64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666(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: 0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666.

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) 0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 × 2 = 1 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332;
  • 2) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 664;
  • 3) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 664 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 328;
  • 4) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 328 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 656;
  • 5) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 656 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 312;
  • 6) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 312 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 624;
  • 7) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 624 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 248;
  • 8) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 248 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 496;
  • 9) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 496 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 992;
  • 10) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 992 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 984;
  • 11) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 984 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 968;
  • 12) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 968 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 936;
  • 13) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 936 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 872;
  • 14) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 872 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 744;
  • 15) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 744 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 488;
  • 16) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 488 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 622 976;
  • 17) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 622 976 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 245 952;
  • 18) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 245 952 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 491 904;
  • 19) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 491 904 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 983 808;
  • 20) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 983 808 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 967 616;
  • 21) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 967 616 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 935 232;
  • 22) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 935 232 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 870 464;
  • 23) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 870 464 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 740 928;
  • 24) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 740 928 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 481 856;
  • 25) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 481 856 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 963 712;
  • 26) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 963 712 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 621 927 424;
  • 27) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 621 927 424 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 243 854 848;
  • 28) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 243 854 848 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 487 709 696;
  • 29) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 487 709 696 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 975 419 392;
  • 30) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 975 419 392 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 950 838 784;
  • 31) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 950 838 784 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 901 677 568;
  • 32) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 901 677 568 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 803 355 136;
  • 33) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 803 355 136 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 606 710 272;
  • 34) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 606 710 272 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 213 420 544;
  • 35) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 213 420 544 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 426 841 088;
  • 36) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 426 841 088 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 620 853 682 176;
  • 37) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 620 853 682 176 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 241 707 364 352;
  • 38) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 241 707 364 352 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 483 414 728 704;
  • 39) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 483 414 728 704 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 966 829 457 408;
  • 40) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 966 829 457 408 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 933 658 914 816;
  • 41) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 933 658 914 816 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 867 317 829 632;
  • 42) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 867 317 829 632 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 734 635 659 264;
  • 43) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 734 635 659 264 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 469 271 318 528;
  • 44) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 469 271 318 528 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 938 542 637 056;
  • 45) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 938 542 637 056 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 877 085 274 112;
  • 46) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 877 085 274 112 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 619 754 170 548 224;
  • 47) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 619 754 170 548 224 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 239 508 341 096 448;
  • 48) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 239 508 341 096 448 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 479 016 682 192 896;
  • 49) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 479 016 682 192 896 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 958 033 364 385 792;
  • 50) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 958 033 364 385 792 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 916 066 728 771 584;
  • 51) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 916 066 728 771 584 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 832 133 457 543 168;
  • 52) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 832 133 457 543 168 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 664 266 915 086 336;
  • 53) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 664 266 915 086 336 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 328 533 830 172 672;

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:


0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666(10) =


0.1001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0(2)


5. Positive number before normalization:

0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666(10) =


0.1001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666(10) =


0.1001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0(2) =


0.1001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0(2) × 20 =


1.0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010(2) × 2-1


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): -1


Mantissa (not normalized):
1.0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


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


(-1 + 1 023)(10) =


1 022(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;
  • 1 022 ÷ 2 = 511 + 0;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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) =


1022(10) =


011 1111 1110(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. 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 =


0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 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 1110


Mantissa (52 bits) =
0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010


The base ten decimal number 0.566 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1111 1110 - 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. 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: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100