Numder of ones in a binary number

[Jesinth]

Is there any way to find out the number of ones in the binary representation of
(3 x 512) + (7 x 64) + (5 x 8 ) + 3 without actual calculation of decimal equicalent and converting into binary.

(The resultant decimal is 2027 with binary equivalent 11111101011 which has 9 ones.)

How to get this value 9 from the expression above?

 

[Sckoarn]

Jesinth,
Take the result and divide it by powers of 2 starting with the highest power that produces biggest divisible number. Then decrease the power each time you divide, the result should be a 1 or a 0. In essence you are converting the number to binary, but you can just count the number of 1's as you loop though the 2^x.

Hope that helps some.
Sckoarn

 

[DXNewcastle]

Yes, there IS a way for you to do that, WITHOUT calculating the result.

The clue is in the fact that 512, 64 and 8 can each be represented by a single bit.
However, we don't need to worry about the actual bit-position these occupy in the result, because they are going to be multiplied (in this case, by 3, 7 and 5).
So all you need to know is how many bits each of those multipliers require in their binary representation.
3 requires 2 "1" bits
7 requires 3 "1" bits
5 requires 2 "1" bits (that's 7 bits so far)
Then there's the last term, 3, which in itself requires 2 "1" bits in its representation. So that's a total of 9 "1" bits.

I should warn you that this is probably some text book example you've presented to us, and is simpler than most real-life situations. Even if you were always presented with power-of-two factors (in this case 512, 64 and 8 ) which conveniently are represented by a single bit in binary notation, there is the risk that the multiplier might increase the value to a magnitude which reaches or exceeds one of the other terms. In the example, 7 x 64 = 448 and 448 is less than 512 so they don't conflict.
But try using this technique if the middle term was (9 x 64). Or if the last term was + 77. These should illustrate the hazards.

 

[lostinxlation]

The simplest solution in this case is
(3 x 512) + (7 x 64) + (5 x 8 ) + 3
= (3 x 8^3) + (7 x 8^2) + (5 x 8 ) + 3
= 3753 (octal)

3 in octal has 2 hot bits, 7 has 3 hot bits, and so on.
Total number of hot bits is 9.