Staff Emeritus
Would I dare say that you are incorrect?

6. Feb 18, 2009

### statdad

It's happened many, many, times in my graduate career and in my careers since finishing my degrees. So, if my comment(s) above were in error, let me know. :D

I expect no lower level of honesty from you.

7. Feb 19, 2009

### HallsofIvy

Staff Emeritus
I agree with you completely. In fact, I dislike the term "decimal fraction" even when working in base 10.

Also, I just noticed that jobyts did not use the term "decimal" himself so you were just giving additional information, not criticizing, and my remark was off base!

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