0.000 020 830 729 321 671 205 134 999 154 509 660 617 452 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 452(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 452(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 452.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 452 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 904;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 904 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 808;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 808 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 939 616;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 939 616 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 879 232;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 879 232 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 758 464;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 758 464 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 516 928;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 516 928 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 033 856;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 033 856 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 067 712;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 067 712 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 135 424;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 135 424 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 270 848;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 270 848 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 541 696;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 541 696 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 889 083 392;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 889 083 392 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 778 166 784;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 778 166 784 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 556 333 568;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 556 333 568 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 112 667 136;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 112 667 136 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 225 334 272;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 225 334 272 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 450 668 544;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 450 668 544 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 901 337 088;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 901 337 088 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 802 674 176;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 802 674 176 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 605 348 352;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 605 348 352 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 783 210 696 704;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 783 210 696 704 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 566 421 393 408;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 566 421 393 408 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 132 842 786 816;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 132 842 786 816 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 265 685 573 632;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 265 685 573 632 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 531 371 147 264;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 531 371 147 264 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 062 742 294 528;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 062 742 294 528 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 125 484 589 056;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 125 484 589 056 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 250 969 178 112;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 250 969 178 112 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 501 938 356 224;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 501 938 356 224 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 817 003 876 712 448;
  • 31) 0.825 297 101 522 529 932 158 401 660 817 003 876 712 448 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 634 007 753 424 896;
  • 32) 0.650 594 203 045 059 864 316 803 321 634 007 753 424 896 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 268 015 506 849 792;
  • 33) 0.301 188 406 090 119 728 633 606 643 268 015 506 849 792 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 536 031 013 699 584;
  • 34) 0.602 376 812 180 239 457 267 213 286 536 031 013 699 584 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 072 062 027 399 168;
  • 35) 0.204 753 624 360 478 914 534 426 573 072 062 027 399 168 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 144 124 054 798 336;
  • 36) 0.409 507 248 720 957 829 068 853 146 144 124 054 798 336 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 288 248 109 596 672;
  • 37) 0.819 014 497 441 915 658 137 706 292 288 248 109 596 672 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 576 496 219 193 344;
  • 38) 0.638 028 994 883 831 316 275 412 584 576 496 219 193 344 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 152 992 438 386 688;
  • 39) 0.276 057 989 767 662 632 550 825 169 152 992 438 386 688 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 305 984 876 773 376;
  • 40) 0.552 115 979 535 325 265 101 650 338 305 984 876 773 376 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 611 969 753 546 752;
  • 41) 0.104 231 959 070 650 530 203 300 676 611 969 753 546 752 × 2 = 0 + 0.208 463 918 141 301 060 406 601 353 223 939 507 093 504;
  • 42) 0.208 463 918 141 301 060 406 601 353 223 939 507 093 504 × 2 = 0 + 0.416 927 836 282 602 120 813 202 706 447 879 014 187 008;
  • 43) 0.416 927 836 282 602 120 813 202 706 447 879 014 187 008 × 2 = 0 + 0.833 855 672 565 204 241 626 405 412 895 758 028 374 016;
  • 44) 0.833 855 672 565 204 241 626 405 412 895 758 028 374 016 × 2 = 1 + 0.667 711 345 130 408 483 252 810 825 791 516 056 748 032;
  • 45) 0.667 711 345 130 408 483 252 810 825 791 516 056 748 032 × 2 = 1 + 0.335 422 690 260 816 966 505 621 651 583 032 113 496 064;
  • 46) 0.335 422 690 260 816 966 505 621 651 583 032 113 496 064 × 2 = 0 + 0.670 845 380 521 633 933 011 243 303 166 064 226 992 128;
  • 47) 0.670 845 380 521 633 933 011 243 303 166 064 226 992 128 × 2 = 1 + 0.341 690 761 043 267 866 022 486 606 332 128 453 984 256;
  • 48) 0.341 690 761 043 267 866 022 486 606 332 128 453 984 256 × 2 = 0 + 0.683 381 522 086 535 732 044 973 212 664 256 907 968 512;
  • 49) 0.683 381 522 086 535 732 044 973 212 664 256 907 968 512 × 2 = 1 + 0.366 763 044 173 071 464 089 946 425 328 513 815 937 024;
  • 50) 0.366 763 044 173 071 464 089 946 425 328 513 815 937 024 × 2 = 0 + 0.733 526 088 346 142 928 179 892 850 657 027 631 874 048;
  • 51) 0.733 526 088 346 142 928 179 892 850 657 027 631 874 048 × 2 = 1 + 0.467 052 176 692 285 856 359 785 701 314 055 263 748 096;
  • 52) 0.467 052 176 692 285 856 359 785 701 314 055 263 748 096 × 2 = 0 + 0.934 104 353 384 571 712 719 571 402 628 110 527 496 192;
  • 53) 0.934 104 353 384 571 712 719 571 402 628 110 527 496 192 × 2 = 1 + 0.868 208 706 769 143 425 439 142 805 256 221 054 992 384;
  • 54) 0.868 208 706 769 143 425 439 142 805 256 221 054 992 384 × 2 = 1 + 0.736 417 413 538 286 850 878 285 610 512 442 109 984 768;
  • 55) 0.736 417 413 538 286 850 878 285 610 512 442 109 984 768 × 2 = 1 + 0.472 834 827 076 573 701 756 571 221 024 884 219 969 536;
  • 56) 0.472 834 827 076 573 701 756 571 221 024 884 219 969 536 × 2 = 0 + 0.945 669 654 153 147 403 513 142 442 049 768 439 939 072;
  • 57) 0.945 669 654 153 147 403 513 142 442 049 768 439 939 072 × 2 = 1 + 0.891 339 308 306 294 807 026 284 884 099 536 879 878 144;
  • 58) 0.891 339 308 306 294 807 026 284 884 099 536 879 878 144 × 2 = 1 + 0.782 678 616 612 589 614 052 569 768 199 073 759 756 288;
  • 59) 0.782 678 616 612 589 614 052 569 768 199 073 759 756 288 × 2 = 1 + 0.565 357 233 225 179 228 105 139 536 398 147 519 512 576;
  • 60) 0.565 357 233 225 179 228 105 139 536 398 147 519 512 576 × 2 = 1 + 0.130 714 466 450 358 456 210 279 072 796 295 039 025 152;
  • 61) 0.130 714 466 450 358 456 210 279 072 796 295 039 025 152 × 2 = 0 + 0.261 428 932 900 716 912 420 558 145 592 590 078 050 304;
  • 62) 0.261 428 932 900 716 912 420 558 145 592 590 078 050 304 × 2 = 0 + 0.522 857 865 801 433 824 841 116 291 185 180 156 100 608;
  • 63) 0.522 857 865 801 433 824 841 116 291 185 180 156 100 608 × 2 = 1 + 0.045 715 731 602 867 649 682 232 582 370 360 312 201 216;
  • 64) 0.045 715 731 602 867 649 682 232 582 370 360 312 201 216 × 2 = 0 + 0.091 431 463 205 735 299 364 465 164 740 720 624 402 432;
  • 65) 0.091 431 463 205 735 299 364 465 164 740 720 624 402 432 × 2 = 0 + 0.182 862 926 411 470 598 728 930 329 481 441 248 804 864;
  • 66) 0.182 862 926 411 470 598 728 930 329 481 441 248 804 864 × 2 = 0 + 0.365 725 852 822 941 197 457 860 658 962 882 497 609 728;
  • 67) 0.365 725 852 822 941 197 457 860 658 962 882 497 609 728 × 2 = 0 + 0.731 451 705 645 882 394 915 721 317 925 764 995 219 456;
  • 68) 0.731 451 705 645 882 394 915 721 317 925 764 995 219 456 × 2 = 1 + 0.462 903 411 291 764 789 831 442 635 851 529 990 438 912;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 452(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 452(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 452(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 452 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