What is the decimal conversion of the binary number 1111111111?

Answer 1

1111111111b = 1023d

When converting a binary number in which every digit is a one, there is an easy trick to doing it: take the number of digits, in this case 10, raise the base to that power 210 = 1024 , then subtract one. 210 - 1 = 1023.

If the reason for this isn't clear, it may be easier to think of it in decimal. Remember that 1 is the highest valued digit in binary. Similarly, 9 is the highest digit in decimal. So in decimal, the number 99999 can also be looked at as 105 - 1 = 100000 - 1 = 99999