0.000 020 830 729 321 671 205 134 999 154 509 660 617 861 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 861(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 861(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 861.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 861 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 235 722;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 235 722 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 471 444;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 471 444 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 942 888;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 942 888 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 885 776;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 885 776 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 771 552;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 771 552 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 543 104;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 543 104 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 086 208;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 086 208 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 172 416;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 172 416 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 344 832;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 344 832 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 689 664;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 689 664 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 945 379 328;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 945 379 328 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 890 758 656;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 890 758 656 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 781 517 312;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 781 517 312 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 563 034 624;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 563 034 624 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 126 069 248;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 126 069 248 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 252 138 496;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 252 138 496 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 504 276 992;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 504 276 992 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 473 008 553 984;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 473 008 553 984 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 946 017 107 968;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 946 017 107 968 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 892 034 215 936;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 892 034 215 936 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 784 068 431 872;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 784 068 431 872 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 568 136 863 744;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 568 136 863 744 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 136 273 727 488;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 136 273 727 488 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 272 547 454 976;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 272 547 454 976 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 545 094 909 952;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 545 094 909 952 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 090 189 819 904;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 090 189 819 904 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 180 379 639 808;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 180 379 639 808 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 360 759 279 616;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 360 759 279 616 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 721 518 559 232;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 721 518 559 232 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 817 443 037 118 464;
  • 31) 0.825 297 101 522 529 932 158 401 660 817 443 037 118 464 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 634 886 074 236 928;
  • 32) 0.650 594 203 045 059 864 316 803 321 634 886 074 236 928 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 269 772 148 473 856;
  • 33) 0.301 188 406 090 119 728 633 606 643 269 772 148 473 856 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 539 544 296 947 712;
  • 34) 0.602 376 812 180 239 457 267 213 286 539 544 296 947 712 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 079 088 593 895 424;
  • 35) 0.204 753 624 360 478 914 534 426 573 079 088 593 895 424 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 158 177 187 790 848;
  • 36) 0.409 507 248 720 957 829 068 853 146 158 177 187 790 848 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 316 354 375 581 696;
  • 37) 0.819 014 497 441 915 658 137 706 292 316 354 375 581 696 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 632 708 751 163 392;
  • 38) 0.638 028 994 883 831 316 275 412 584 632 708 751 163 392 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 265 417 502 326 784;
  • 39) 0.276 057 989 767 662 632 550 825 169 265 417 502 326 784 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 530 835 004 653 568;
  • 40) 0.552 115 979 535 325 265 101 650 338 530 835 004 653 568 × 2 = 1 + 0.104 231 959 070 650 530 203 300 677 061 670 009 307 136;
  • 41) 0.104 231 959 070 650 530 203 300 677 061 670 009 307 136 × 2 = 0 + 0.208 463 918 141 301 060 406 601 354 123 340 018 614 272;
  • 42) 0.208 463 918 141 301 060 406 601 354 123 340 018 614 272 × 2 = 0 + 0.416 927 836 282 602 120 813 202 708 246 680 037 228 544;
  • 43) 0.416 927 836 282 602 120 813 202 708 246 680 037 228 544 × 2 = 0 + 0.833 855 672 565 204 241 626 405 416 493 360 074 457 088;
  • 44) 0.833 855 672 565 204 241 626 405 416 493 360 074 457 088 × 2 = 1 + 0.667 711 345 130 408 483 252 810 832 986 720 148 914 176;
  • 45) 0.667 711 345 130 408 483 252 810 832 986 720 148 914 176 × 2 = 1 + 0.335 422 690 260 816 966 505 621 665 973 440 297 828 352;
  • 46) 0.335 422 690 260 816 966 505 621 665 973 440 297 828 352 × 2 = 0 + 0.670 845 380 521 633 933 011 243 331 946 880 595 656 704;
  • 47) 0.670 845 380 521 633 933 011 243 331 946 880 595 656 704 × 2 = 1 + 0.341 690 761 043 267 866 022 486 663 893 761 191 313 408;
  • 48) 0.341 690 761 043 267 866 022 486 663 893 761 191 313 408 × 2 = 0 + 0.683 381 522 086 535 732 044 973 327 787 522 382 626 816;
  • 49) 0.683 381 522 086 535 732 044 973 327 787 522 382 626 816 × 2 = 1 + 0.366 763 044 173 071 464 089 946 655 575 044 765 253 632;
  • 50) 0.366 763 044 173 071 464 089 946 655 575 044 765 253 632 × 2 = 0 + 0.733 526 088 346 142 928 179 893 311 150 089 530 507 264;
  • 51) 0.733 526 088 346 142 928 179 893 311 150 089 530 507 264 × 2 = 1 + 0.467 052 176 692 285 856 359 786 622 300 179 061 014 528;
  • 52) 0.467 052 176 692 285 856 359 786 622 300 179 061 014 528 × 2 = 0 + 0.934 104 353 384 571 712 719 573 244 600 358 122 029 056;
  • 53) 0.934 104 353 384 571 712 719 573 244 600 358 122 029 056 × 2 = 1 + 0.868 208 706 769 143 425 439 146 489 200 716 244 058 112;
  • 54) 0.868 208 706 769 143 425 439 146 489 200 716 244 058 112 × 2 = 1 + 0.736 417 413 538 286 850 878 292 978 401 432 488 116 224;
  • 55) 0.736 417 413 538 286 850 878 292 978 401 432 488 116 224 × 2 = 1 + 0.472 834 827 076 573 701 756 585 956 802 864 976 232 448;
  • 56) 0.472 834 827 076 573 701 756 585 956 802 864 976 232 448 × 2 = 0 + 0.945 669 654 153 147 403 513 171 913 605 729 952 464 896;
  • 57) 0.945 669 654 153 147 403 513 171 913 605 729 952 464 896 × 2 = 1 + 0.891 339 308 306 294 807 026 343 827 211 459 904 929 792;
  • 58) 0.891 339 308 306 294 807 026 343 827 211 459 904 929 792 × 2 = 1 + 0.782 678 616 612 589 614 052 687 654 422 919 809 859 584;
  • 59) 0.782 678 616 612 589 614 052 687 654 422 919 809 859 584 × 2 = 1 + 0.565 357 233 225 179 228 105 375 308 845 839 619 719 168;
  • 60) 0.565 357 233 225 179 228 105 375 308 845 839 619 719 168 × 2 = 1 + 0.130 714 466 450 358 456 210 750 617 691 679 239 438 336;
  • 61) 0.130 714 466 450 358 456 210 750 617 691 679 239 438 336 × 2 = 0 + 0.261 428 932 900 716 912 421 501 235 383 358 478 876 672;
  • 62) 0.261 428 932 900 716 912 421 501 235 383 358 478 876 672 × 2 = 0 + 0.522 857 865 801 433 824 843 002 470 766 716 957 753 344;
  • 63) 0.522 857 865 801 433 824 843 002 470 766 716 957 753 344 × 2 = 1 + 0.045 715 731 602 867 649 686 004 941 533 433 915 506 688;
  • 64) 0.045 715 731 602 867 649 686 004 941 533 433 915 506 688 × 2 = 0 + 0.091 431 463 205 735 299 372 009 883 066 867 831 013 376;
  • 65) 0.091 431 463 205 735 299 372 009 883 066 867 831 013 376 × 2 = 0 + 0.182 862 926 411 470 598 744 019 766 133 735 662 026 752;
  • 66) 0.182 862 926 411 470 598 744 019 766 133 735 662 026 752 × 2 = 0 + 0.365 725 852 822 941 197 488 039 532 267 471 324 053 504;
  • 67) 0.365 725 852 822 941 197 488 039 532 267 471 324 053 504 × 2 = 0 + 0.731 451 705 645 882 394 976 079 064 534 942 648 107 008;
  • 68) 0.731 451 705 645 882 394 976 079 064 534 942 648 107 008 × 2 = 1 + 0.462 903 411 291 764 789 952 158 129 069 885 296 214 016;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 861(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 861(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 861(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 861 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