0.000 020 830 729 321 671 205 134 999 154 509 660 615 16 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 615 16(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
0.000 020 830 729 321 671 205 134 999 154 509 660 615 16(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 615 16.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 615 16 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 230 32;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 230 32 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 460 64;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 460 64 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 921 28;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 921 28 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 842 56;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 842 56 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 685 12;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 685 12 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 370 24;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 370 24 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 558 740 48;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 558 740 48 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 117 480 96;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 117 480 96 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 234 961 92;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 234 961 92 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 469 923 84;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 469 923 84 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 939 847 68;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 939 847 68 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 879 695 36;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 879 695 36 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 759 390 72;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 759 390 72 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 518 781 44;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 518 781 44 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 037 562 88;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 037 562 88 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 075 125 76;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 075 125 76 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 150 251 52;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 150 251 52 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 300 503 04;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 300 503 04 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 944 601 006 08;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 944 601 006 08 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 889 202 012 16;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 889 202 012 16 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 778 404 024 32;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 778 404 024 32 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 556 808 048 64;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 556 808 048 64 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 113 616 097 28;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 113 616 097 28 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 227 232 194 56;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 227 232 194 56 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 454 464 389 12;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 454 464 389 12 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 800 908 928 778 24;
  • 27) 0.926 581 068 845 158 120 759 900 103 800 908 928 778 24 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 601 817 857 556 48;
  • 28) 0.853 162 137 690 316 241 519 800 207 601 817 857 556 48 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 203 635 715 112 96;
  • 29) 0.706 324 275 380 632 483 039 600 415 203 635 715 112 96 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 407 271 430 225 92;
  • 30) 0.412 648 550 761 264 966 079 200 830 407 271 430 225 92 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 814 542 860 451 84;
  • 31) 0.825 297 101 522 529 932 158 401 660 814 542 860 451 84 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 629 085 720 903 68;
  • 32) 0.650 594 203 045 059 864 316 803 321 629 085 720 903 68 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 258 171 441 807 36;
  • 33) 0.301 188 406 090 119 728 633 606 643 258 171 441 807 36 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 516 342 883 614 72;
  • 34) 0.602 376 812 180 239 457 267 213 286 516 342 883 614 72 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 032 685 767 229 44;
  • 35) 0.204 753 624 360 478 914 534 426 573 032 685 767 229 44 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 065 371 534 458 88;
  • 36) 0.409 507 248 720 957 829 068 853 146 065 371 534 458 88 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 130 743 068 917 76;
  • 37) 0.819 014 497 441 915 658 137 706 292 130 743 068 917 76 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 261 486 137 835 52;
  • 38) 0.638 028 994 883 831 316 275 412 584 261 486 137 835 52 × 2 = 1 + 0.276 057 989 767 662 632 550 825 168 522 972 275 671 04;
  • 39) 0.276 057 989 767 662 632 550 825 168 522 972 275 671 04 × 2 = 0 + 0.552 115 979 535 325 265 101 650 337 045 944 551 342 08;
  • 40) 0.552 115 979 535 325 265 101 650 337 045 944 551 342 08 × 2 = 1 + 0.104 231 959 070 650 530 203 300 674 091 889 102 684 16;
  • 41) 0.104 231 959 070 650 530 203 300 674 091 889 102 684 16 × 2 = 0 + 0.208 463 918 141 301 060 406 601 348 183 778 205 368 32;
  • 42) 0.208 463 918 141 301 060 406 601 348 183 778 205 368 32 × 2 = 0 + 0.416 927 836 282 602 120 813 202 696 367 556 410 736 64;
  • 43) 0.416 927 836 282 602 120 813 202 696 367 556 410 736 64 × 2 = 0 + 0.833 855 672 565 204 241 626 405 392 735 112 821 473 28;
  • 44) 0.833 855 672 565 204 241 626 405 392 735 112 821 473 28 × 2 = 1 + 0.667 711 345 130 408 483 252 810 785 470 225 642 946 56;
  • 45) 0.667 711 345 130 408 483 252 810 785 470 225 642 946 56 × 2 = 1 + 0.335 422 690 260 816 966 505 621 570 940 451 285 893 12;
  • 46) 0.335 422 690 260 816 966 505 621 570 940 451 285 893 12 × 2 = 0 + 0.670 845 380 521 633 933 011 243 141 880 902 571 786 24;
  • 47) 0.670 845 380 521 633 933 011 243 141 880 902 571 786 24 × 2 = 1 + 0.341 690 761 043 267 866 022 486 283 761 805 143 572 48;
  • 48) 0.341 690 761 043 267 866 022 486 283 761 805 143 572 48 × 2 = 0 + 0.683 381 522 086 535 732 044 972 567 523 610 287 144 96;
  • 49) 0.683 381 522 086 535 732 044 972 567 523 610 287 144 96 × 2 = 1 + 0.366 763 044 173 071 464 089 945 135 047 220 574 289 92;
  • 50) 0.366 763 044 173 071 464 089 945 135 047 220 574 289 92 × 2 = 0 + 0.733 526 088 346 142 928 179 890 270 094 441 148 579 84;
  • 51) 0.733 526 088 346 142 928 179 890 270 094 441 148 579 84 × 2 = 1 + 0.467 052 176 692 285 856 359 780 540 188 882 297 159 68;
  • 52) 0.467 052 176 692 285 856 359 780 540 188 882 297 159 68 × 2 = 0 + 0.934 104 353 384 571 712 719 561 080 377 764 594 319 36;
  • 53) 0.934 104 353 384 571 712 719 561 080 377 764 594 319 36 × 2 = 1 + 0.868 208 706 769 143 425 439 122 160 755 529 188 638 72;
  • 54) 0.868 208 706 769 143 425 439 122 160 755 529 188 638 72 × 2 = 1 + 0.736 417 413 538 286 850 878 244 321 511 058 377 277 44;
  • 55) 0.736 417 413 538 286 850 878 244 321 511 058 377 277 44 × 2 = 1 + 0.472 834 827 076 573 701 756 488 643 022 116 754 554 88;
  • 56) 0.472 834 827 076 573 701 756 488 643 022 116 754 554 88 × 2 = 0 + 0.945 669 654 153 147 403 512 977 286 044 233 509 109 76;
  • 57) 0.945 669 654 153 147 403 512 977 286 044 233 509 109 76 × 2 = 1 + 0.891 339 308 306 294 807 025 954 572 088 467 018 219 52;
  • 58) 0.891 339 308 306 294 807 025 954 572 088 467 018 219 52 × 2 = 1 + 0.782 678 616 612 589 614 051 909 144 176 934 036 439 04;
  • 59) 0.782 678 616 612 589 614 051 909 144 176 934 036 439 04 × 2 = 1 + 0.565 357 233 225 179 228 103 818 288 353 868 072 878 08;
  • 60) 0.565 357 233 225 179 228 103 818 288 353 868 072 878 08 × 2 = 1 + 0.130 714 466 450 358 456 207 636 576 707 736 145 756 16;
  • 61) 0.130 714 466 450 358 456 207 636 576 707 736 145 756 16 × 2 = 0 + 0.261 428 932 900 716 912 415 273 153 415 472 291 512 32;
  • 62) 0.261 428 932 900 716 912 415 273 153 415 472 291 512 32 × 2 = 0 + 0.522 857 865 801 433 824 830 546 306 830 944 583 024 64;
  • 63) 0.522 857 865 801 433 824 830 546 306 830 944 583 024 64 × 2 = 1 + 0.045 715 731 602 867 649 661 092 613 661 889 166 049 28;
  • 64) 0.045 715 731 602 867 649 661 092 613 661 889 166 049 28 × 2 = 0 + 0.091 431 463 205 735 299 322 185 227 323 778 332 098 56;
  • 65) 0.091 431 463 205 735 299 322 185 227 323 778 332 098 56 × 2 = 0 + 0.182 862 926 411 470 598 644 370 454 647 556 664 197 12;
  • 66) 0.182 862 926 411 470 598 644 370 454 647 556 664 197 12 × 2 = 0 + 0.365 725 852 822 941 197 288 740 909 295 113 328 394 24;
  • 67) 0.365 725 852 822 941 197 288 740 909 295 113 328 394 24 × 2 = 0 + 0.731 451 705 645 882 394 577 481 818 590 226 656 788 48;
  • 68) 0.731 451 705 645 882 394 577 481 818 590 226 656 788 48 × 2 = 1 + 0.462 903 411 291 764 789 154 963 637 180 453 313 576 96;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 615 16(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 615 16(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 615 16(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 0.000 020 830 729 321 671 205 134 999 154 509 660 615 16 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