64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.007 85 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.007 85(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.


0(10) =


0(2)


3. Convert to binary (base 2) the fractional part: 0.007 85.

Multiply it repeatedly by 2.


Keep track of each integer part of the results.


Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.007 85 × 2 = 0 + 0.015 7;
  • 2) 0.015 7 × 2 = 0 + 0.031 4;
  • 3) 0.031 4 × 2 = 0 + 0.062 8;
  • 4) 0.062 8 × 2 = 0 + 0.125 6;
  • 5) 0.125 6 × 2 = 0 + 0.251 2;
  • 6) 0.251 2 × 2 = 0 + 0.502 4;
  • 7) 0.502 4 × 2 = 1 + 0.004 8;
  • 8) 0.004 8 × 2 = 0 + 0.009 6;
  • 9) 0.009 6 × 2 = 0 + 0.019 2;
  • 10) 0.019 2 × 2 = 0 + 0.038 4;
  • 11) 0.038 4 × 2 = 0 + 0.076 8;
  • 12) 0.076 8 × 2 = 0 + 0.153 6;
  • 13) 0.153 6 × 2 = 0 + 0.307 2;
  • 14) 0.307 2 × 2 = 0 + 0.614 4;
  • 15) 0.614 4 × 2 = 1 + 0.228 8;
  • 16) 0.228 8 × 2 = 0 + 0.457 6;
  • 17) 0.457 6 × 2 = 0 + 0.915 2;
  • 18) 0.915 2 × 2 = 1 + 0.830 4;
  • 19) 0.830 4 × 2 = 1 + 0.660 8;
  • 20) 0.660 8 × 2 = 1 + 0.321 6;
  • 21) 0.321 6 × 2 = 0 + 0.643 2;
  • 22) 0.643 2 × 2 = 1 + 0.286 4;
  • 23) 0.286 4 × 2 = 0 + 0.572 8;
  • 24) 0.572 8 × 2 = 1 + 0.145 6;
  • 25) 0.145 6 × 2 = 0 + 0.291 2;
  • 26) 0.291 2 × 2 = 0 + 0.582 4;
  • 27) 0.582 4 × 2 = 1 + 0.164 8;
  • 28) 0.164 8 × 2 = 0 + 0.329 6;
  • 29) 0.329 6 × 2 = 0 + 0.659 2;
  • 30) 0.659 2 × 2 = 1 + 0.318 4;
  • 31) 0.318 4 × 2 = 0 + 0.636 8;
  • 32) 0.636 8 × 2 = 1 + 0.273 6;
  • 33) 0.273 6 × 2 = 0 + 0.547 2;
  • 34) 0.547 2 × 2 = 1 + 0.094 4;
  • 35) 0.094 4 × 2 = 0 + 0.188 8;
  • 36) 0.188 8 × 2 = 0 + 0.377 6;
  • 37) 0.377 6 × 2 = 0 + 0.755 2;
  • 38) 0.755 2 × 2 = 1 + 0.510 4;
  • 39) 0.510 4 × 2 = 1 + 0.020 8;
  • 40) 0.020 8 × 2 = 0 + 0.041 6;
  • 41) 0.041 6 × 2 = 0 + 0.083 2;
  • 42) 0.083 2 × 2 = 0 + 0.166 4;
  • 43) 0.166 4 × 2 = 0 + 0.332 8;
  • 44) 0.332 8 × 2 = 0 + 0.665 6;
  • 45) 0.665 6 × 2 = 1 + 0.331 2;
  • 46) 0.331 2 × 2 = 0 + 0.662 4;
  • 47) 0.662 4 × 2 = 1 + 0.324 8;
  • 48) 0.324 8 × 2 = 0 + 0.649 6;
  • 49) 0.649 6 × 2 = 1 + 0.299 2;
  • 50) 0.299 2 × 2 = 0 + 0.598 4;
  • 51) 0.598 4 × 2 = 1 + 0.196 8;
  • 52) 0.196 8 × 2 = 0 + 0.393 6;
  • 53) 0.393 6 × 2 = 0 + 0.787 2;
  • 54) 0.787 2 × 2 = 1 + 0.574 4;
  • 55) 0.574 4 × 2 = 1 + 0.148 8;
  • 56) 0.148 8 × 2 = 0 + 0.297 6;
  • 57) 0.297 6 × 2 = 0 + 0.595 2;
  • 58) 0.595 2 × 2 = 1 + 0.190 4;
  • 59) 0.190 4 × 2 = 0 + 0.380 8;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (losing precision...)


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.007 85(10) =


0.0000 0010 0000 0010 0111 0101 0010 0101 0100 0110 0000 1010 1010 0110 010(2)


5. Positive number before normalization:

0.007 85(10) =


0.0000 0010 0000 0010 0111 0101 0010 0101 0100 0110 0000 1010 1010 0110 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.007 85(10) =


0.0000 0010 0000 0010 0111 0101 0010 0101 0100 0110 0000 1010 1010 0110 010(2) =


0.0000 0010 0000 0010 0111 0101 0010 0101 0100 0110 0000 1010 1010 0110 010(2) × 20 =


1.0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010(2) × 2-7


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)


Exponent (unadjusted): -7


Mantissa (not normalized):
1.0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


Exponent (unadjusted) + 2(11-1) - 1 =


-7 + 2(11-1) - 1 =


(-7 + 1 023)(10) =


1 016(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1016(10) =


011 1111 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.


b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =


1. 0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010 =


0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)


Exponent (11 bits) =
011 1111 1000


Mantissa (52 bits) =
0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010


The base ten decimal number 0.007 85 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1111 1000 - 0000 0001 0011 1010 1001 0010 1010 0011 0000 0101 0101 0011 0010

The latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation