0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9(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
0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9(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: 0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9.

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) 0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9 × 2 = 0 + 0.000 000 000 222 022 220 220 222 200 000 022 202 002 222 291 8;
  • 2) 0.000 000 000 222 022 220 220 222 200 000 022 202 002 222 291 8 × 2 = 0 + 0.000 000 000 444 044 440 440 444 400 000 044 404 004 444 583 6;
  • 3) 0.000 000 000 444 044 440 440 444 400 000 044 404 004 444 583 6 × 2 = 0 + 0.000 000 000 888 088 880 880 888 800 000 088 808 008 889 167 2;
  • 4) 0.000 000 000 888 088 880 880 888 800 000 088 808 008 889 167 2 × 2 = 0 + 0.000 000 001 776 177 761 761 777 600 000 177 616 017 778 334 4;
  • 5) 0.000 000 001 776 177 761 761 777 600 000 177 616 017 778 334 4 × 2 = 0 + 0.000 000 003 552 355 523 523 555 200 000 355 232 035 556 668 8;
  • 6) 0.000 000 003 552 355 523 523 555 200 000 355 232 035 556 668 8 × 2 = 0 + 0.000 000 007 104 711 047 047 110 400 000 710 464 071 113 337 6;
  • 7) 0.000 000 007 104 711 047 047 110 400 000 710 464 071 113 337 6 × 2 = 0 + 0.000 000 014 209 422 094 094 220 800 001 420 928 142 226 675 2;
  • 8) 0.000 000 014 209 422 094 094 220 800 001 420 928 142 226 675 2 × 2 = 0 + 0.000 000 028 418 844 188 188 441 600 002 841 856 284 453 350 4;
  • 9) 0.000 000 028 418 844 188 188 441 600 002 841 856 284 453 350 4 × 2 = 0 + 0.000 000 056 837 688 376 376 883 200 005 683 712 568 906 700 8;
  • 10) 0.000 000 056 837 688 376 376 883 200 005 683 712 568 906 700 8 × 2 = 0 + 0.000 000 113 675 376 752 753 766 400 011 367 425 137 813 401 6;
  • 11) 0.000 000 113 675 376 752 753 766 400 011 367 425 137 813 401 6 × 2 = 0 + 0.000 000 227 350 753 505 507 532 800 022 734 850 275 626 803 2;
  • 12) 0.000 000 227 350 753 505 507 532 800 022 734 850 275 626 803 2 × 2 = 0 + 0.000 000 454 701 507 011 015 065 600 045 469 700 551 253 606 4;
  • 13) 0.000 000 454 701 507 011 015 065 600 045 469 700 551 253 606 4 × 2 = 0 + 0.000 000 909 403 014 022 030 131 200 090 939 401 102 507 212 8;
  • 14) 0.000 000 909 403 014 022 030 131 200 090 939 401 102 507 212 8 × 2 = 0 + 0.000 001 818 806 028 044 060 262 400 181 878 802 205 014 425 6;
  • 15) 0.000 001 818 806 028 044 060 262 400 181 878 802 205 014 425 6 × 2 = 0 + 0.000 003 637 612 056 088 120 524 800 363 757 604 410 028 851 2;
  • 16) 0.000 003 637 612 056 088 120 524 800 363 757 604 410 028 851 2 × 2 = 0 + 0.000 007 275 224 112 176 241 049 600 727 515 208 820 057 702 4;
  • 17) 0.000 007 275 224 112 176 241 049 600 727 515 208 820 057 702 4 × 2 = 0 + 0.000 014 550 448 224 352 482 099 201 455 030 417 640 115 404 8;
  • 18) 0.000 014 550 448 224 352 482 099 201 455 030 417 640 115 404 8 × 2 = 0 + 0.000 029 100 896 448 704 964 198 402 910 060 835 280 230 809 6;
  • 19) 0.000 029 100 896 448 704 964 198 402 910 060 835 280 230 809 6 × 2 = 0 + 0.000 058 201 792 897 409 928 396 805 820 121 670 560 461 619 2;
  • 20) 0.000 058 201 792 897 409 928 396 805 820 121 670 560 461 619 2 × 2 = 0 + 0.000 116 403 585 794 819 856 793 611 640 243 341 120 923 238 4;
  • 21) 0.000 116 403 585 794 819 856 793 611 640 243 341 120 923 238 4 × 2 = 0 + 0.000 232 807 171 589 639 713 587 223 280 486 682 241 846 476 8;
  • 22) 0.000 232 807 171 589 639 713 587 223 280 486 682 241 846 476 8 × 2 = 0 + 0.000 465 614 343 179 279 427 174 446 560 973 364 483 692 953 6;
  • 23) 0.000 465 614 343 179 279 427 174 446 560 973 364 483 692 953 6 × 2 = 0 + 0.000 931 228 686 358 558 854 348 893 121 946 728 967 385 907 2;
  • 24) 0.000 931 228 686 358 558 854 348 893 121 946 728 967 385 907 2 × 2 = 0 + 0.001 862 457 372 717 117 708 697 786 243 893 457 934 771 814 4;
  • 25) 0.001 862 457 372 717 117 708 697 786 243 893 457 934 771 814 4 × 2 = 0 + 0.003 724 914 745 434 235 417 395 572 487 786 915 869 543 628 8;
  • 26) 0.003 724 914 745 434 235 417 395 572 487 786 915 869 543 628 8 × 2 = 0 + 0.007 449 829 490 868 470 834 791 144 975 573 831 739 087 257 6;
  • 27) 0.007 449 829 490 868 470 834 791 144 975 573 831 739 087 257 6 × 2 = 0 + 0.014 899 658 981 736 941 669 582 289 951 147 663 478 174 515 2;
  • 28) 0.014 899 658 981 736 941 669 582 289 951 147 663 478 174 515 2 × 2 = 0 + 0.029 799 317 963 473 883 339 164 579 902 295 326 956 349 030 4;
  • 29) 0.029 799 317 963 473 883 339 164 579 902 295 326 956 349 030 4 × 2 = 0 + 0.059 598 635 926 947 766 678 329 159 804 590 653 912 698 060 8;
  • 30) 0.059 598 635 926 947 766 678 329 159 804 590 653 912 698 060 8 × 2 = 0 + 0.119 197 271 853 895 533 356 658 319 609 181 307 825 396 121 6;
  • 31) 0.119 197 271 853 895 533 356 658 319 609 181 307 825 396 121 6 × 2 = 0 + 0.238 394 543 707 791 066 713 316 639 218 362 615 650 792 243 2;
  • 32) 0.238 394 543 707 791 066 713 316 639 218 362 615 650 792 243 2 × 2 = 0 + 0.476 789 087 415 582 133 426 633 278 436 725 231 301 584 486 4;
  • 33) 0.476 789 087 415 582 133 426 633 278 436 725 231 301 584 486 4 × 2 = 0 + 0.953 578 174 831 164 266 853 266 556 873 450 462 603 168 972 8;
  • 34) 0.953 578 174 831 164 266 853 266 556 873 450 462 603 168 972 8 × 2 = 1 + 0.907 156 349 662 328 533 706 533 113 746 900 925 206 337 945 6;
  • 35) 0.907 156 349 662 328 533 706 533 113 746 900 925 206 337 945 6 × 2 = 1 + 0.814 312 699 324 657 067 413 066 227 493 801 850 412 675 891 2;
  • 36) 0.814 312 699 324 657 067 413 066 227 493 801 850 412 675 891 2 × 2 = 1 + 0.628 625 398 649 314 134 826 132 454 987 603 700 825 351 782 4;
  • 37) 0.628 625 398 649 314 134 826 132 454 987 603 700 825 351 782 4 × 2 = 1 + 0.257 250 797 298 628 269 652 264 909 975 207 401 650 703 564 8;
  • 38) 0.257 250 797 298 628 269 652 264 909 975 207 401 650 703 564 8 × 2 = 0 + 0.514 501 594 597 256 539 304 529 819 950 414 803 301 407 129 6;
  • 39) 0.514 501 594 597 256 539 304 529 819 950 414 803 301 407 129 6 × 2 = 1 + 0.029 003 189 194 513 078 609 059 639 900 829 606 602 814 259 2;
  • 40) 0.029 003 189 194 513 078 609 059 639 900 829 606 602 814 259 2 × 2 = 0 + 0.058 006 378 389 026 157 218 119 279 801 659 213 205 628 518 4;
  • 41) 0.058 006 378 389 026 157 218 119 279 801 659 213 205 628 518 4 × 2 = 0 + 0.116 012 756 778 052 314 436 238 559 603 318 426 411 257 036 8;
  • 42) 0.116 012 756 778 052 314 436 238 559 603 318 426 411 257 036 8 × 2 = 0 + 0.232 025 513 556 104 628 872 477 119 206 636 852 822 514 073 6;
  • 43) 0.232 025 513 556 104 628 872 477 119 206 636 852 822 514 073 6 × 2 = 0 + 0.464 051 027 112 209 257 744 954 238 413 273 705 645 028 147 2;
  • 44) 0.464 051 027 112 209 257 744 954 238 413 273 705 645 028 147 2 × 2 = 0 + 0.928 102 054 224 418 515 489 908 476 826 547 411 290 056 294 4;
  • 45) 0.928 102 054 224 418 515 489 908 476 826 547 411 290 056 294 4 × 2 = 1 + 0.856 204 108 448 837 030 979 816 953 653 094 822 580 112 588 8;
  • 46) 0.856 204 108 448 837 030 979 816 953 653 094 822 580 112 588 8 × 2 = 1 + 0.712 408 216 897 674 061 959 633 907 306 189 645 160 225 177 6;
  • 47) 0.712 408 216 897 674 061 959 633 907 306 189 645 160 225 177 6 × 2 = 1 + 0.424 816 433 795 348 123 919 267 814 612 379 290 320 450 355 2;
  • 48) 0.424 816 433 795 348 123 919 267 814 612 379 290 320 450 355 2 × 2 = 0 + 0.849 632 867 590 696 247 838 535 629 224 758 580 640 900 710 4;
  • 49) 0.849 632 867 590 696 247 838 535 629 224 758 580 640 900 710 4 × 2 = 1 + 0.699 265 735 181 392 495 677 071 258 449 517 161 281 801 420 8;
  • 50) 0.699 265 735 181 392 495 677 071 258 449 517 161 281 801 420 8 × 2 = 1 + 0.398 531 470 362 784 991 354 142 516 899 034 322 563 602 841 6;
  • 51) 0.398 531 470 362 784 991 354 142 516 899 034 322 563 602 841 6 × 2 = 0 + 0.797 062 940 725 569 982 708 285 033 798 068 645 127 205 683 2;
  • 52) 0.797 062 940 725 569 982 708 285 033 798 068 645 127 205 683 2 × 2 = 1 + 0.594 125 881 451 139 965 416 570 067 596 137 290 254 411 366 4;
  • 53) 0.594 125 881 451 139 965 416 570 067 596 137 290 254 411 366 4 × 2 = 1 + 0.188 251 762 902 279 930 833 140 135 192 274 580 508 822 732 8;
  • 54) 0.188 251 762 902 279 930 833 140 135 192 274 580 508 822 732 8 × 2 = 0 + 0.376 503 525 804 559 861 666 280 270 384 549 161 017 645 465 6;
  • 55) 0.376 503 525 804 559 861 666 280 270 384 549 161 017 645 465 6 × 2 = 0 + 0.753 007 051 609 119 723 332 560 540 769 098 322 035 290 931 2;
  • 56) 0.753 007 051 609 119 723 332 560 540 769 098 322 035 290 931 2 × 2 = 1 + 0.506 014 103 218 239 446 665 121 081 538 196 644 070 581 862 4;
  • 57) 0.506 014 103 218 239 446 665 121 081 538 196 644 070 581 862 4 × 2 = 1 + 0.012 028 206 436 478 893 330 242 163 076 393 288 141 163 724 8;
  • 58) 0.012 028 206 436 478 893 330 242 163 076 393 288 141 163 724 8 × 2 = 0 + 0.024 056 412 872 957 786 660 484 326 152 786 576 282 327 449 6;
  • 59) 0.024 056 412 872 957 786 660 484 326 152 786 576 282 327 449 6 × 2 = 0 + 0.048 112 825 745 915 573 320 968 652 305 573 152 564 654 899 2;
  • 60) 0.048 112 825 745 915 573 320 968 652 305 573 152 564 654 899 2 × 2 = 0 + 0.096 225 651 491 831 146 641 937 304 611 146 305 129 309 798 4;
  • 61) 0.096 225 651 491 831 146 641 937 304 611 146 305 129 309 798 4 × 2 = 0 + 0.192 451 302 983 662 293 283 874 609 222 292 610 258 619 596 8;
  • 62) 0.192 451 302 983 662 293 283 874 609 222 292 610 258 619 596 8 × 2 = 0 + 0.384 902 605 967 324 586 567 749 218 444 585 220 517 239 193 6;
  • 63) 0.384 902 605 967 324 586 567 749 218 444 585 220 517 239 193 6 × 2 = 0 + 0.769 805 211 934 649 173 135 498 436 889 170 441 034 478 387 2;
  • 64) 0.769 805 211 934 649 173 135 498 436 889 170 441 034 478 387 2 × 2 = 1 + 0.539 610 423 869 298 346 270 996 873 778 340 882 068 956 774 4;
  • 65) 0.539 610 423 869 298 346 270 996 873 778 340 882 068 956 774 4 × 2 = 1 + 0.079 220 847 738 596 692 541 993 747 556 681 764 137 913 548 8;
  • 66) 0.079 220 847 738 596 692 541 993 747 556 681 764 137 913 548 8 × 2 = 0 + 0.158 441 695 477 193 385 083 987 495 113 363 528 275 827 097 6;
  • 67) 0.158 441 695 477 193 385 083 987 495 113 363 528 275 827 097 6 × 2 = 0 + 0.316 883 390 954 386 770 167 974 990 226 727 056 551 654 195 2;
  • 68) 0.316 883 390 954 386 770 167 974 990 226 727 056 551 654 195 2 × 2 = 0 + 0.633 766 781 908 773 540 335 949 980 453 454 113 103 308 390 4;
  • 69) 0.633 766 781 908 773 540 335 949 980 453 454 113 103 308 390 4 × 2 = 1 + 0.267 533 563 817 547 080 671 899 960 906 908 226 206 616 780 8;
  • 70) 0.267 533 563 817 547 080 671 899 960 906 908 226 206 616 780 8 × 2 = 0 + 0.535 067 127 635 094 161 343 799 921 813 816 452 413 233 561 6;
  • 71) 0.535 067 127 635 094 161 343 799 921 813 816 452 413 233 561 6 × 2 = 1 + 0.070 134 255 270 188 322 687 599 843 627 632 904 826 467 123 2;
  • 72) 0.070 134 255 270 188 322 687 599 843 627 632 904 826 467 123 2 × 2 = 0 + 0.140 268 510 540 376 645 375 199 687 255 265 809 652 934 246 4;
  • 73) 0.140 268 510 540 376 645 375 199 687 255 265 809 652 934 246 4 × 2 = 0 + 0.280 537 021 080 753 290 750 399 374 510 531 619 305 868 492 8;
  • 74) 0.280 537 021 080 753 290 750 399 374 510 531 619 305 868 492 8 × 2 = 0 + 0.561 074 042 161 506 581 500 798 749 021 063 238 611 736 985 6;
  • 75) 0.561 074 042 161 506 581 500 798 749 021 063 238 611 736 985 6 × 2 = 1 + 0.122 148 084 323 013 163 001 597 498 042 126 477 223 473 971 2;
  • 76) 0.122 148 084 323 013 163 001 597 498 042 126 477 223 473 971 2 × 2 = 0 + 0.244 296 168 646 026 326 003 194 996 084 252 954 446 947 942 4;
  • 77) 0.244 296 168 646 026 326 003 194 996 084 252 954 446 947 942 4 × 2 = 0 + 0.488 592 337 292 052 652 006 389 992 168 505 908 893 895 884 8;
  • 78) 0.488 592 337 292 052 652 006 389 992 168 505 908 893 895 884 8 × 2 = 0 + 0.977 184 674 584 105 304 012 779 984 337 011 817 787 791 769 6;
  • 79) 0.977 184 674 584 105 304 012 779 984 337 011 817 787 791 769 6 × 2 = 1 + 0.954 369 349 168 210 608 025 559 968 674 023 635 575 583 539 2;
  • 80) 0.954 369 349 168 210 608 025 559 968 674 023 635 575 583 539 2 × 2 = 1 + 0.908 738 698 336 421 216 051 119 937 348 047 271 151 167 078 4;
  • 81) 0.908 738 698 336 421 216 051 119 937 348 047 271 151 167 078 4 × 2 = 1 + 0.817 477 396 672 842 432 102 239 874 696 094 542 302 334 156 8;
  • 82) 0.817 477 396 672 842 432 102 239 874 696 094 542 302 334 156 8 × 2 = 1 + 0.634 954 793 345 684 864 204 479 749 392 189 084 604 668 313 6;
  • 83) 0.634 954 793 345 684 864 204 479 749 392 189 084 604 668 313 6 × 2 = 1 + 0.269 909 586 691 369 728 408 959 498 784 378 169 209 336 627 2;
  • 84) 0.269 909 586 691 369 728 408 959 498 784 378 169 209 336 627 2 × 2 = 0 + 0.539 819 173 382 739 456 817 918 997 568 756 338 418 673 254 4;
  • 85) 0.539 819 173 382 739 456 817 918 997 568 756 338 418 673 254 4 × 2 = 1 + 0.079 638 346 765 478 913 635 837 995 137 512 676 837 346 508 8;
  • 86) 0.079 638 346 765 478 913 635 837 995 137 512 676 837 346 508 8 × 2 = 0 + 0.159 276 693 530 957 827 271 675 990 275 025 353 674 693 017 6;

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:


0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0111 1010 0000 1110 1101 1001 1000 0001 1000 1010 0010 0011 1110 10(2)

5. Positive number before normalization:

0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0111 1010 0000 1110 1101 1001 1000 0001 1000 1010 0010 0011 1110 10(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 34 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0111 1010 0000 1110 1101 1001 1000 0001 1000 1010 0010 0011 1110 10(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0111 1010 0000 1110 1101 1001 1000 0001 1000 1010 0010 0011 1110 10(2) × 20 =

1.1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010(2) × 2-34


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): -34

Mantissa (not normalized):
1.1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-34 + 2(11-1) - 1 =

(-34 + 1 023)(10) =

989(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;
  • 989 ÷ 2 = 494 + 1;
  • 494 ÷ 2 = 247 + 0;
  • 247 ÷ 2 = 123 + 1;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

989(10) =

011 1101 1101(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. 1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010 =

1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010


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 1101 1101

Mantissa (52 bits) =
1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010


Decimal number 0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 145 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1101 - 1110 1000 0011 1011 0110 0110 0000 0110 0010 1000 1000 1111 1010

» Convert decimal 0.000 000 000 111 011 110 110 111 100 000 011 101 001 111 147 6 to 64 bit double precision IEEE 754 binary floating point representation standard

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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