0.000 020 830 729 321 671 205 134 999 145 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 145(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 020 830 729 321 671 205 134 999 145(10) to 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: 0.000 020 830 729 321 671 205 134 999 145.

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) 0.000 020 830 729 321 671 205 134 999 145 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 29;
  • 2) 0.000 041 661 458 643 342 410 269 998 29 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 58;
  • 3) 0.000 083 322 917 286 684 820 539 996 58 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 16;
  • 4) 0.000 166 645 834 573 369 641 079 993 16 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 32;
  • 5) 0.000 333 291 669 146 739 282 159 986 32 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 64;
  • 6) 0.000 666 583 338 293 478 564 319 972 64 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 28;
  • 7) 0.001 333 166 676 586 957 128 639 945 28 × 2 = 0 + 0.002 666 333 353 173 914 257 279 890 56;
  • 8) 0.002 666 333 353 173 914 257 279 890 56 × 2 = 0 + 0.005 332 666 706 347 828 514 559 781 12;
  • 9) 0.005 332 666 706 347 828 514 559 781 12 × 2 = 0 + 0.010 665 333 412 695 657 029 119 562 24;
  • 10) 0.010 665 333 412 695 657 029 119 562 24 × 2 = 0 + 0.021 330 666 825 391 314 058 239 124 48;
  • 11) 0.021 330 666 825 391 314 058 239 124 48 × 2 = 0 + 0.042 661 333 650 782 628 116 478 248 96;
  • 12) 0.042 661 333 650 782 628 116 478 248 96 × 2 = 0 + 0.085 322 667 301 565 256 232 956 497 92;
  • 13) 0.085 322 667 301 565 256 232 956 497 92 × 2 = 0 + 0.170 645 334 603 130 512 465 912 995 84;
  • 14) 0.170 645 334 603 130 512 465 912 995 84 × 2 = 0 + 0.341 290 669 206 261 024 931 825 991 68;
  • 15) 0.341 290 669 206 261 024 931 825 991 68 × 2 = 0 + 0.682 581 338 412 522 049 863 651 983 36;
  • 16) 0.682 581 338 412 522 049 863 651 983 36 × 2 = 1 + 0.365 162 676 825 044 099 727 303 966 72;
  • 17) 0.365 162 676 825 044 099 727 303 966 72 × 2 = 0 + 0.730 325 353 650 088 199 454 607 933 44;
  • 18) 0.730 325 353 650 088 199 454 607 933 44 × 2 = 1 + 0.460 650 707 300 176 398 909 215 866 88;
  • 19) 0.460 650 707 300 176 398 909 215 866 88 × 2 = 0 + 0.921 301 414 600 352 797 818 431 733 76;
  • 20) 0.921 301 414 600 352 797 818 431 733 76 × 2 = 1 + 0.842 602 829 200 705 595 636 863 467 52;
  • 21) 0.842 602 829 200 705 595 636 863 467 52 × 2 = 1 + 0.685 205 658 401 411 191 273 726 935 04;
  • 22) 0.685 205 658 401 411 191 273 726 935 04 × 2 = 1 + 0.370 411 316 802 822 382 547 453 870 08;
  • 23) 0.370 411 316 802 822 382 547 453 870 08 × 2 = 0 + 0.740 822 633 605 644 765 094 907 740 16;
  • 24) 0.740 822 633 605 644 765 094 907 740 16 × 2 = 1 + 0.481 645 267 211 289 530 189 815 480 32;
  • 25) 0.481 645 267 211 289 530 189 815 480 32 × 2 = 0 + 0.963 290 534 422 579 060 379 630 960 64;
  • 26) 0.963 290 534 422 579 060 379 630 960 64 × 2 = 1 + 0.926 581 068 845 158 120 759 261 921 28;
  • 27) 0.926 581 068 845 158 120 759 261 921 28 × 2 = 1 + 0.853 162 137 690 316 241 518 523 842 56;
  • 28) 0.853 162 137 690 316 241 518 523 842 56 × 2 = 1 + 0.706 324 275 380 632 483 037 047 685 12;
  • 29) 0.706 324 275 380 632 483 037 047 685 12 × 2 = 1 + 0.412 648 550 761 264 966 074 095 370 24;
  • 30) 0.412 648 550 761 264 966 074 095 370 24 × 2 = 0 + 0.825 297 101 522 529 932 148 190 740 48;
  • 31) 0.825 297 101 522 529 932 148 190 740 48 × 2 = 1 + 0.650 594 203 045 059 864 296 381 480 96;
  • 32) 0.650 594 203 045 059 864 296 381 480 96 × 2 = 1 + 0.301 188 406 090 119 728 592 762 961 92;
  • 33) 0.301 188 406 090 119 728 592 762 961 92 × 2 = 0 + 0.602 376 812 180 239 457 185 525 923 84;
  • 34) 0.602 376 812 180 239 457 185 525 923 84 × 2 = 1 + 0.204 753 624 360 478 914 371 051 847 68;
  • 35) 0.204 753 624 360 478 914 371 051 847 68 × 2 = 0 + 0.409 507 248 720 957 828 742 103 695 36;
  • 36) 0.409 507 248 720 957 828 742 103 695 36 × 2 = 0 + 0.819 014 497 441 915 657 484 207 390 72;
  • 37) 0.819 014 497 441 915 657 484 207 390 72 × 2 = 1 + 0.638 028 994 883 831 314 968 414 781 44;
  • 38) 0.638 028 994 883 831 314 968 414 781 44 × 2 = 1 + 0.276 057 989 767 662 629 936 829 562 88;
  • 39) 0.276 057 989 767 662 629 936 829 562 88 × 2 = 0 + 0.552 115 979 535 325 259 873 659 125 76;
  • 40) 0.552 115 979 535 325 259 873 659 125 76 × 2 = 1 + 0.104 231 959 070 650 519 747 318 251 52;
  • 41) 0.104 231 959 070 650 519 747 318 251 52 × 2 = 0 + 0.208 463 918 141 301 039 494 636 503 04;
  • 42) 0.208 463 918 141 301 039 494 636 503 04 × 2 = 0 + 0.416 927 836 282 602 078 989 273 006 08;
  • 43) 0.416 927 836 282 602 078 989 273 006 08 × 2 = 0 + 0.833 855 672 565 204 157 978 546 012 16;
  • 44) 0.833 855 672 565 204 157 978 546 012 16 × 2 = 1 + 0.667 711 345 130 408 315 957 092 024 32;
  • 45) 0.667 711 345 130 408 315 957 092 024 32 × 2 = 1 + 0.335 422 690 260 816 631 914 184 048 64;
  • 46) 0.335 422 690 260 816 631 914 184 048 64 × 2 = 0 + 0.670 845 380 521 633 263 828 368 097 28;
  • 47) 0.670 845 380 521 633 263 828 368 097 28 × 2 = 1 + 0.341 690 761 043 266 527 656 736 194 56;
  • 48) 0.341 690 761 043 266 527 656 736 194 56 × 2 = 0 + 0.683 381 522 086 533 055 313 472 389 12;
  • 49) 0.683 381 522 086 533 055 313 472 389 12 × 2 = 1 + 0.366 763 044 173 066 110 626 944 778 24;
  • 50) 0.366 763 044 173 066 110 626 944 778 24 × 2 = 0 + 0.733 526 088 346 132 221 253 889 556 48;
  • 51) 0.733 526 088 346 132 221 253 889 556 48 × 2 = 1 + 0.467 052 176 692 264 442 507 779 112 96;
  • 52) 0.467 052 176 692 264 442 507 779 112 96 × 2 = 0 + 0.934 104 353 384 528 885 015 558 225 92;
  • 53) 0.934 104 353 384 528 885 015 558 225 92 × 2 = 1 + 0.868 208 706 769 057 770 031 116 451 84;
  • 54) 0.868 208 706 769 057 770 031 116 451 84 × 2 = 1 + 0.736 417 413 538 115 540 062 232 903 68;
  • 55) 0.736 417 413 538 115 540 062 232 903 68 × 2 = 1 + 0.472 834 827 076 231 080 124 465 807 36;
  • 56) 0.472 834 827 076 231 080 124 465 807 36 × 2 = 0 + 0.945 669 654 152 462 160 248 931 614 72;
  • 57) 0.945 669 654 152 462 160 248 931 614 72 × 2 = 1 + 0.891 339 308 304 924 320 497 863 229 44;
  • 58) 0.891 339 308 304 924 320 497 863 229 44 × 2 = 1 + 0.782 678 616 609 848 640 995 726 458 88;
  • 59) 0.782 678 616 609 848 640 995 726 458 88 × 2 = 1 + 0.565 357 233 219 697 281 991 452 917 76;
  • 60) 0.565 357 233 219 697 281 991 452 917 76 × 2 = 1 + 0.130 714 466 439 394 563 982 905 835 52;
  • 61) 0.130 714 466 439 394 563 982 905 835 52 × 2 = 0 + 0.261 428 932 878 789 127 965 811 671 04;
  • 62) 0.261 428 932 878 789 127 965 811 671 04 × 2 = 0 + 0.522 857 865 757 578 255 931 623 342 08;
  • 63) 0.522 857 865 757 578 255 931 623 342 08 × 2 = 1 + 0.045 715 731 515 156 511 863 246 684 16;
  • 64) 0.045 715 731 515 156 511 863 246 684 16 × 2 = 0 + 0.091 431 463 030 313 023 726 493 368 32;
  • 65) 0.091 431 463 030 313 023 726 493 368 32 × 2 = 0 + 0.182 862 926 060 626 047 452 986 736 64;
  • 66) 0.182 862 926 060 626 047 452 986 736 64 × 2 = 0 + 0.365 725 852 121 252 094 905 973 473 28;
  • 67) 0.365 725 852 121 252 094 905 973 473 28 × 2 = 0 + 0.731 451 704 242 504 189 811 946 946 56;
  • 68) 0.731 451 704 242 504 189 811 946 946 56 × 2 = 1 + 0.462 903 408 485 008 379 623 893 893 12;

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 - the converted number we get in the end will be just a very good approximation of the initial one).


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:


0.000 020 830 729 321 671 205 134 999 145(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 145(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 145(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 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) =

1007(10) =

011 1110 1111(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. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


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 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 145 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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