64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.000 000 127 591 192 722 320 3 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.000 000 127 591 192 722 320 3(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: 0.000 000 127 591 192 722 320 3.

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) 0.000 000 127 591 192 722 320 3 × 2 = 0 + 0.000 000 255 182 385 444 640 6;
  • 2) 0.000 000 255 182 385 444 640 6 × 2 = 0 + 0.000 000 510 364 770 889 281 2;
  • 3) 0.000 000 510 364 770 889 281 2 × 2 = 0 + 0.000 001 020 729 541 778 562 4;
  • 4) 0.000 001 020 729 541 778 562 4 × 2 = 0 + 0.000 002 041 459 083 557 124 8;
  • 5) 0.000 002 041 459 083 557 124 8 × 2 = 0 + 0.000 004 082 918 167 114 249 6;
  • 6) 0.000 004 082 918 167 114 249 6 × 2 = 0 + 0.000 008 165 836 334 228 499 2;
  • 7) 0.000 008 165 836 334 228 499 2 × 2 = 0 + 0.000 016 331 672 668 456 998 4;
  • 8) 0.000 016 331 672 668 456 998 4 × 2 = 0 + 0.000 032 663 345 336 913 996 8;
  • 9) 0.000 032 663 345 336 913 996 8 × 2 = 0 + 0.000 065 326 690 673 827 993 6;
  • 10) 0.000 065 326 690 673 827 993 6 × 2 = 0 + 0.000 130 653 381 347 655 987 2;
  • 11) 0.000 130 653 381 347 655 987 2 × 2 = 0 + 0.000 261 306 762 695 311 974 4;
  • 12) 0.000 261 306 762 695 311 974 4 × 2 = 0 + 0.000 522 613 525 390 623 948 8;
  • 13) 0.000 522 613 525 390 623 948 8 × 2 = 0 + 0.001 045 227 050 781 247 897 6;
  • 14) 0.001 045 227 050 781 247 897 6 × 2 = 0 + 0.002 090 454 101 562 495 795 2;
  • 15) 0.002 090 454 101 562 495 795 2 × 2 = 0 + 0.004 180 908 203 124 991 590 4;
  • 16) 0.004 180 908 203 124 991 590 4 × 2 = 0 + 0.008 361 816 406 249 983 180 8;
  • 17) 0.008 361 816 406 249 983 180 8 × 2 = 0 + 0.016 723 632 812 499 966 361 6;
  • 18) 0.016 723 632 812 499 966 361 6 × 2 = 0 + 0.033 447 265 624 999 932 723 2;
  • 19) 0.033 447 265 624 999 932 723 2 × 2 = 0 + 0.066 894 531 249 999 865 446 4;
  • 20) 0.066 894 531 249 999 865 446 4 × 2 = 0 + 0.133 789 062 499 999 730 892 8;
  • 21) 0.133 789 062 499 999 730 892 8 × 2 = 0 + 0.267 578 124 999 999 461 785 6;
  • 22) 0.267 578 124 999 999 461 785 6 × 2 = 0 + 0.535 156 249 999 998 923 571 2;
  • 23) 0.535 156 249 999 998 923 571 2 × 2 = 1 + 0.070 312 499 999 997 847 142 4;
  • 24) 0.070 312 499 999 997 847 142 4 × 2 = 0 + 0.140 624 999 999 995 694 284 8;
  • 25) 0.140 624 999 999 995 694 284 8 × 2 = 0 + 0.281 249 999 999 991 388 569 6;
  • 26) 0.281 249 999 999 991 388 569 6 × 2 = 0 + 0.562 499 999 999 982 777 139 2;
  • 27) 0.562 499 999 999 982 777 139 2 × 2 = 1 + 0.124 999 999 999 965 554 278 4;
  • 28) 0.124 999 999 999 965 554 278 4 × 2 = 0 + 0.249 999 999 999 931 108 556 8;
  • 29) 0.249 999 999 999 931 108 556 8 × 2 = 0 + 0.499 999 999 999 862 217 113 6;
  • 30) 0.499 999 999 999 862 217 113 6 × 2 = 0 + 0.999 999 999 999 724 434 227 2;
  • 31) 0.999 999 999 999 724 434 227 2 × 2 = 1 + 0.999 999 999 999 448 868 454 4;
  • 32) 0.999 999 999 999 448 868 454 4 × 2 = 1 + 0.999 999 999 998 897 736 908 8;
  • 33) 0.999 999 999 998 897 736 908 8 × 2 = 1 + 0.999 999 999 997 795 473 817 6;
  • 34) 0.999 999 999 997 795 473 817 6 × 2 = 1 + 0.999 999 999 995 590 947 635 2;
  • 35) 0.999 999 999 995 590 947 635 2 × 2 = 1 + 0.999 999 999 991 181 895 270 4;
  • 36) 0.999 999 999 991 181 895 270 4 × 2 = 1 + 0.999 999 999 982 363 790 540 8;
  • 37) 0.999 999 999 982 363 790 540 8 × 2 = 1 + 0.999 999 999 964 727 581 081 6;
  • 38) 0.999 999 999 964 727 581 081 6 × 2 = 1 + 0.999 999 999 929 455 162 163 2;
  • 39) 0.999 999 999 929 455 162 163 2 × 2 = 1 + 0.999 999 999 858 910 324 326 4;
  • 40) 0.999 999 999 858 910 324 326 4 × 2 = 1 + 0.999 999 999 717 820 648 652 8;
  • 41) 0.999 999 999 717 820 648 652 8 × 2 = 1 + 0.999 999 999 435 641 297 305 6;
  • 42) 0.999 999 999 435 641 297 305 6 × 2 = 1 + 0.999 999 998 871 282 594 611 2;
  • 43) 0.999 999 998 871 282 594 611 2 × 2 = 1 + 0.999 999 997 742 565 189 222 4;
  • 44) 0.999 999 997 742 565 189 222 4 × 2 = 1 + 0.999 999 995 485 130 378 444 8;
  • 45) 0.999 999 995 485 130 378 444 8 × 2 = 1 + 0.999 999 990 970 260 756 889 6;
  • 46) 0.999 999 990 970 260 756 889 6 × 2 = 1 + 0.999 999 981 940 521 513 779 2;
  • 47) 0.999 999 981 940 521 513 779 2 × 2 = 1 + 0.999 999 963 881 043 027 558 4;
  • 48) 0.999 999 963 881 043 027 558 4 × 2 = 1 + 0.999 999 927 762 086 055 116 8;
  • 49) 0.999 999 927 762 086 055 116 8 × 2 = 1 + 0.999 999 855 524 172 110 233 6;
  • 50) 0.999 999 855 524 172 110 233 6 × 2 = 1 + 0.999 999 711 048 344 220 467 2;
  • 51) 0.999 999 711 048 344 220 467 2 × 2 = 1 + 0.999 999 422 096 688 440 934 4;
  • 52) 0.999 999 422 096 688 440 934 4 × 2 = 1 + 0.999 998 844 193 376 881 868 8;
  • 53) 0.999 998 844 193 376 881 868 8 × 2 = 1 + 0.999 997 688 386 753 763 737 6;
  • 54) 0.999 997 688 386 753 763 737 6 × 2 = 1 + 0.999 995 376 773 507 527 475 2;
  • 55) 0.999 995 376 773 507 527 475 2 × 2 = 1 + 0.999 990 753 547 015 054 950 4;
  • 56) 0.999 990 753 547 015 054 950 4 × 2 = 1 + 0.999 981 507 094 030 109 900 8;
  • 57) 0.999 981 507 094 030 109 900 8 × 2 = 1 + 0.999 963 014 188 060 219 801 6;
  • 58) 0.999 963 014 188 060 219 801 6 × 2 = 1 + 0.999 926 028 376 120 439 603 2;
  • 59) 0.999 926 028 376 120 439 603 2 × 2 = 1 + 0.999 852 056 752 240 879 206 4;
  • 60) 0.999 852 056 752 240 879 206 4 × 2 = 1 + 0.999 704 113 504 481 758 412 8;
  • 61) 0.999 704 113 504 481 758 412 8 × 2 = 1 + 0.999 408 227 008 963 516 825 6;
  • 62) 0.999 408 227 008 963 516 825 6 × 2 = 1 + 0.998 816 454 017 927 033 651 2;
  • 63) 0.998 816 454 017 927 033 651 2 × 2 = 1 + 0.997 632 908 035 854 067 302 4;
  • 64) 0.997 632 908 035 854 067 302 4 × 2 = 1 + 0.995 265 816 071 708 134 604 8;
  • 65) 0.995 265 816 071 708 134 604 8 × 2 = 1 + 0.990 531 632 143 416 269 209 6;
  • 66) 0.990 531 632 143 416 269 209 6 × 2 = 1 + 0.981 063 264 286 832 538 419 2;
  • 67) 0.981 063 264 286 832 538 419 2 × 2 = 1 + 0.962 126 528 573 665 076 838 4;
  • 68) 0.962 126 528 573 665 076 838 4 × 2 = 1 + 0.924 253 057 147 330 153 676 8;
  • 69) 0.924 253 057 147 330 153 676 8 × 2 = 1 + 0.848 506 114 294 660 307 353 6;
  • 70) 0.848 506 114 294 660 307 353 6 × 2 = 1 + 0.697 012 228 589 320 614 707 2;
  • 71) 0.697 012 228 589 320 614 707 2 × 2 = 1 + 0.394 024 457 178 641 229 414 4;
  • 72) 0.394 024 457 178 641 229 414 4 × 2 = 0 + 0.788 048 914 357 282 458 828 8;
  • 73) 0.788 048 914 357 282 458 828 8 × 2 = 1 + 0.576 097 828 714 564 917 657 6;
  • 74) 0.576 097 828 714 564 917 657 6 × 2 = 1 + 0.152 195 657 429 129 835 315 2;
  • 75) 0.152 195 657 429 129 835 315 2 × 2 = 0 + 0.304 391 314 858 259 670 630 4;

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:


0.000 000 127 591 192 722 320 3(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 110(2)


5. Positive number before normalization:

0.000 000 127 591 192 722 320 3(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 110(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 23 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 127 591 192 722 320 3(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 110(2) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 110(2) × 20 =

1.0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110(2) × 2-23


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): -23

Mantissa (not normalized):
1.0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-23 + 2(11-1) - 1 =

(-23 + 1 023)(10) =

1 000(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;
  • 1 000 ÷ 2 = 500 + 0;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 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) =

1000(10) =

011 1110 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. 0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110 =

0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110


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 1110 1000

Mantissa (52 bits) =
0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110


The base ten decimal number 0.000 000 127 591 192 722 320 3 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1110 1000 - 0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110

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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