Number base converter with step-by-step working

The first division by the base gives the remainder that is the lowest digit — the units. Each next division works on the quotient, which is the number shifted one place right, so its remainder is the next digit to the left. The division goes on until the quotient is 0, and the last remainder is the highest digit (some schools stop one step earlier and write the last quotient, smaller than the base, as that digit). That is why the answer starts from the bottom. Reading the remainders from the top gives the digits in reverse order — the most common mistake in these problems.

A fraction ends in base p only when its denominator, in lowest terms, has no prime factors other than those of p. 0.1 is 1/10, and 10 = 2·5: the factor 5 never cancels in base 2. When the fractional part is multiplied by 2 again and again, the value 0.2 comes back after four steps, and from there the digits 0011 repeat forever: 0.1 = 0.0(0011)₂. The same number is a finite 0.1 in base 10 and 0.1(9) in base 16.

Because 8 = 2³ and 16 = 2⁴: one octal digit holds exactly three bits, one hex digit exactly four. So the conversion needs no division — split the binary number into groups from the point, left for the integer part and right for the fraction, pad the outer groups with zeros and replace each group with its digit. Between 8 and 16 there is no such relation, so the route goes through binary.

Flipping all bits gives the ones' complement, and a number plus its ones' complement is a row of 1s, which is 2ʷ − 1. One more makes exactly 2ʷ, which does not fit into w bits and turns into zero. So the two's complement of x behaves like −x in ordinary addition, and the processor uses the same adder for subtraction. As a bonus there is only one zero, and the range goes one further on the negative side: from −128 to 127 on 8 bits.

A float keeps 24 significant bits (23 stored and a leading 1 that is implied). The binary 0.0(0011) is infinite, so it is cut to 24 bits and rounded to the nearest value that fits — here upward. The stored number is 0.100000001490116119384765625 exactly; that finite decimal exists because the stored value is a fraction with a power of two below. The IEEE 754 tab shows this value and the rounding error for any number.

Turn every term into a power of two: 4²⁰²⁰ = 2⁴⁰⁴⁰. A power 2ⁿ is 1 followed by n zeros. For 2²⁰¹⁷ − 15, write 15 as 16 − 1: 2²⁰¹⁷ − 16 = 2²⁰¹⁷ − 2⁴ is 2013 ones followed by 4 zeros, and adding 1 back makes the last zero a one. Together with the single 1 of 2⁴⁰⁴⁰ that gives 1 + 2013 + 1 = 2015 ones. The Exam problems mode shows each such step and checks the count on the exact value.