Math Help - Conversion of bases

1. Conversion of bases

So in base 10 we can represent 200 in base two as follows.

200/2 = 100 R 0
100/2 = 50 R 0
50/2 = 25 R0
25/2 = 12 R1
12/2 = 6 R0
6/2 = 3 R0
3/2 = 1 R1
1/2 = 0 R1

So 200 in base 2 = 11001000.

I understand how that works. But to get back we do this.

Base 10 digits 0 1 2 3 4 5 6 7
Base 2 digits 0 0 0 1 0 0 1 1

where there are ones 2^3 + 2^6 + 2^7 = 200. Could someone explain how that works?

Thanks.

2. Originally Posted by mark090480
So in base 10 we can represent 200 in base two as follows.

200/2 = 100 R 0
100/2 = 50 R 0
50/2 = 25 R0
25/2 = 12 R1
12/2 = 6 R0
6/2 = 3 R0
3/2 = 1 R1
1/2 = 0 R1

So 200 in base 2 = 11001000.

I understand how that works. But to get back we do this.

Base 10 digits 0 1 2 3 4 5 6 7
Base 2 digits 0 0 0 1 0 0 1 1

where there are ones 2^3 + 2^6 + 2^7 = 200. Could someone explain how that works?

Thanks.
That is precisely the meaning of "writing in basis x": you express numbers as a sum of powers of x multiplied by integers between 0 and x.

For example, $306= 3\cdot 10^2+0\cdot 10^1+6\cdot 10^0=3\cdot 10^2+6$

That is why, thus, in base 2 you have to sum up the powers of 2 corresponding to the 1's in the expression of the number in binary.

Tonio

3. But why does doing the power of 2^x, where x is a remainder give the exact value?