-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2(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. Start with the positive version of the number:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 464 4;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 464 4 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 928 8;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 928 8 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 857 6;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 857 6 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 715 2;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 715 2 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 430 4;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 430 4 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 860 8;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 860 8 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 477 721 6;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 477 721 6 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 955 443 2;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 955 443 2 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 910 886 4;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 910 886 4 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 821 772 8;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 821 772 8 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 643 545 6;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 643 545 6 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 287 091 2;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 287 091 2 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 574 182 4;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 574 182 4 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 613 148 364 8;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 613 148 364 8 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 226 296 729 6;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 226 296 729 6 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 452 593 459 2;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 452 593 459 2 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 905 186 918 4;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 905 186 918 4 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 810 373 836 8;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 810 373 836 8 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 620 747 673 6;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 620 747 673 6 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 241 495 347 2;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 241 495 347 2 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 482 990 694 4;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 482 990 694 4 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 965 981 388 8;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 965 981 388 8 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 931 962 777 6;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 931 962 777 6 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 863 925 555 2;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 863 925 555 2 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 727 851 110 4;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 727 851 110 4 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 543 455 702 220 8;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 543 455 702 220 8 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 086 911 404 441 6;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 086 911 404 441 6 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 173 822 808 883 2;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 173 822 808 883 2 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 347 645 617 766 4;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 347 645 617 766 4 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 695 291 235 532 8;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 695 291 235 532 8 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 390 582 471 065 6;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 390 582 471 065 6 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 781 164 942 131 2;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 781 164 942 131 2 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 562 329 884 262 4;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 562 329 884 262 4 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 571 124 659 768 524 8;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 571 124 659 768 524 8 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 142 249 319 537 049 6;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 142 249 319 537 049 6 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 284 498 639 074 099 2;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 284 498 639 074 099 2 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 568 997 278 148 198 4;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 568 997 278 148 198 4 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 137 994 556 296 396 8;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 137 994 556 296 396 8 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 275 989 112 592 793 6;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 275 989 112 592 793 6 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 551 978 225 185 587 2;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 551 978 225 185 587 2 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 103 956 450 371 174 4;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 103 956 450 371 174 4 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 207 912 900 742 348 8;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 207 912 900 742 348 8 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 415 825 801 484 697 6;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 415 825 801 484 697 6 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 831 651 602 969 395 2;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 831 651 602 969 395 2 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 663 303 205 938 790 4;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 663 303 205 938 790 4 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 499 326 606 411 877 580 8;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 499 326 606 411 877 580 8 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 998 653 212 823 755 161 6;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 998 653 212 823 755 161 6 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 997 306 425 647 510 323 2;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 997 306 425 647 510 323 2 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 994 612 851 295 020 646 4;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 994 612 851 295 020 646 4 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 989 225 702 590 041 292 8;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 989 225 702 590 041 292 8 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 978 451 405 180 082 585 6;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 978 451 405 180 082 585 6 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 956 902 810 360 165 171 2;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 956 902 810 360 165 171 2 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 913 805 620 720 330 342 4;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 913 805 620 720 330 342 4 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 827 611 241 440 660 684 8;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 827 611 241 440 660 684 8 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 655 222 482 881 321 369 6;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 655 222 482 881 321 369 6 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 631 310 444 965 762 642 739 2;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 631 310 444 965 762 642 739 2 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 262 620 889 931 525 285 478 4;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 262 620 889 931 525 285 478 4 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 525 241 779 863 050 570 956 8;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 525 241 779 863 050 570 956 8 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 050 483 559 726 101 141 913 6;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 050 483 559 726 101 141 913 6 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 100 967 119 452 202 283 827 2;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 100 967 119 452 202 283 827 2 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 201 934 238 904 404 567 654 4;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 201 934 238 904 404 567 654 4 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 403 868 477 808 809 135 308 8;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 403 868 477 808 809 135 308 8 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 241 856 992 996 704 976 807 736 955 617 618 270 617 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).


5. 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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 11 positions to the right, so that only one non zero digit remains to the left of it:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) × 20 =

1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010(2) × 2-11


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): -11

Mantissa (not normalized):
1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-11 + 1 023)(10) =

1 012(10)


10. 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;
  • 1 012 ÷ 2 = 506 + 0;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1012(10) =

011 1111 0100(2)


12. 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. 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010 =

1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1111 0100

Mantissa (52 bits) =
1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


Decimal number -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 732 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0100 - 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010

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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