0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31(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
0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31(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: 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31.

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) 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31 × 2 = 0 + 0.018 469 135 621 974 469 134 196 665 975 308 642 001 975 397 530 62;
  • 2) 0.018 469 135 621 974 469 134 196 665 975 308 642 001 975 397 530 62 × 2 = 0 + 0.036 938 271 243 948 938 268 393 331 950 617 284 003 950 795 061 24;
  • 3) 0.036 938 271 243 948 938 268 393 331 950 617 284 003 950 795 061 24 × 2 = 0 + 0.073 876 542 487 897 876 536 786 663 901 234 568 007 901 590 122 48;
  • 4) 0.073 876 542 487 897 876 536 786 663 901 234 568 007 901 590 122 48 × 2 = 0 + 0.147 753 084 975 795 753 073 573 327 802 469 136 015 803 180 244 96;
  • 5) 0.147 753 084 975 795 753 073 573 327 802 469 136 015 803 180 244 96 × 2 = 0 + 0.295 506 169 951 591 506 147 146 655 604 938 272 031 606 360 489 92;
  • 6) 0.295 506 169 951 591 506 147 146 655 604 938 272 031 606 360 489 92 × 2 = 0 + 0.591 012 339 903 183 012 294 293 311 209 876 544 063 212 720 979 84;
  • 7) 0.591 012 339 903 183 012 294 293 311 209 876 544 063 212 720 979 84 × 2 = 1 + 0.182 024 679 806 366 024 588 586 622 419 753 088 126 425 441 959 68;
  • 8) 0.182 024 679 806 366 024 588 586 622 419 753 088 126 425 441 959 68 × 2 = 0 + 0.364 049 359 612 732 049 177 173 244 839 506 176 252 850 883 919 36;
  • 9) 0.364 049 359 612 732 049 177 173 244 839 506 176 252 850 883 919 36 × 2 = 0 + 0.728 098 719 225 464 098 354 346 489 679 012 352 505 701 767 838 72;
  • 10) 0.728 098 719 225 464 098 354 346 489 679 012 352 505 701 767 838 72 × 2 = 1 + 0.456 197 438 450 928 196 708 692 979 358 024 705 011 403 535 677 44;
  • 11) 0.456 197 438 450 928 196 708 692 979 358 024 705 011 403 535 677 44 × 2 = 0 + 0.912 394 876 901 856 393 417 385 958 716 049 410 022 807 071 354 88;
  • 12) 0.912 394 876 901 856 393 417 385 958 716 049 410 022 807 071 354 88 × 2 = 1 + 0.824 789 753 803 712 786 834 771 917 432 098 820 045 614 142 709 76;
  • 13) 0.824 789 753 803 712 786 834 771 917 432 098 820 045 614 142 709 76 × 2 = 1 + 0.649 579 507 607 425 573 669 543 834 864 197 640 091 228 285 419 52;
  • 14) 0.649 579 507 607 425 573 669 543 834 864 197 640 091 228 285 419 52 × 2 = 1 + 0.299 159 015 214 851 147 339 087 669 728 395 280 182 456 570 839 04;
  • 15) 0.299 159 015 214 851 147 339 087 669 728 395 280 182 456 570 839 04 × 2 = 0 + 0.598 318 030 429 702 294 678 175 339 456 790 560 364 913 141 678 08;
  • 16) 0.598 318 030 429 702 294 678 175 339 456 790 560 364 913 141 678 08 × 2 = 1 + 0.196 636 060 859 404 589 356 350 678 913 581 120 729 826 283 356 16;
  • 17) 0.196 636 060 859 404 589 356 350 678 913 581 120 729 826 283 356 16 × 2 = 0 + 0.393 272 121 718 809 178 712 701 357 827 162 241 459 652 566 712 32;
  • 18) 0.393 272 121 718 809 178 712 701 357 827 162 241 459 652 566 712 32 × 2 = 0 + 0.786 544 243 437 618 357 425 402 715 654 324 482 919 305 133 424 64;
  • 19) 0.786 544 243 437 618 357 425 402 715 654 324 482 919 305 133 424 64 × 2 = 1 + 0.573 088 486 875 236 714 850 805 431 308 648 965 838 610 266 849 28;
  • 20) 0.573 088 486 875 236 714 850 805 431 308 648 965 838 610 266 849 28 × 2 = 1 + 0.146 176 973 750 473 429 701 610 862 617 297 931 677 220 533 698 56;
  • 21) 0.146 176 973 750 473 429 701 610 862 617 297 931 677 220 533 698 56 × 2 = 0 + 0.292 353 947 500 946 859 403 221 725 234 595 863 354 441 067 397 12;
  • 22) 0.292 353 947 500 946 859 403 221 725 234 595 863 354 441 067 397 12 × 2 = 0 + 0.584 707 895 001 893 718 806 443 450 469 191 726 708 882 134 794 24;
  • 23) 0.584 707 895 001 893 718 806 443 450 469 191 726 708 882 134 794 24 × 2 = 1 + 0.169 415 790 003 787 437 612 886 900 938 383 453 417 764 269 588 48;
  • 24) 0.169 415 790 003 787 437 612 886 900 938 383 453 417 764 269 588 48 × 2 = 0 + 0.338 831 580 007 574 875 225 773 801 876 766 906 835 528 539 176 96;
  • 25) 0.338 831 580 007 574 875 225 773 801 876 766 906 835 528 539 176 96 × 2 = 0 + 0.677 663 160 015 149 750 451 547 603 753 533 813 671 057 078 353 92;
  • 26) 0.677 663 160 015 149 750 451 547 603 753 533 813 671 057 078 353 92 × 2 = 1 + 0.355 326 320 030 299 500 903 095 207 507 067 627 342 114 156 707 84;
  • 27) 0.355 326 320 030 299 500 903 095 207 507 067 627 342 114 156 707 84 × 2 = 0 + 0.710 652 640 060 599 001 806 190 415 014 135 254 684 228 313 415 68;
  • 28) 0.710 652 640 060 599 001 806 190 415 014 135 254 684 228 313 415 68 × 2 = 1 + 0.421 305 280 121 198 003 612 380 830 028 270 509 368 456 626 831 36;
  • 29) 0.421 305 280 121 198 003 612 380 830 028 270 509 368 456 626 831 36 × 2 = 0 + 0.842 610 560 242 396 007 224 761 660 056 541 018 736 913 253 662 72;
  • 30) 0.842 610 560 242 396 007 224 761 660 056 541 018 736 913 253 662 72 × 2 = 1 + 0.685 221 120 484 792 014 449 523 320 113 082 037 473 826 507 325 44;
  • 31) 0.685 221 120 484 792 014 449 523 320 113 082 037 473 826 507 325 44 × 2 = 1 + 0.370 442 240 969 584 028 899 046 640 226 164 074 947 653 014 650 88;
  • 32) 0.370 442 240 969 584 028 899 046 640 226 164 074 947 653 014 650 88 × 2 = 0 + 0.740 884 481 939 168 057 798 093 280 452 328 149 895 306 029 301 76;
  • 33) 0.740 884 481 939 168 057 798 093 280 452 328 149 895 306 029 301 76 × 2 = 1 + 0.481 768 963 878 336 115 596 186 560 904 656 299 790 612 058 603 52;
  • 34) 0.481 768 963 878 336 115 596 186 560 904 656 299 790 612 058 603 52 × 2 = 0 + 0.963 537 927 756 672 231 192 373 121 809 312 599 581 224 117 207 04;
  • 35) 0.963 537 927 756 672 231 192 373 121 809 312 599 581 224 117 207 04 × 2 = 1 + 0.927 075 855 513 344 462 384 746 243 618 625 199 162 448 234 414 08;
  • 36) 0.927 075 855 513 344 462 384 746 243 618 625 199 162 448 234 414 08 × 2 = 1 + 0.854 151 711 026 688 924 769 492 487 237 250 398 324 896 468 828 16;
  • 37) 0.854 151 711 026 688 924 769 492 487 237 250 398 324 896 468 828 16 × 2 = 1 + 0.708 303 422 053 377 849 538 984 974 474 500 796 649 792 937 656 32;
  • 38) 0.708 303 422 053 377 849 538 984 974 474 500 796 649 792 937 656 32 × 2 = 1 + 0.416 606 844 106 755 699 077 969 948 949 001 593 299 585 875 312 64;
  • 39) 0.416 606 844 106 755 699 077 969 948 949 001 593 299 585 875 312 64 × 2 = 0 + 0.833 213 688 213 511 398 155 939 897 898 003 186 599 171 750 625 28;
  • 40) 0.833 213 688 213 511 398 155 939 897 898 003 186 599 171 750 625 28 × 2 = 1 + 0.666 427 376 427 022 796 311 879 795 796 006 373 198 343 501 250 56;
  • 41) 0.666 427 376 427 022 796 311 879 795 796 006 373 198 343 501 250 56 × 2 = 1 + 0.332 854 752 854 045 592 623 759 591 592 012 746 396 687 002 501 12;
  • 42) 0.332 854 752 854 045 592 623 759 591 592 012 746 396 687 002 501 12 × 2 = 0 + 0.665 709 505 708 091 185 247 519 183 184 025 492 793 374 005 002 24;
  • 43) 0.665 709 505 708 091 185 247 519 183 184 025 492 793 374 005 002 24 × 2 = 1 + 0.331 419 011 416 182 370 495 038 366 368 050 985 586 748 010 004 48;
  • 44) 0.331 419 011 416 182 370 495 038 366 368 050 985 586 748 010 004 48 × 2 = 0 + 0.662 838 022 832 364 740 990 076 732 736 101 971 173 496 020 008 96;
  • 45) 0.662 838 022 832 364 740 990 076 732 736 101 971 173 496 020 008 96 × 2 = 1 + 0.325 676 045 664 729 481 980 153 465 472 203 942 346 992 040 017 92;
  • 46) 0.325 676 045 664 729 481 980 153 465 472 203 942 346 992 040 017 92 × 2 = 0 + 0.651 352 091 329 458 963 960 306 930 944 407 884 693 984 080 035 84;
  • 47) 0.651 352 091 329 458 963 960 306 930 944 407 884 693 984 080 035 84 × 2 = 1 + 0.302 704 182 658 917 927 920 613 861 888 815 769 387 968 160 071 68;
  • 48) 0.302 704 182 658 917 927 920 613 861 888 815 769 387 968 160 071 68 × 2 = 0 + 0.605 408 365 317 835 855 841 227 723 777 631 538 775 936 320 143 36;
  • 49) 0.605 408 365 317 835 855 841 227 723 777 631 538 775 936 320 143 36 × 2 = 1 + 0.210 816 730 635 671 711 682 455 447 555 263 077 551 872 640 286 72;
  • 50) 0.210 816 730 635 671 711 682 455 447 555 263 077 551 872 640 286 72 × 2 = 0 + 0.421 633 461 271 343 423 364 910 895 110 526 155 103 745 280 573 44;
  • 51) 0.421 633 461 271 343 423 364 910 895 110 526 155 103 745 280 573 44 × 2 = 0 + 0.843 266 922 542 686 846 729 821 790 221 052 310 207 490 561 146 88;
  • 52) 0.843 266 922 542 686 846 729 821 790 221 052 310 207 490 561 146 88 × 2 = 1 + 0.686 533 845 085 373 693 459 643 580 442 104 620 414 981 122 293 76;
  • 53) 0.686 533 845 085 373 693 459 643 580 442 104 620 414 981 122 293 76 × 2 = 1 + 0.373 067 690 170 747 386 919 287 160 884 209 240 829 962 244 587 52;
  • 54) 0.373 067 690 170 747 386 919 287 160 884 209 240 829 962 244 587 52 × 2 = 0 + 0.746 135 380 341 494 773 838 574 321 768 418 481 659 924 489 175 04;
  • 55) 0.746 135 380 341 494 773 838 574 321 768 418 481 659 924 489 175 04 × 2 = 1 + 0.492 270 760 682 989 547 677 148 643 536 836 963 319 848 978 350 08;
  • 56) 0.492 270 760 682 989 547 677 148 643 536 836 963 319 848 978 350 08 × 2 = 0 + 0.984 541 521 365 979 095 354 297 287 073 673 926 639 697 956 700 16;
  • 57) 0.984 541 521 365 979 095 354 297 287 073 673 926 639 697 956 700 16 × 2 = 1 + 0.969 083 042 731 958 190 708 594 574 147 347 853 279 395 913 400 32;
  • 58) 0.969 083 042 731 958 190 708 594 574 147 347 853 279 395 913 400 32 × 2 = 1 + 0.938 166 085 463 916 381 417 189 148 294 695 706 558 791 826 800 64;
  • 59) 0.938 166 085 463 916 381 417 189 148 294 695 706 558 791 826 800 64 × 2 = 1 + 0.876 332 170 927 832 762 834 378 296 589 391 413 117 583 653 601 28;

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:


0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2)

5. Positive number before normalization:

0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2) × 20 =

1.0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111(2) × 2-7


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): -7

Mantissa (not normalized):
1.0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-7 + 2(11-1) - 1 =

(-7 + 1 023)(10) =

1 016(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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1016(10) =

011 1111 1000(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. 0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111 =

0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


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 1111 1000

Mantissa (52 bits) =
0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


Decimal number 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 31 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1000 - 0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111

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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