Convert 751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510 to 64 Bit Double Precision IEEE 754 Binary Floating Point Standard, From a Number in Base 10 Decimal System

How to convert the decimal number 751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510(10)
to
64 bit double precision IEEE 754 binary floating point
(1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510 ÷ 2 = 375 570 585 575 550 555 520 555 570 575 535 606 605 755 905 055 805 405 555 505 555 755 + 0;
  • 375 570 585 575 550 555 520 555 570 575 535 606 605 755 905 055 805 405 555 505 555 755 ÷ 2 = 187 785 292 787 775 277 760 277 785 287 767 803 302 877 952 527 902 702 777 752 777 877 + 1;
  • 187 785 292 787 775 277 760 277 785 287 767 803 302 877 952 527 902 702 777 752 777 877 ÷ 2 = 93 892 646 393 887 638 880 138 892 643 883 901 651 438 976 263 951 351 388 876 388 938 + 1;
  • 93 892 646 393 887 638 880 138 892 643 883 901 651 438 976 263 951 351 388 876 388 938 ÷ 2 = 46 946 323 196 943 819 440 069 446 321 941 950 825 719 488 131 975 675 694 438 194 469 + 0;
  • 46 946 323 196 943 819 440 069 446 321 941 950 825 719 488 131 975 675 694 438 194 469 ÷ 2 = 23 473 161 598 471 909 720 034 723 160 970 975 412 859 744 065 987 837 847 219 097 234 + 1;
  • 23 473 161 598 471 909 720 034 723 160 970 975 412 859 744 065 987 837 847 219 097 234 ÷ 2 = 11 736 580 799 235 954 860 017 361 580 485 487 706 429 872 032 993 918 923 609 548 617 + 0;
  • 11 736 580 799 235 954 860 017 361 580 485 487 706 429 872 032 993 918 923 609 548 617 ÷ 2 = 5 868 290 399 617 977 430 008 680 790 242 743 853 214 936 016 496 959 461 804 774 308 + 1;
  • 5 868 290 399 617 977 430 008 680 790 242 743 853 214 936 016 496 959 461 804 774 308 ÷ 2 = 2 934 145 199 808 988 715 004 340 395 121 371 926 607 468 008 248 479 730 902 387 154 + 0;
  • 2 934 145 199 808 988 715 004 340 395 121 371 926 607 468 008 248 479 730 902 387 154 ÷ 2 = 1 467 072 599 904 494 357 502 170 197 560 685 963 303 734 004 124 239 865 451 193 577 + 0;
  • 1 467 072 599 904 494 357 502 170 197 560 685 963 303 734 004 124 239 865 451 193 577 ÷ 2 = 733 536 299 952 247 178 751 085 098 780 342 981 651 867 002 062 119 932 725 596 788 + 1;
  • 733 536 299 952 247 178 751 085 098 780 342 981 651 867 002 062 119 932 725 596 788 ÷ 2 = 366 768 149 976 123 589 375 542 549 390 171 490 825 933 501 031 059 966 362 798 394 + 0;
  • 366 768 149 976 123 589 375 542 549 390 171 490 825 933 501 031 059 966 362 798 394 ÷ 2 = 183 384 074 988 061 794 687 771 274 695 085 745 412 966 750 515 529 983 181 399 197 + 0;
  • 183 384 074 988 061 794 687 771 274 695 085 745 412 966 750 515 529 983 181 399 197 ÷ 2 = 91 692 037 494 030 897 343 885 637 347 542 872 706 483 375 257 764 991 590 699 598 + 1;
  • 91 692 037 494 030 897 343 885 637 347 542 872 706 483 375 257 764 991 590 699 598 ÷ 2 = 45 846 018 747 015 448 671 942 818 673 771 436 353 241 687 628 882 495 795 349 799 + 0;
  • 45 846 018 747 015 448 671 942 818 673 771 436 353 241 687 628 882 495 795 349 799 ÷ 2 = 22 923 009 373 507 724 335 971 409 336 885 718 176 620 843 814 441 247 897 674 899 + 1;
  • 22 923 009 373 507 724 335 971 409 336 885 718 176 620 843 814 441 247 897 674 899 ÷ 2 = 11 461 504 686 753 862 167 985 704 668 442 859 088 310 421 907 220 623 948 837 449 + 1;
  • 11 461 504 686 753 862 167 985 704 668 442 859 088 310 421 907 220 623 948 837 449 ÷ 2 = 5 730 752 343 376 931 083 992 852 334 221 429 544 155 210 953 610 311 974 418 724 + 1;
  • 5 730 752 343 376 931 083 992 852 334 221 429 544 155 210 953 610 311 974 418 724 ÷ 2 = 2 865 376 171 688 465 541 996 426 167 110 714 772 077 605 476 805 155 987 209 362 + 0;
  • 2 865 376 171 688 465 541 996 426 167 110 714 772 077 605 476 805 155 987 209 362 ÷ 2 = 1 432 688 085 844 232 770 998 213 083 555 357 386 038 802 738 402 577 993 604 681 + 0;
  • 1 432 688 085 844 232 770 998 213 083 555 357 386 038 802 738 402 577 993 604 681 ÷ 2 = 716 344 042 922 116 385 499 106 541 777 678 693 019 401 369 201 288 996 802 340 + 1;
  • 716 344 042 922 116 385 499 106 541 777 678 693 019 401 369 201 288 996 802 340 ÷ 2 = 358 172 021 461 058 192 749 553 270 888 839 346 509 700 684 600 644 498 401 170 + 0;
  • 358 172 021 461 058 192 749 553 270 888 839 346 509 700 684 600 644 498 401 170 ÷ 2 = 179 086 010 730 529 096 374 776 635 444 419 673 254 850 342 300 322 249 200 585 + 0;
  • 179 086 010 730 529 096 374 776 635 444 419 673 254 850 342 300 322 249 200 585 ÷ 2 = 89 543 005 365 264 548 187 388 317 722 209 836 627 425 171 150 161 124 600 292 + 1;
  • 89 543 005 365 264 548 187 388 317 722 209 836 627 425 171 150 161 124 600 292 ÷ 2 = 44 771 502 682 632 274 093 694 158 861 104 918 313 712 585 575 080 562 300 146 + 0;
  • 44 771 502 682 632 274 093 694 158 861 104 918 313 712 585 575 080 562 300 146 ÷ 2 = 22 385 751 341 316 137 046 847 079 430 552 459 156 856 292 787 540 281 150 073 + 0;
  • 22 385 751 341 316 137 046 847 079 430 552 459 156 856 292 787 540 281 150 073 ÷ 2 = 11 192 875 670 658 068 523 423 539 715 276 229 578 428 146 393 770 140 575 036 + 1;
  • 11 192 875 670 658 068 523 423 539 715 276 229 578 428 146 393 770 140 575 036 ÷ 2 = 5 596 437 835 329 034 261 711 769 857 638 114 789 214 073 196 885 070 287 518 + 0;
  • 5 596 437 835 329 034 261 711 769 857 638 114 789 214 073 196 885 070 287 518 ÷ 2 = 2 798 218 917 664 517 130 855 884 928 819 057 394 607 036 598 442 535 143 759 + 0;
  • 2 798 218 917 664 517 130 855 884 928 819 057 394 607 036 598 442 535 143 759 ÷ 2 = 1 399 109 458 832 258 565 427 942 464 409 528 697 303 518 299 221 267 571 879 + 1;
  • 1 399 109 458 832 258 565 427 942 464 409 528 697 303 518 299 221 267 571 879 ÷ 2 = 699 554 729 416 129 282 713 971 232 204 764 348 651 759 149 610 633 785 939 + 1;
  • 699 554 729 416 129 282 713 971 232 204 764 348 651 759 149 610 633 785 939 ÷ 2 = 349 777 364 708 064 641 356 985 616 102 382 174 325 879 574 805 316 892 969 + 1;
  • 349 777 364 708 064 641 356 985 616 102 382 174 325 879 574 805 316 892 969 ÷ 2 = 174 888 682 354 032 320 678 492 808 051 191 087 162 939 787 402 658 446 484 + 1;
  • 174 888 682 354 032 320 678 492 808 051 191 087 162 939 787 402 658 446 484 ÷ 2 = 87 444 341 177 016 160 339 246 404 025 595 543 581 469 893 701 329 223 242 + 0;
  • 87 444 341 177 016 160 339 246 404 025 595 543 581 469 893 701 329 223 242 ÷ 2 = 43 722 170 588 508 080 169 623 202 012 797 771 790 734 946 850 664 611 621 + 0;
  • 43 722 170 588 508 080 169 623 202 012 797 771 790 734 946 850 664 611 621 ÷ 2 = 21 861 085 294 254 040 084 811 601 006 398 885 895 367 473 425 332 305 810 + 1;
  • 21 861 085 294 254 040 084 811 601 006 398 885 895 367 473 425 332 305 810 ÷ 2 = 10 930 542 647 127 020 042 405 800 503 199 442 947 683 736 712 666 152 905 + 0;
  • 10 930 542 647 127 020 042 405 800 503 199 442 947 683 736 712 666 152 905 ÷ 2 = 5 465 271 323 563 510 021 202 900 251 599 721 473 841 868 356 333 076 452 + 1;
  • 5 465 271 323 563 510 021 202 900 251 599 721 473 841 868 356 333 076 452 ÷ 2 = 2 732 635 661 781 755 010 601 450 125 799 860 736 920 934 178 166 538 226 + 0;
  • 2 732 635 661 781 755 010 601 450 125 799 860 736 920 934 178 166 538 226 ÷ 2 = 1 366 317 830 890 877 505 300 725 062 899 930 368 460 467 089 083 269 113 + 0;
  • 1 366 317 830 890 877 505 300 725 062 899 930 368 460 467 089 083 269 113 ÷ 2 = 683 158 915 445 438 752 650 362 531 449 965 184 230 233 544 541 634 556 + 1;
  • 683 158 915 445 438 752 650 362 531 449 965 184 230 233 544 541 634 556 ÷ 2 = 341 579 457 722 719 376 325 181 265 724 982 592 115 116 772 270 817 278 + 0;
  • 341 579 457 722 719 376 325 181 265 724 982 592 115 116 772 270 817 278 ÷ 2 = 170 789 728 861 359 688 162 590 632 862 491 296 057 558 386 135 408 639 + 0;
  • 170 789 728 861 359 688 162 590 632 862 491 296 057 558 386 135 408 639 ÷ 2 = 85 394 864 430 679 844 081 295 316 431 245 648 028 779 193 067 704 319 + 1;
  • 85 394 864 430 679 844 081 295 316 431 245 648 028 779 193 067 704 319 ÷ 2 = 42 697 432 215 339 922 040 647 658 215 622 824 014 389 596 533 852 159 + 1;
  • 42 697 432 215 339 922 040 647 658 215 622 824 014 389 596 533 852 159 ÷ 2 = 21 348 716 107 669 961 020 323 829 107 811 412 007 194 798 266 926 079 + 1;
  • 21 348 716 107 669 961 020 323 829 107 811 412 007 194 798 266 926 079 ÷ 2 = 10 674 358 053 834 980 510 161 914 553 905 706 003 597 399 133 463 039 + 1;
  • 10 674 358 053 834 980 510 161 914 553 905 706 003 597 399 133 463 039 ÷ 2 = 5 337 179 026 917 490 255 080 957 276 952 853 001 798 699 566 731 519 + 1;
  • 5 337 179 026 917 490 255 080 957 276 952 853 001 798 699 566 731 519 ÷ 2 = 2 668 589 513 458 745 127 540 478 638 476 426 500 899 349 783 365 759 + 1;
  • 2 668 589 513 458 745 127 540 478 638 476 426 500 899 349 783 365 759 ÷ 2 = 1 334 294 756 729 372 563 770 239 319 238 213 250 449 674 891 682 879 + 1;
  • 1 334 294 756 729 372 563 770 239 319 238 213 250 449 674 891 682 879 ÷ 2 = 667 147 378 364 686 281 885 119 659 619 106 625 224 837 445 841 439 + 1;
  • 667 147 378 364 686 281 885 119 659 619 106 625 224 837 445 841 439 ÷ 2 = 333 573 689 182 343 140 942 559 829 809 553 312 612 418 722 920 719 + 1;
  • 333 573 689 182 343 140 942 559 829 809 553 312 612 418 722 920 719 ÷ 2 = 166 786 844 591 171 570 471 279 914 904 776 656 306 209 361 460 359 + 1;
  • 166 786 844 591 171 570 471 279 914 904 776 656 306 209 361 460 359 ÷ 2 = 83 393 422 295 585 785 235 639 957 452 388 328 153 104 680 730 179 + 1;
  • 83 393 422 295 585 785 235 639 957 452 388 328 153 104 680 730 179 ÷ 2 = 41 696 711 147 792 892 617 819 978 726 194 164 076 552 340 365 089 + 1;
  • 41 696 711 147 792 892 617 819 978 726 194 164 076 552 340 365 089 ÷ 2 = 20 848 355 573 896 446 308 909 989 363 097 082 038 276 170 182 544 + 1;
  • 20 848 355 573 896 446 308 909 989 363 097 082 038 276 170 182 544 ÷ 2 = 10 424 177 786 948 223 154 454 994 681 548 541 019 138 085 091 272 + 0;
  • 10 424 177 786 948 223 154 454 994 681 548 541 019 138 085 091 272 ÷ 2 = 5 212 088 893 474 111 577 227 497 340 774 270 509 569 042 545 636 + 0;
  • 5 212 088 893 474 111 577 227 497 340 774 270 509 569 042 545 636 ÷ 2 = 2 606 044 446 737 055 788 613 748 670 387 135 254 784 521 272 818 + 0;
  • 2 606 044 446 737 055 788 613 748 670 387 135 254 784 521 272 818 ÷ 2 = 1 303 022 223 368 527 894 306 874 335 193 567 627 392 260 636 409 + 0;
  • 1 303 022 223 368 527 894 306 874 335 193 567 627 392 260 636 409 ÷ 2 = 651 511 111 684 263 947 153 437 167 596 783 813 696 130 318 204 + 1;
  • 651 511 111 684 263 947 153 437 167 596 783 813 696 130 318 204 ÷ 2 = 325 755 555 842 131 973 576 718 583 798 391 906 848 065 159 102 + 0;
  • 325 755 555 842 131 973 576 718 583 798 391 906 848 065 159 102 ÷ 2 = 162 877 777 921 065 986 788 359 291 899 195 953 424 032 579 551 + 0;
  • 162 877 777 921 065 986 788 359 291 899 195 953 424 032 579 551 ÷ 2 = 81 438 888 960 532 993 394 179 645 949 597 976 712 016 289 775 + 1;
  • 81 438 888 960 532 993 394 179 645 949 597 976 712 016 289 775 ÷ 2 = 40 719 444 480 266 496 697 089 822 974 798 988 356 008 144 887 + 1;
  • 40 719 444 480 266 496 697 089 822 974 798 988 356 008 144 887 ÷ 2 = 20 359 722 240 133 248 348 544 911 487 399 494 178 004 072 443 + 1;
  • 20 359 722 240 133 248 348 544 911 487 399 494 178 004 072 443 ÷ 2 = 10 179 861 120 066 624 174 272 455 743 699 747 089 002 036 221 + 1;
  • 10 179 861 120 066 624 174 272 455 743 699 747 089 002 036 221 ÷ 2 = 5 089 930 560 033 312 087 136 227 871 849 873 544 501 018 110 + 1;
  • 5 089 930 560 033 312 087 136 227 871 849 873 544 501 018 110 ÷ 2 = 2 544 965 280 016 656 043 568 113 935 924 936 772 250 509 055 + 0;
  • 2 544 965 280 016 656 043 568 113 935 924 936 772 250 509 055 ÷ 2 = 1 272 482 640 008 328 021 784 056 967 962 468 386 125 254 527 + 1;
  • 1 272 482 640 008 328 021 784 056 967 962 468 386 125 254 527 ÷ 2 = 636 241 320 004 164 010 892 028 483 981 234 193 062 627 263 + 1;
  • 636 241 320 004 164 010 892 028 483 981 234 193 062 627 263 ÷ 2 = 318 120 660 002 082 005 446 014 241 990 617 096 531 313 631 + 1;
  • 318 120 660 002 082 005 446 014 241 990 617 096 531 313 631 ÷ 2 = 159 060 330 001 041 002 723 007 120 995 308 548 265 656 815 + 1;
  • 159 060 330 001 041 002 723 007 120 995 308 548 265 656 815 ÷ 2 = 79 530 165 000 520 501 361 503 560 497 654 274 132 828 407 + 1;
  • 79 530 165 000 520 501 361 503 560 497 654 274 132 828 407 ÷ 2 = 39 765 082 500 260 250 680 751 780 248 827 137 066 414 203 + 1;
  • 39 765 082 500 260 250 680 751 780 248 827 137 066 414 203 ÷ 2 = 19 882 541 250 130 125 340 375 890 124 413 568 533 207 101 + 1;
  • 19 882 541 250 130 125 340 375 890 124 413 568 533 207 101 ÷ 2 = 9 941 270 625 065 062 670 187 945 062 206 784 266 603 550 + 1;
  • 9 941 270 625 065 062 670 187 945 062 206 784 266 603 550 ÷ 2 = 4 970 635 312 532 531 335 093 972 531 103 392 133 301 775 + 0;
  • 4 970 635 312 532 531 335 093 972 531 103 392 133 301 775 ÷ 2 = 2 485 317 656 266 265 667 546 986 265 551 696 066 650 887 + 1;
  • 2 485 317 656 266 265 667 546 986 265 551 696 066 650 887 ÷ 2 = 1 242 658 828 133 132 833 773 493 132 775 848 033 325 443 + 1;
  • 1 242 658 828 133 132 833 773 493 132 775 848 033 325 443 ÷ 2 = 621 329 414 066 566 416 886 746 566 387 924 016 662 721 + 1;
  • 621 329 414 066 566 416 886 746 566 387 924 016 662 721 ÷ 2 = 310 664 707 033 283 208 443 373 283 193 962 008 331 360 + 1;
  • 310 664 707 033 283 208 443 373 283 193 962 008 331 360 ÷ 2 = 155 332 353 516 641 604 221 686 641 596 981 004 165 680 + 0;
  • 155 332 353 516 641 604 221 686 641 596 981 004 165 680 ÷ 2 = 77 666 176 758 320 802 110 843 320 798 490 502 082 840 + 0;
  • 77 666 176 758 320 802 110 843 320 798 490 502 082 840 ÷ 2 = 38 833 088 379 160 401 055 421 660 399 245 251 041 420 + 0;
  • 38 833 088 379 160 401 055 421 660 399 245 251 041 420 ÷ 2 = 19 416 544 189 580 200 527 710 830 199 622 625 520 710 + 0;
  • 19 416 544 189 580 200 527 710 830 199 622 625 520 710 ÷ 2 = 9 708 272 094 790 100 263 855 415 099 811 312 760 355 + 0;
  • 9 708 272 094 790 100 263 855 415 099 811 312 760 355 ÷ 2 = 4 854 136 047 395 050 131 927 707 549 905 656 380 177 + 1;
  • 4 854 136 047 395 050 131 927 707 549 905 656 380 177 ÷ 2 = 2 427 068 023 697 525 065 963 853 774 952 828 190 088 + 1;
  • 2 427 068 023 697 525 065 963 853 774 952 828 190 088 ÷ 2 = 1 213 534 011 848 762 532 981 926 887 476 414 095 044 + 0;
  • 1 213 534 011 848 762 532 981 926 887 476 414 095 044 ÷ 2 = 606 767 005 924 381 266 490 963 443 738 207 047 522 + 0;
  • 606 767 005 924 381 266 490 963 443 738 207 047 522 ÷ 2 = 303 383 502 962 190 633 245 481 721 869 103 523 761 + 0;
  • 303 383 502 962 190 633 245 481 721 869 103 523 761 ÷ 2 = 151 691 751 481 095 316 622 740 860 934 551 761 880 + 1;
  • 151 691 751 481 095 316 622 740 860 934 551 761 880 ÷ 2 = 75 845 875 740 547 658 311 370 430 467 275 880 940 + 0;
  • 75 845 875 740 547 658 311 370 430 467 275 880 940 ÷ 2 = 37 922 937 870 273 829 155 685 215 233 637 940 470 + 0;
  • 37 922 937 870 273 829 155 685 215 233 637 940 470 ÷ 2 = 18 961 468 935 136 914 577 842 607 616 818 970 235 + 0;
  • 18 961 468 935 136 914 577 842 607 616 818 970 235 ÷ 2 = 9 480 734 467 568 457 288 921 303 808 409 485 117 + 1;
  • 9 480 734 467 568 457 288 921 303 808 409 485 117 ÷ 2 = 4 740 367 233 784 228 644 460 651 904 204 742 558 + 1;
  • 4 740 367 233 784 228 644 460 651 904 204 742 558 ÷ 2 = 2 370 183 616 892 114 322 230 325 952 102 371 279 + 0;
  • 2 370 183 616 892 114 322 230 325 952 102 371 279 ÷ 2 = 1 185 091 808 446 057 161 115 162 976 051 185 639 + 1;
  • 1 185 091 808 446 057 161 115 162 976 051 185 639 ÷ 2 = 592 545 904 223 028 580 557 581 488 025 592 819 + 1;
  • 592 545 904 223 028 580 557 581 488 025 592 819 ÷ 2 = 296 272 952 111 514 290 278 790 744 012 796 409 + 1;
  • 296 272 952 111 514 290 278 790 744 012 796 409 ÷ 2 = 148 136 476 055 757 145 139 395 372 006 398 204 + 1;
  • 148 136 476 055 757 145 139 395 372 006 398 204 ÷ 2 = 74 068 238 027 878 572 569 697 686 003 199 102 + 0;
  • 74 068 238 027 878 572 569 697 686 003 199 102 ÷ 2 = 37 034 119 013 939 286 284 848 843 001 599 551 + 0;
  • 37 034 119 013 939 286 284 848 843 001 599 551 ÷ 2 = 18 517 059 506 969 643 142 424 421 500 799 775 + 1;
  • 18 517 059 506 969 643 142 424 421 500 799 775 ÷ 2 = 9 258 529 753 484 821 571 212 210 750 399 887 + 1;
  • 9 258 529 753 484 821 571 212 210 750 399 887 ÷ 2 = 4 629 264 876 742 410 785 606 105 375 199 943 + 1;
  • 4 629 264 876 742 410 785 606 105 375 199 943 ÷ 2 = 2 314 632 438 371 205 392 803 052 687 599 971 + 1;
  • 2 314 632 438 371 205 392 803 052 687 599 971 ÷ 2 = 1 157 316 219 185 602 696 401 526 343 799 985 + 1;
  • 1 157 316 219 185 602 696 401 526 343 799 985 ÷ 2 = 578 658 109 592 801 348 200 763 171 899 992 + 1;
  • 578 658 109 592 801 348 200 763 171 899 992 ÷ 2 = 289 329 054 796 400 674 100 381 585 949 996 + 0;
  • 289 329 054 796 400 674 100 381 585 949 996 ÷ 2 = 144 664 527 398 200 337 050 190 792 974 998 + 0;
  • 144 664 527 398 200 337 050 190 792 974 998 ÷ 2 = 72 332 263 699 100 168 525 095 396 487 499 + 0;
  • 72 332 263 699 100 168 525 095 396 487 499 ÷ 2 = 36 166 131 849 550 084 262 547 698 243 749 + 1;
  • 36 166 131 849 550 084 262 547 698 243 749 ÷ 2 = 18 083 065 924 775 042 131 273 849 121 874 + 1;
  • 18 083 065 924 775 042 131 273 849 121 874 ÷ 2 = 9 041 532 962 387 521 065 636 924 560 937 + 0;
  • 9 041 532 962 387 521 065 636 924 560 937 ÷ 2 = 4 520 766 481 193 760 532 818 462 280 468 + 1;
  • 4 520 766 481 193 760 532 818 462 280 468 ÷ 2 = 2 260 383 240 596 880 266 409 231 140 234 + 0;
  • 2 260 383 240 596 880 266 409 231 140 234 ÷ 2 = 1 130 191 620 298 440 133 204 615 570 117 + 0;
  • 1 130 191 620 298 440 133 204 615 570 117 ÷ 2 = 565 095 810 149 220 066 602 307 785 058 + 1;
  • 565 095 810 149 220 066 602 307 785 058 ÷ 2 = 282 547 905 074 610 033 301 153 892 529 + 0;
  • 282 547 905 074 610 033 301 153 892 529 ÷ 2 = 141 273 952 537 305 016 650 576 946 264 + 1;
  • 141 273 952 537 305 016 650 576 946 264 ÷ 2 = 70 636 976 268 652 508 325 288 473 132 + 0;
  • 70 636 976 268 652 508 325 288 473 132 ÷ 2 = 35 318 488 134 326 254 162 644 236 566 + 0;
  • 35 318 488 134 326 254 162 644 236 566 ÷ 2 = 17 659 244 067 163 127 081 322 118 283 + 0;
  • 17 659 244 067 163 127 081 322 118 283 ÷ 2 = 8 829 622 033 581 563 540 661 059 141 + 1;
  • 8 829 622 033 581 563 540 661 059 141 ÷ 2 = 4 414 811 016 790 781 770 330 529 570 + 1;
  • 4 414 811 016 790 781 770 330 529 570 ÷ 2 = 2 207 405 508 395 390 885 165 264 785 + 0;
  • 2 207 405 508 395 390 885 165 264 785 ÷ 2 = 1 103 702 754 197 695 442 582 632 392 + 1;
  • 1 103 702 754 197 695 442 582 632 392 ÷ 2 = 551 851 377 098 847 721 291 316 196 + 0;
  • 551 851 377 098 847 721 291 316 196 ÷ 2 = 275 925 688 549 423 860 645 658 098 + 0;
  • 275 925 688 549 423 860 645 658 098 ÷ 2 = 137 962 844 274 711 930 322 829 049 + 0;
  • 137 962 844 274 711 930 322 829 049 ÷ 2 = 68 981 422 137 355 965 161 414 524 + 1;
  • 68 981 422 137 355 965 161 414 524 ÷ 2 = 34 490 711 068 677 982 580 707 262 + 0;
  • 34 490 711 068 677 982 580 707 262 ÷ 2 = 17 245 355 534 338 991 290 353 631 + 0;
  • 17 245 355 534 338 991 290 353 631 ÷ 2 = 8 622 677 767 169 495 645 176 815 + 1;
  • 8 622 677 767 169 495 645 176 815 ÷ 2 = 4 311 338 883 584 747 822 588 407 + 1;
  • 4 311 338 883 584 747 822 588 407 ÷ 2 = 2 155 669 441 792 373 911 294 203 + 1;
  • 2 155 669 441 792 373 911 294 203 ÷ 2 = 1 077 834 720 896 186 955 647 101 + 1;
  • 1 077 834 720 896 186 955 647 101 ÷ 2 = 538 917 360 448 093 477 823 550 + 1;
  • 538 917 360 448 093 477 823 550 ÷ 2 = 269 458 680 224 046 738 911 775 + 0;
  • 269 458 680 224 046 738 911 775 ÷ 2 = 134 729 340 112 023 369 455 887 + 1;
  • 134 729 340 112 023 369 455 887 ÷ 2 = 67 364 670 056 011 684 727 943 + 1;
  • 67 364 670 056 011 684 727 943 ÷ 2 = 33 682 335 028 005 842 363 971 + 1;
  • 33 682 335 028 005 842 363 971 ÷ 2 = 16 841 167 514 002 921 181 985 + 1;
  • 16 841 167 514 002 921 181 985 ÷ 2 = 8 420 583 757 001 460 590 992 + 1;
  • 8 420 583 757 001 460 590 992 ÷ 2 = 4 210 291 878 500 730 295 496 + 0;
  • 4 210 291 878 500 730 295 496 ÷ 2 = 2 105 145 939 250 365 147 748 + 0;
  • 2 105 145 939 250 365 147 748 ÷ 2 = 1 052 572 969 625 182 573 874 + 0;
  • 1 052 572 969 625 182 573 874 ÷ 2 = 526 286 484 812 591 286 937 + 0;
  • 526 286 484 812 591 286 937 ÷ 2 = 263 143 242 406 295 643 468 + 1;
  • 263 143 242 406 295 643 468 ÷ 2 = 131 571 621 203 147 821 734 + 0;
  • 131 571 621 203 147 821 734 ÷ 2 = 65 785 810 601 573 910 867 + 0;
  • 65 785 810 601 573 910 867 ÷ 2 = 32 892 905 300 786 955 433 + 1;
  • 32 892 905 300 786 955 433 ÷ 2 = 16 446 452 650 393 477 716 + 1;
  • 16 446 452 650 393 477 716 ÷ 2 = 8 223 226 325 196 738 858 + 0;
  • 8 223 226 325 196 738 858 ÷ 2 = 4 111 613 162 598 369 429 + 0;
  • 4 111 613 162 598 369 429 ÷ 2 = 2 055 806 581 299 184 714 + 1;
  • 2 055 806 581 299 184 714 ÷ 2 = 1 027 903 290 649 592 357 + 0;
  • 1 027 903 290 649 592 357 ÷ 2 = 513 951 645 324 796 178 + 1;
  • 513 951 645 324 796 178 ÷ 2 = 256 975 822 662 398 089 + 0;
  • 256 975 822 662 398 089 ÷ 2 = 128 487 911 331 199 044 + 1;
  • 128 487 911 331 199 044 ÷ 2 = 64 243 955 665 599 522 + 0;
  • 64 243 955 665 599 522 ÷ 2 = 32 121 977 832 799 761 + 0;
  • 32 121 977 832 799 761 ÷ 2 = 16 060 988 916 399 880 + 1;
  • 16 060 988 916 399 880 ÷ 2 = 8 030 494 458 199 940 + 0;
  • 8 030 494 458 199 940 ÷ 2 = 4 015 247 229 099 970 + 0;
  • 4 015 247 229 099 970 ÷ 2 = 2 007 623 614 549 985 + 0;
  • 2 007 623 614 549 985 ÷ 2 = 1 003 811 807 274 992 + 1;
  • 1 003 811 807 274 992 ÷ 2 = 501 905 903 637 496 + 0;
  • 501 905 903 637 496 ÷ 2 = 250 952 951 818 748 + 0;
  • 250 952 951 818 748 ÷ 2 = 125 476 475 909 374 + 0;
  • 125 476 475 909 374 ÷ 2 = 62 738 237 954 687 + 0;
  • 62 738 237 954 687 ÷ 2 = 31 369 118 977 343 + 1;
  • 31 369 118 977 343 ÷ 2 = 15 684 559 488 671 + 1;
  • 15 684 559 488 671 ÷ 2 = 7 842 279 744 335 + 1;
  • 7 842 279 744 335 ÷ 2 = 3 921 139 872 167 + 1;
  • 3 921 139 872 167 ÷ 2 = 1 960 569 936 083 + 1;
  • 1 960 569 936 083 ÷ 2 = 980 284 968 041 + 1;
  • 980 284 968 041 ÷ 2 = 490 142 484 020 + 1;
  • 490 142 484 020 ÷ 2 = 245 071 242 010 + 0;
  • 245 071 242 010 ÷ 2 = 122 535 621 005 + 0;
  • 122 535 621 005 ÷ 2 = 61 267 810 502 + 1;
  • 61 267 810 502 ÷ 2 = 30 633 905 251 + 0;
  • 30 633 905 251 ÷ 2 = 15 316 952 625 + 1;
  • 15 316 952 625 ÷ 2 = 7 658 476 312 + 1;
  • 7 658 476 312 ÷ 2 = 3 829 238 156 + 0;
  • 3 829 238 156 ÷ 2 = 1 914 619 078 + 0;
  • 1 914 619 078 ÷ 2 = 957 309 539 + 0;
  • 957 309 539 ÷ 2 = 478 654 769 + 1;
  • 478 654 769 ÷ 2 = 239 327 384 + 1;
  • 239 327 384 ÷ 2 = 119 663 692 + 0;
  • 119 663 692 ÷ 2 = 59 831 846 + 0;
  • 59 831 846 ÷ 2 = 29 915 923 + 0;
  • 29 915 923 ÷ 2 = 14 957 961 + 1;
  • 14 957 961 ÷ 2 = 7 478 980 + 1;
  • 7 478 980 ÷ 2 = 3 739 490 + 0;
  • 3 739 490 ÷ 2 = 1 869 745 + 0;
  • 1 869 745 ÷ 2 = 934 872 + 1;
  • 934 872 ÷ 2 = 467 436 + 0;
  • 467 436 ÷ 2 = 233 718 + 0;
  • 233 718 ÷ 2 = 116 859 + 0;
  • 116 859 ÷ 2 = 58 429 + 1;
  • 58 429 ÷ 2 = 29 214 + 1;
  • 29 214 ÷ 2 = 14 607 + 0;
  • 14 607 ÷ 2 = 7 303 + 1;
  • 7 303 ÷ 2 = 3 651 + 1;
  • 3 651 ÷ 2 = 1 825 + 1;
  • 1 825 ÷ 2 = 912 + 1;
  • 912 ÷ 2 = 456 + 0;
  • 456 ÷ 2 = 228 + 0;
  • 228 ÷ 2 = 114 + 0;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the integer part of the number, by taking all the remainders starting from the bottom of the list constructed above:

751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510(10) =


111 0010 0001 1110 1100 0100 1100 0110 0011 0100 1111 1110 0001 0001 0010 1010 0110 0100 0011 1110 1111 1001 0001 0110 0010 1001 0110 0011 1111 0011 1101 1000 1000 1100 0001 1110 1111 1111 0111 1100 1000 0111 1111 1111 1100 1001 0100 1111 0010 0100 1001 1101 0010 0101 0110(2)

3. Normalize the binary representation of the number, shifting the decimal mark 218 positions to the left so that only one non zero digit remains to the left of it:

751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510(10) =


111 0010 0001 1110 1100 0100 1100 0110 0011 0100 1111 1110 0001 0001 0010 1010 0110 0100 0011 1110 1111 1001 0001 0110 0010 1001 0110 0011 1111 0011 1101 1000 1000 1100 0001 1110 1111 1111 0111 1100 1000 0111 1111 1111 1100 1001 0100 1111 0010 0100 1001 1101 0010 0101 0110(2) =


111 0010 0001 1110 1100 0100 1100 0110 0011 0100 1111 1110 0001 0001 0010 1010 0110 0100 0011 1110 1111 1001 0001 0110 0010 1001 0110 0011 1111 0011 1101 1000 1000 1100 0001 1110 1111 1111 0111 1100 1000 0111 1111 1111 1100 1001 0100 1111 0010 0100 1001 1101 0010 0101 0110(2) × 20 =


1.1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100 0100 1010 1001 1001 0000 1111 1011 1110 0100 0101 1000 1010 0101 1000 1111 1100 1111 0110 0010 0011 0000 0111 1011 1111 1101 1111 0010 0001 1111 1111 1111 0010 0101 0011 1100 1001 0010 0111 0100 1001 0101 10(2) × 2218

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): 218


Mantissa (not normalized): 1.1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100 0100 1010 1001 1001 0000 1111 1011 1110 0100 0101 1000 1010 0101 1000 1111 1100 1111 0110 0010 0011 0000 0111 1011 1111 1101 1111 0010 0001 1111 1111 1111 0010 0101 0011 1100 1001 0010 0111 0100 1001 0101 10

4. 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:

Exponent (adjusted) =


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


218 + 2(11-1) - 1 =


(218 + 1 023)(10) =


1 241(10)


  • division = quotient + remainder;
  • 1 241 ÷ 2 = 620 + 1;
  • 620 ÷ 2 = 310 + 0;
  • 310 ÷ 2 = 155 + 0;
  • 155 ÷ 2 = 77 + 1;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

Exponent (adjusted) =


1241(10) =


100 1101 1001(2)

5. Normalize mantissa, remove the leading (the leftmost) bit, since it's allways 1 (and the decimal point, if the case) then adjust its length to 52 bits, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...):

Mantissa (normalized) =


1. 1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100 01 0010 1010 0110 0100 0011 1110 1111 1001 0001 0110 0010 1001 0110 0011 1111 0011 1101 1000 1000 1100 0001 1110 1111 1111 0111 1100 1000 0111 1111 1111 1100 1001 0100 1111 0010 0100 1001 1101 0010 0101 0110 =


1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100

Conclusion:

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) =
100 1101 1001


Mantissa (52 bits) =
1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100

Number 751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510 converted from decimal system (base 10)
to
64 bit double precision IEEE 754 binary floating point:
0 - 100 1101 1001 - 1100 1000 0111 1011 0001 0011 0001 1000 1101 0011 1111 1000 0100

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 1

      59
    • 1

      58
    • 0

      57
    • 1

      56
    • 1

      55
    • 0

      54
    • 0

      53
    • 1

      52
  • Mantissa (52 bits):

    • 1

      51
    • 1

      50
    • 0

      49
    • 0

      48
    • 1

      47
    • 0

      46
    • 0

      45
    • 0

      44
    • 0

      43
    • 1

      42
    • 1

      41
    • 1

      40
    • 1

      39
    • 0

      38
    • 1

      37
    • 1

      36
    • 0

      35
    • 0

      34
    • 0

      33
    • 1

      32
    • 0

      31
    • 0

      30
    • 1

      29
    • 1

      28
    • 0

      27
    • 0

      26
    • 0

      25
    • 1

      24
    • 1

      23
    • 0

      22
    • 0

      21
    • 0

      20
    • 1

      19
    • 1

      18
    • 0

      17
    • 1

      16
    • 0

      15
    • 0

      14
    • 1

      13
    • 1

      12
    • 1

      11
    • 1

      10
    • 1

      9
    • 1

      8
    • 1

      7
    • 0

      6
    • 0

      5
    • 0

      4
    • 0

      3
    • 1

      2
    • 0

      1
    • 0

      0

751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510 = ? ... 751 141 171 151 101 111 041 111 141 151 071 213 211 511 810 111 610 811 111 011 111 510 = ?


Convert to 64 bit double precision IEEE 754 binary floating point standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes one bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

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