0.000 020 830 729 321 671 205 134 999 154 509 660 617 981 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 981(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 981(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 981.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 981 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 235 962;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 235 962 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 471 924;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 471 924 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 943 848;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 943 848 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 887 696;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 887 696 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 775 392;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 775 392 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 550 784;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 550 784 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 101 568;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 101 568 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 203 136;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 203 136 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 406 272;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 406 272 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 812 544;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 812 544 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 945 625 088;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 945 625 088 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 891 250 176;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 891 250 176 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 782 500 352;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 782 500 352 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 565 000 704;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 565 000 704 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 130 001 408;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 130 001 408 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 260 002 816;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 260 002 816 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 520 005 632;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 520 005 632 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 473 040 011 264;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 473 040 011 264 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 946 080 022 528;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 946 080 022 528 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 892 160 045 056;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 892 160 045 056 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 784 320 090 112;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 784 320 090 112 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 568 640 180 224;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 568 640 180 224 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 137 280 360 448;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 137 280 360 448 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 274 560 720 896;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 274 560 720 896 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 549 121 441 792;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 549 121 441 792 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 098 242 883 584;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 098 242 883 584 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 196 485 767 168;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 196 485 767 168 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 392 971 534 336;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 392 971 534 336 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 785 943 068 672;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 785 943 068 672 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 817 571 886 137 344;
  • 31) 0.825 297 101 522 529 932 158 401 660 817 571 886 137 344 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 635 143 772 274 688;
  • 32) 0.650 594 203 045 059 864 316 803 321 635 143 772 274 688 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 270 287 544 549 376;
  • 33) 0.301 188 406 090 119 728 633 606 643 270 287 544 549 376 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 540 575 089 098 752;
  • 34) 0.602 376 812 180 239 457 267 213 286 540 575 089 098 752 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 081 150 178 197 504;
  • 35) 0.204 753 624 360 478 914 534 426 573 081 150 178 197 504 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 162 300 356 395 008;
  • 36) 0.409 507 248 720 957 829 068 853 146 162 300 356 395 008 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 324 600 712 790 016;
  • 37) 0.819 014 497 441 915 658 137 706 292 324 600 712 790 016 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 649 201 425 580 032;
  • 38) 0.638 028 994 883 831 316 275 412 584 649 201 425 580 032 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 298 402 851 160 064;
  • 39) 0.276 057 989 767 662 632 550 825 169 298 402 851 160 064 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 596 805 702 320 128;
  • 40) 0.552 115 979 535 325 265 101 650 338 596 805 702 320 128 × 2 = 1 + 0.104 231 959 070 650 530 203 300 677 193 611 404 640 256;
  • 41) 0.104 231 959 070 650 530 203 300 677 193 611 404 640 256 × 2 = 0 + 0.208 463 918 141 301 060 406 601 354 387 222 809 280 512;
  • 42) 0.208 463 918 141 301 060 406 601 354 387 222 809 280 512 × 2 = 0 + 0.416 927 836 282 602 120 813 202 708 774 445 618 561 024;
  • 43) 0.416 927 836 282 602 120 813 202 708 774 445 618 561 024 × 2 = 0 + 0.833 855 672 565 204 241 626 405 417 548 891 237 122 048;
  • 44) 0.833 855 672 565 204 241 626 405 417 548 891 237 122 048 × 2 = 1 + 0.667 711 345 130 408 483 252 810 835 097 782 474 244 096;
  • 45) 0.667 711 345 130 408 483 252 810 835 097 782 474 244 096 × 2 = 1 + 0.335 422 690 260 816 966 505 621 670 195 564 948 488 192;
  • 46) 0.335 422 690 260 816 966 505 621 670 195 564 948 488 192 × 2 = 0 + 0.670 845 380 521 633 933 011 243 340 391 129 896 976 384;
  • 47) 0.670 845 380 521 633 933 011 243 340 391 129 896 976 384 × 2 = 1 + 0.341 690 761 043 267 866 022 486 680 782 259 793 952 768;
  • 48) 0.341 690 761 043 267 866 022 486 680 782 259 793 952 768 × 2 = 0 + 0.683 381 522 086 535 732 044 973 361 564 519 587 905 536;
  • 49) 0.683 381 522 086 535 732 044 973 361 564 519 587 905 536 × 2 = 1 + 0.366 763 044 173 071 464 089 946 723 129 039 175 811 072;
  • 50) 0.366 763 044 173 071 464 089 946 723 129 039 175 811 072 × 2 = 0 + 0.733 526 088 346 142 928 179 893 446 258 078 351 622 144;
  • 51) 0.733 526 088 346 142 928 179 893 446 258 078 351 622 144 × 2 = 1 + 0.467 052 176 692 285 856 359 786 892 516 156 703 244 288;
  • 52) 0.467 052 176 692 285 856 359 786 892 516 156 703 244 288 × 2 = 0 + 0.934 104 353 384 571 712 719 573 785 032 313 406 488 576;
  • 53) 0.934 104 353 384 571 712 719 573 785 032 313 406 488 576 × 2 = 1 + 0.868 208 706 769 143 425 439 147 570 064 626 812 977 152;
  • 54) 0.868 208 706 769 143 425 439 147 570 064 626 812 977 152 × 2 = 1 + 0.736 417 413 538 286 850 878 295 140 129 253 625 954 304;
  • 55) 0.736 417 413 538 286 850 878 295 140 129 253 625 954 304 × 2 = 1 + 0.472 834 827 076 573 701 756 590 280 258 507 251 908 608;
  • 56) 0.472 834 827 076 573 701 756 590 280 258 507 251 908 608 × 2 = 0 + 0.945 669 654 153 147 403 513 180 560 517 014 503 817 216;
  • 57) 0.945 669 654 153 147 403 513 180 560 517 014 503 817 216 × 2 = 1 + 0.891 339 308 306 294 807 026 361 121 034 029 007 634 432;
  • 58) 0.891 339 308 306 294 807 026 361 121 034 029 007 634 432 × 2 = 1 + 0.782 678 616 612 589 614 052 722 242 068 058 015 268 864;
  • 59) 0.782 678 616 612 589 614 052 722 242 068 058 015 268 864 × 2 = 1 + 0.565 357 233 225 179 228 105 444 484 136 116 030 537 728;
  • 60) 0.565 357 233 225 179 228 105 444 484 136 116 030 537 728 × 2 = 1 + 0.130 714 466 450 358 456 210 888 968 272 232 061 075 456;
  • 61) 0.130 714 466 450 358 456 210 888 968 272 232 061 075 456 × 2 = 0 + 0.261 428 932 900 716 912 421 777 936 544 464 122 150 912;
  • 62) 0.261 428 932 900 716 912 421 777 936 544 464 122 150 912 × 2 = 0 + 0.522 857 865 801 433 824 843 555 873 088 928 244 301 824;
  • 63) 0.522 857 865 801 433 824 843 555 873 088 928 244 301 824 × 2 = 1 + 0.045 715 731 602 867 649 687 111 746 177 856 488 603 648;
  • 64) 0.045 715 731 602 867 649 687 111 746 177 856 488 603 648 × 2 = 0 + 0.091 431 463 205 735 299 374 223 492 355 712 977 207 296;
  • 65) 0.091 431 463 205 735 299 374 223 492 355 712 977 207 296 × 2 = 0 + 0.182 862 926 411 470 598 748 446 984 711 425 954 414 592;
  • 66) 0.182 862 926 411 470 598 748 446 984 711 425 954 414 592 × 2 = 0 + 0.365 725 852 822 941 197 496 893 969 422 851 908 829 184;
  • 67) 0.365 725 852 822 941 197 496 893 969 422 851 908 829 184 × 2 = 0 + 0.731 451 705 645 882 394 993 787 938 845 703 817 658 368;
  • 68) 0.731 451 705 645 882 394 993 787 938 845 703 817 658 368 × 2 = 1 + 0.462 903 411 291 764 789 987 575 877 691 407 635 316 736;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 981(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 981(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 981(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 981 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