Question about representation of numbers in different bases

[Arian.D]

I'm thinking of expanding the numerical value of a real number in different bases. I want you guys to ensure some things for me so I know I'm not wrong.

If X is the representation of a in the base B and Y is the representation of b in the same base(radix), can we say that X+Y = a+b?

In particular I wonder if I can evalute the binary expansion of -4.125 in the following way:
First I write -4.125 = -5 + 0.875, then I find the binary expansion of -5, to do that first I see that 5 = (101)₂ and then I add the minus sign to it. Right?
Now I find the binary expansion of 0.875, which is easy because I have an algorithm that gives me the binary expansion of any positive real number less than one.

All I have said is easily generalized to other bases. But the question is, how should I expand negative numbers in a bases? How do I expand -4.625 in base-10 for example?
I'm confused.

 

[Muphrid]

I'm not sure why you would think you'd need to "expand" -4.625 in base 10. It's...already written in base 10. -4 - 6/10 - 2/100 - 5/1000. That's what decimals mean.

Similarly, you could take -4.125 and just recursively divide it by 2, keeping track of how many times you've done so, until you get a number smaller in magnitude than 1. Take that result, shunt it into your algorithm that finds the binary expansion of any number less than 1, tack on the minus sign if needed and a 0 on the end for each time you had to divide by 2. This repeated dividing by 2 just moved the "decimal point" (but remember, it's still binary) of the number until you got something smaller than 1.

 

[haruspex]

If X is the representation of a in the base B and Y is the representation of b in the same base(radix), can we say that X+Y = a+b?

Addition of numbers in a given representation is defined to be the mechanical procedure that gives the right result, so by definition, yes.

But the question is, how should I expand negative numbers in a bases? How do I expand -4.625 in base-10 for example?

Representation of negative numbers can be done with a minus sign in any base.
There are other possible representations. Computers mostly use "2's complement". If you wanted to get really confused you could adopt this in written form e.g. by defining that the leading digit is negative, the rest positive. -4.625 would be written 5.375 (-5 + 0.375). All positive numbers would be written with a leading zero (as in "2's complement"). Addition would proceed thus: 5.375 + 0.813 = 4.188.