0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5(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
0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5(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: 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5.

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) 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5 × 2 = 0 + 0.019 999 888 889 222 221 655 329 410 570 357 194 582 130 423 854 319 487 441 307;
  • 2) 0.019 999 888 889 222 221 655 329 410 570 357 194 582 130 423 854 319 487 441 307 × 2 = 0 + 0.039 999 777 778 444 443 310 658 821 140 714 389 164 260 847 708 638 974 882 614;
  • 3) 0.039 999 777 778 444 443 310 658 821 140 714 389 164 260 847 708 638 974 882 614 × 2 = 0 + 0.079 999 555 556 888 886 621 317 642 281 428 778 328 521 695 417 277 949 765 228;
  • 4) 0.079 999 555 556 888 886 621 317 642 281 428 778 328 521 695 417 277 949 765 228 × 2 = 0 + 0.159 999 111 113 777 773 242 635 284 562 857 556 657 043 390 834 555 899 530 456;
  • 5) 0.159 999 111 113 777 773 242 635 284 562 857 556 657 043 390 834 555 899 530 456 × 2 = 0 + 0.319 998 222 227 555 546 485 270 569 125 715 113 314 086 781 669 111 799 060 912;
  • 6) 0.319 998 222 227 555 546 485 270 569 125 715 113 314 086 781 669 111 799 060 912 × 2 = 0 + 0.639 996 444 455 111 092 970 541 138 251 430 226 628 173 563 338 223 598 121 824;
  • 7) 0.639 996 444 455 111 092 970 541 138 251 430 226 628 173 563 338 223 598 121 824 × 2 = 1 + 0.279 992 888 910 222 185 941 082 276 502 860 453 256 347 126 676 447 196 243 648;
  • 8) 0.279 992 888 910 222 185 941 082 276 502 860 453 256 347 126 676 447 196 243 648 × 2 = 0 + 0.559 985 777 820 444 371 882 164 553 005 720 906 512 694 253 352 894 392 487 296;
  • 9) 0.559 985 777 820 444 371 882 164 553 005 720 906 512 694 253 352 894 392 487 296 × 2 = 1 + 0.119 971 555 640 888 743 764 329 106 011 441 813 025 388 506 705 788 784 974 592;
  • 10) 0.119 971 555 640 888 743 764 329 106 011 441 813 025 388 506 705 788 784 974 592 × 2 = 0 + 0.239 943 111 281 777 487 528 658 212 022 883 626 050 777 013 411 577 569 949 184;
  • 11) 0.239 943 111 281 777 487 528 658 212 022 883 626 050 777 013 411 577 569 949 184 × 2 = 0 + 0.479 886 222 563 554 975 057 316 424 045 767 252 101 554 026 823 155 139 898 368;
  • 12) 0.479 886 222 563 554 975 057 316 424 045 767 252 101 554 026 823 155 139 898 368 × 2 = 0 + 0.959 772 445 127 109 950 114 632 848 091 534 504 203 108 053 646 310 279 796 736;
  • 13) 0.959 772 445 127 109 950 114 632 848 091 534 504 203 108 053 646 310 279 796 736 × 2 = 1 + 0.919 544 890 254 219 900 229 265 696 183 069 008 406 216 107 292 620 559 593 472;
  • 14) 0.919 544 890 254 219 900 229 265 696 183 069 008 406 216 107 292 620 559 593 472 × 2 = 1 + 0.839 089 780 508 439 800 458 531 392 366 138 016 812 432 214 585 241 119 186 944;
  • 15) 0.839 089 780 508 439 800 458 531 392 366 138 016 812 432 214 585 241 119 186 944 × 2 = 1 + 0.678 179 561 016 879 600 917 062 784 732 276 033 624 864 429 170 482 238 373 888;
  • 16) 0.678 179 561 016 879 600 917 062 784 732 276 033 624 864 429 170 482 238 373 888 × 2 = 1 + 0.356 359 122 033 759 201 834 125 569 464 552 067 249 728 858 340 964 476 747 776;
  • 17) 0.356 359 122 033 759 201 834 125 569 464 552 067 249 728 858 340 964 476 747 776 × 2 = 0 + 0.712 718 244 067 518 403 668 251 138 929 104 134 499 457 716 681 928 953 495 552;
  • 18) 0.712 718 244 067 518 403 668 251 138 929 104 134 499 457 716 681 928 953 495 552 × 2 = 1 + 0.425 436 488 135 036 807 336 502 277 858 208 268 998 915 433 363 857 906 991 104;
  • 19) 0.425 436 488 135 036 807 336 502 277 858 208 268 998 915 433 363 857 906 991 104 × 2 = 0 + 0.850 872 976 270 073 614 673 004 555 716 416 537 997 830 866 727 715 813 982 208;
  • 20) 0.850 872 976 270 073 614 673 004 555 716 416 537 997 830 866 727 715 813 982 208 × 2 = 1 + 0.701 745 952 540 147 229 346 009 111 432 833 075 995 661 733 455 431 627 964 416;
  • 21) 0.701 745 952 540 147 229 346 009 111 432 833 075 995 661 733 455 431 627 964 416 × 2 = 1 + 0.403 491 905 080 294 458 692 018 222 865 666 151 991 323 466 910 863 255 928 832;
  • 22) 0.403 491 905 080 294 458 692 018 222 865 666 151 991 323 466 910 863 255 928 832 × 2 = 0 + 0.806 983 810 160 588 917 384 036 445 731 332 303 982 646 933 821 726 511 857 664;
  • 23) 0.806 983 810 160 588 917 384 036 445 731 332 303 982 646 933 821 726 511 857 664 × 2 = 1 + 0.613 967 620 321 177 834 768 072 891 462 664 607 965 293 867 643 453 023 715 328;
  • 24) 0.613 967 620 321 177 834 768 072 891 462 664 607 965 293 867 643 453 023 715 328 × 2 = 1 + 0.227 935 240 642 355 669 536 145 782 925 329 215 930 587 735 286 906 047 430 656;
  • 25) 0.227 935 240 642 355 669 536 145 782 925 329 215 930 587 735 286 906 047 430 656 × 2 = 0 + 0.455 870 481 284 711 339 072 291 565 850 658 431 861 175 470 573 812 094 861 312;
  • 26) 0.455 870 481 284 711 339 072 291 565 850 658 431 861 175 470 573 812 094 861 312 × 2 = 0 + 0.911 740 962 569 422 678 144 583 131 701 316 863 722 350 941 147 624 189 722 624;
  • 27) 0.911 740 962 569 422 678 144 583 131 701 316 863 722 350 941 147 624 189 722 624 × 2 = 1 + 0.823 481 925 138 845 356 289 166 263 402 633 727 444 701 882 295 248 379 445 248;
  • 28) 0.823 481 925 138 845 356 289 166 263 402 633 727 444 701 882 295 248 379 445 248 × 2 = 1 + 0.646 963 850 277 690 712 578 332 526 805 267 454 889 403 764 590 496 758 890 496;
  • 29) 0.646 963 850 277 690 712 578 332 526 805 267 454 889 403 764 590 496 758 890 496 × 2 = 1 + 0.293 927 700 555 381 425 156 665 053 610 534 909 778 807 529 180 993 517 780 992;
  • 30) 0.293 927 700 555 381 425 156 665 053 610 534 909 778 807 529 180 993 517 780 992 × 2 = 0 + 0.587 855 401 110 762 850 313 330 107 221 069 819 557 615 058 361 987 035 561 984;
  • 31) 0.587 855 401 110 762 850 313 330 107 221 069 819 557 615 058 361 987 035 561 984 × 2 = 1 + 0.175 710 802 221 525 700 626 660 214 442 139 639 115 230 116 723 974 071 123 968;
  • 32) 0.175 710 802 221 525 700 626 660 214 442 139 639 115 230 116 723 974 071 123 968 × 2 = 0 + 0.351 421 604 443 051 401 253 320 428 884 279 278 230 460 233 447 948 142 247 936;
  • 33) 0.351 421 604 443 051 401 253 320 428 884 279 278 230 460 233 447 948 142 247 936 × 2 = 0 + 0.702 843 208 886 102 802 506 640 857 768 558 556 460 920 466 895 896 284 495 872;
  • 34) 0.702 843 208 886 102 802 506 640 857 768 558 556 460 920 466 895 896 284 495 872 × 2 = 1 + 0.405 686 417 772 205 605 013 281 715 537 117 112 921 840 933 791 792 568 991 744;
  • 35) 0.405 686 417 772 205 605 013 281 715 537 117 112 921 840 933 791 792 568 991 744 × 2 = 0 + 0.811 372 835 544 411 210 026 563 431 074 234 225 843 681 867 583 585 137 983 488;
  • 36) 0.811 372 835 544 411 210 026 563 431 074 234 225 843 681 867 583 585 137 983 488 × 2 = 1 + 0.622 745 671 088 822 420 053 126 862 148 468 451 687 363 735 167 170 275 966 976;
  • 37) 0.622 745 671 088 822 420 053 126 862 148 468 451 687 363 735 167 170 275 966 976 × 2 = 1 + 0.245 491 342 177 644 840 106 253 724 296 936 903 374 727 470 334 340 551 933 952;
  • 38) 0.245 491 342 177 644 840 106 253 724 296 936 903 374 727 470 334 340 551 933 952 × 2 = 0 + 0.490 982 684 355 289 680 212 507 448 593 873 806 749 454 940 668 681 103 867 904;
  • 39) 0.490 982 684 355 289 680 212 507 448 593 873 806 749 454 940 668 681 103 867 904 × 2 = 0 + 0.981 965 368 710 579 360 425 014 897 187 747 613 498 909 881 337 362 207 735 808;
  • 40) 0.981 965 368 710 579 360 425 014 897 187 747 613 498 909 881 337 362 207 735 808 × 2 = 1 + 0.963 930 737 421 158 720 850 029 794 375 495 226 997 819 762 674 724 415 471 616;
  • 41) 0.963 930 737 421 158 720 850 029 794 375 495 226 997 819 762 674 724 415 471 616 × 2 = 1 + 0.927 861 474 842 317 441 700 059 588 750 990 453 995 639 525 349 448 830 943 232;
  • 42) 0.927 861 474 842 317 441 700 059 588 750 990 453 995 639 525 349 448 830 943 232 × 2 = 1 + 0.855 722 949 684 634 883 400 119 177 501 980 907 991 279 050 698 897 661 886 464;
  • 43) 0.855 722 949 684 634 883 400 119 177 501 980 907 991 279 050 698 897 661 886 464 × 2 = 1 + 0.711 445 899 369 269 766 800 238 355 003 961 815 982 558 101 397 795 323 772 928;
  • 44) 0.711 445 899 369 269 766 800 238 355 003 961 815 982 558 101 397 795 323 772 928 × 2 = 1 + 0.422 891 798 738 539 533 600 476 710 007 923 631 965 116 202 795 590 647 545 856;
  • 45) 0.422 891 798 738 539 533 600 476 710 007 923 631 965 116 202 795 590 647 545 856 × 2 = 0 + 0.845 783 597 477 079 067 200 953 420 015 847 263 930 232 405 591 181 295 091 712;
  • 46) 0.845 783 597 477 079 067 200 953 420 015 847 263 930 232 405 591 181 295 091 712 × 2 = 1 + 0.691 567 194 954 158 134 401 906 840 031 694 527 860 464 811 182 362 590 183 424;
  • 47) 0.691 567 194 954 158 134 401 906 840 031 694 527 860 464 811 182 362 590 183 424 × 2 = 1 + 0.383 134 389 908 316 268 803 813 680 063 389 055 720 929 622 364 725 180 366 848;
  • 48) 0.383 134 389 908 316 268 803 813 680 063 389 055 720 929 622 364 725 180 366 848 × 2 = 0 + 0.766 268 779 816 632 537 607 627 360 126 778 111 441 859 244 729 450 360 733 696;
  • 49) 0.766 268 779 816 632 537 607 627 360 126 778 111 441 859 244 729 450 360 733 696 × 2 = 1 + 0.532 537 559 633 265 075 215 254 720 253 556 222 883 718 489 458 900 721 467 392;
  • 50) 0.532 537 559 633 265 075 215 254 720 253 556 222 883 718 489 458 900 721 467 392 × 2 = 1 + 0.065 075 119 266 530 150 430 509 440 507 112 445 767 436 978 917 801 442 934 784;
  • 51) 0.065 075 119 266 530 150 430 509 440 507 112 445 767 436 978 917 801 442 934 784 × 2 = 0 + 0.130 150 238 533 060 300 861 018 881 014 224 891 534 873 957 835 602 885 869 568;
  • 52) 0.130 150 238 533 060 300 861 018 881 014 224 891 534 873 957 835 602 885 869 568 × 2 = 0 + 0.260 300 477 066 120 601 722 037 762 028 449 783 069 747 915 671 205 771 739 136;
  • 53) 0.260 300 477 066 120 601 722 037 762 028 449 783 069 747 915 671 205 771 739 136 × 2 = 0 + 0.520 600 954 132 241 203 444 075 524 056 899 566 139 495 831 342 411 543 478 272;
  • 54) 0.520 600 954 132 241 203 444 075 524 056 899 566 139 495 831 342 411 543 478 272 × 2 = 1 + 0.041 201 908 264 482 406 888 151 048 113 799 132 278 991 662 684 823 086 956 544;
  • 55) 0.041 201 908 264 482 406 888 151 048 113 799 132 278 991 662 684 823 086 956 544 × 2 = 0 + 0.082 403 816 528 964 813 776 302 096 227 598 264 557 983 325 369 646 173 913 088;
  • 56) 0.082 403 816 528 964 813 776 302 096 227 598 264 557 983 325 369 646 173 913 088 × 2 = 0 + 0.164 807 633 057 929 627 552 604 192 455 196 529 115 966 650 739 292 347 826 176;
  • 57) 0.164 807 633 057 929 627 552 604 192 455 196 529 115 966 650 739 292 347 826 176 × 2 = 0 + 0.329 615 266 115 859 255 105 208 384 910 393 058 231 933 301 478 584 695 652 352;
  • 58) 0.329 615 266 115 859 255 105 208 384 910 393 058 231 933 301 478 584 695 652 352 × 2 = 0 + 0.659 230 532 231 718 510 210 416 769 820 786 116 463 866 602 957 169 391 304 704;
  • 59) 0.659 230 532 231 718 510 210 416 769 820 786 116 463 866 602 957 169 391 304 704 × 2 = 1 + 0.318 461 064 463 437 020 420 833 539 641 572 232 927 733 205 914 338 782 609 408;

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:


0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2)

5. Positive number before normalization:

0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(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:


0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2) × 20 =

1.0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001(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):
1.0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001


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. 0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001 =

0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 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 1111 1000

Mantissa (52 bits) =
0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001


Decimal number 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 653 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1000 - 0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 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