-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 444 090 505 085 762;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 444 090 505 085 762 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 888 181 010 171 524;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 888 181 010 171 524 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 776 362 020 343 048;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 776 362 020 343 048 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 552 724 040 686 096;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 552 724 040 686 096 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 105 448 081 372 192;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 105 448 081 372 192 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 210 896 162 744 384;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 210 896 162 744 384 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 476 421 792 325 488 768;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 476 421 792 325 488 768 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 952 843 584 650 977 536;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 952 843 584 650 977 536 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 905 687 169 301 955 072;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 905 687 169 301 955 072 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 811 374 338 603 910 144;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 811 374 338 603 910 144 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 622 748 677 207 820 288;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 622 748 677 207 820 288 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 245 497 354 415 640 576;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 245 497 354 415 640 576 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 490 994 708 831 281 152;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 490 994 708 831 281 152 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 612 981 989 417 662 562 304;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 612 981 989 417 662 562 304 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 225 963 978 835 325 124 608;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 225 963 978 835 325 124 608 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 451 927 957 670 650 249 216;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 451 927 957 670 650 249 216 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 903 855 915 341 300 498 432;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 903 855 915 341 300 498 432 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 807 711 830 682 600 996 864;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 807 711 830 682 600 996 864 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 615 423 661 365 201 993 728;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 615 423 661 365 201 993 728 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 230 847 322 730 403 987 456;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 230 847 322 730 403 987 456 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 461 694 645 460 807 974 912;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 461 694 645 460 807 974 912 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 923 389 290 921 615 949 824;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 923 389 290 921 615 949 824 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 846 778 581 843 231 899 648;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 846 778 581 843 231 899 648 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 693 557 163 686 463 799 296;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 693 557 163 686 463 799 296 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 387 114 327 372 927 598 592;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 387 114 327 372 927 598 592 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 542 774 228 654 745 855 197 184;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 542 774 228 654 745 855 197 184 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 085 548 457 309 491 710 394 368;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 085 548 457 309 491 710 394 368 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 171 096 914 618 983 420 788 736;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 171 096 914 618 983 420 788 736 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 342 193 829 237 966 841 577 472;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 342 193 829 237 966 841 577 472 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 684 387 658 475 933 683 154 944;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 684 387 658 475 933 683 154 944 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 368 775 316 951 867 366 309 888;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 368 775 316 951 867 366 309 888 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 737 550 633 903 734 732 619 776;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 737 550 633 903 734 732 619 776 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 475 101 267 807 469 465 239 552;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 475 101 267 807 469 465 239 552 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 570 950 202 535 614 938 930 479 104;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 570 950 202 535 614 938 930 479 104 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 141 900 405 071 229 877 860 958 208;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 141 900 405 071 229 877 860 958 208 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 283 800 810 142 459 755 721 916 416;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 283 800 810 142 459 755 721 916 416 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 567 601 620 284 919 511 443 832 832;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 567 601 620 284 919 511 443 832 832 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 135 203 240 569 839 022 887 665 664;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 135 203 240 569 839 022 887 665 664 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 270 406 481 139 678 045 775 331 328;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 270 406 481 139 678 045 775 331 328 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 540 812 962 279 356 091 550 662 656;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 540 812 962 279 356 091 550 662 656 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 081 625 924 558 712 183 101 325 312;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 081 625 924 558 712 183 101 325 312 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 163 251 849 117 424 366 202 650 624;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 163 251 849 117 424 366 202 650 624 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 326 503 698 234 848 732 405 301 248;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 326 503 698 234 848 732 405 301 248 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 653 007 396 469 697 464 810 602 496;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 653 007 396 469 697 464 810 602 496 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 306 014 792 939 394 929 621 204 992;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 306 014 792 939 394 929 621 204 992 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 498 612 029 585 878 789 859 242 409 984;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 498 612 029 585 878 789 859 242 409 984 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 997 224 059 171 757 579 718 484 819 968;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 997 224 059 171 757 579 718 484 819 968 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 994 448 118 343 515 159 436 969 639 936;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 994 448 118 343 515 159 436 969 639 936 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 988 896 236 687 030 318 873 939 279 872;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 988 896 236 687 030 318 873 939 279 872 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 977 792 473 374 060 637 747 878 559 744;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 977 792 473 374 060 637 747 878 559 744 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 955 584 946 748 121 275 495 757 119 488;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 955 584 946 748 121 275 495 757 119 488 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 911 169 893 496 242 550 991 514 238 976;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 911 169 893 496 242 550 991 514 238 976 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 822 339 786 992 485 101 983 028 477 952;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 822 339 786 992 485 101 983 028 477 952 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 644 679 573 984 970 203 966 056 955 904;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 644 679 573 984 970 203 966 056 955 904 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 289 359 147 969 940 407 932 113 911 808;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 289 359 147 969 940 407 932 113 911 808 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 630 578 718 295 939 880 815 864 227 823 616;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 630 578 718 295 939 880 815 864 227 823 616 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 261 157 436 591 879 761 631 728 455 647 232;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 261 157 436 591 879 761 631 728 455 647 232 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 522 314 873 183 759 523 263 456 911 294 464;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 522 314 873 183 759 523 263 456 911 294 464 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 044 629 746 367 519 046 526 913 822 588 928;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 044 629 746 367 519 046 526 913 822 588 928 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 089 259 492 735 038 093 053 827 645 177 856;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 089 259 492 735 038 093 053 827 645 177 856 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 178 518 985 470 076 186 107 655 290 355 712;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 178 518 985 470 076 186 107 655 290 355 712 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 357 037 970 940 152 372 215 310 580 711 424;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 357 037 970 940 152 372 215 310 580 711 424 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 241 856 992 996 704 976 714 075 941 880 304 744 430 621 161 422 848;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 045 252 542 881 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