0.000 000 000 634 751 367 889 197 801 221 261 632 94 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 632 94(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
0.000 000 000 634 751 367 889 197 801 221 261 632 94(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: 0.000 000 000 634 751 367 889 197 801 221 261 632 94.

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) 0.000 000 000 634 751 367 889 197 801 221 261 632 94 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 265 88;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 265 88 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 531 76;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 531 76 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 063 52;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 063 52 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 127 04;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 127 04 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 254 08;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 254 08 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 508 16;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 508 16 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 016 32;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 016 32 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 032 64;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 032 64 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 065 28;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 065 28 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 130 56;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 130 56 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 824 261 12;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 824 261 12 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 648 522 24;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 648 522 24 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 297 044 48;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 297 044 48 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 594 088 96;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 594 088 96 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 188 177 92;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 188 177 92 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 376 355 84;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 376 355 84 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 752 711 68;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 752 711 68 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 505 423 36;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 505 423 36 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 010 846 72;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 010 846 72 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 021 693 44;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 021 693 44 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 276 043 386 88;
  • 22) 0.001 331 170 100 671 566 947 226 771 276 043 386 88 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 552 086 773 76;
  • 23) 0.002 662 340 201 343 133 894 453 542 552 086 773 76 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 104 173 547 52;
  • 24) 0.005 324 680 402 686 267 788 907 085 104 173 547 52 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 208 347 095 04;
  • 25) 0.010 649 360 805 372 535 577 814 170 208 347 095 04 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 416 694 190 08;
  • 26) 0.021 298 721 610 745 071 155 628 340 416 694 190 08 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 833 388 380 16;
  • 27) 0.042 597 443 221 490 142 311 256 680 833 388 380 16 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 666 776 760 32;
  • 28) 0.085 194 886 442 980 284 622 513 361 666 776 760 32 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 333 553 520 64;
  • 29) 0.170 389 772 885 960 569 245 026 723 333 553 520 64 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 667 107 041 28;
  • 30) 0.340 779 545 771 921 138 490 053 446 667 107 041 28 × 2 = 0 + 0.681 559 091 543 842 276 980 106 893 334 214 082 56;
  • 31) 0.681 559 091 543 842 276 980 106 893 334 214 082 56 × 2 = 1 + 0.363 118 183 087 684 553 960 213 786 668 428 165 12;
  • 32) 0.363 118 183 087 684 553 960 213 786 668 428 165 12 × 2 = 0 + 0.726 236 366 175 369 107 920 427 573 336 856 330 24;
  • 33) 0.726 236 366 175 369 107 920 427 573 336 856 330 24 × 2 = 1 + 0.452 472 732 350 738 215 840 855 146 673 712 660 48;
  • 34) 0.452 472 732 350 738 215 840 855 146 673 712 660 48 × 2 = 0 + 0.904 945 464 701 476 431 681 710 293 347 425 320 96;
  • 35) 0.904 945 464 701 476 431 681 710 293 347 425 320 96 × 2 = 1 + 0.809 890 929 402 952 863 363 420 586 694 850 641 92;
  • 36) 0.809 890 929 402 952 863 363 420 586 694 850 641 92 × 2 = 1 + 0.619 781 858 805 905 726 726 841 173 389 701 283 84;
  • 37) 0.619 781 858 805 905 726 726 841 173 389 701 283 84 × 2 = 1 + 0.239 563 717 611 811 453 453 682 346 779 402 567 68;
  • 38) 0.239 563 717 611 811 453 453 682 346 779 402 567 68 × 2 = 0 + 0.479 127 435 223 622 906 907 364 693 558 805 135 36;
  • 39) 0.479 127 435 223 622 906 907 364 693 558 805 135 36 × 2 = 0 + 0.958 254 870 447 245 813 814 729 387 117 610 270 72;
  • 40) 0.958 254 870 447 245 813 814 729 387 117 610 270 72 × 2 = 1 + 0.916 509 740 894 491 627 629 458 774 235 220 541 44;
  • 41) 0.916 509 740 894 491 627 629 458 774 235 220 541 44 × 2 = 1 + 0.833 019 481 788 983 255 258 917 548 470 441 082 88;
  • 42) 0.833 019 481 788 983 255 258 917 548 470 441 082 88 × 2 = 1 + 0.666 038 963 577 966 510 517 835 096 940 882 165 76;
  • 43) 0.666 038 963 577 966 510 517 835 096 940 882 165 76 × 2 = 1 + 0.332 077 927 155 933 021 035 670 193 881 764 331 52;
  • 44) 0.332 077 927 155 933 021 035 670 193 881 764 331 52 × 2 = 0 + 0.664 155 854 311 866 042 071 340 387 763 528 663 04;
  • 45) 0.664 155 854 311 866 042 071 340 387 763 528 663 04 × 2 = 1 + 0.328 311 708 623 732 084 142 680 775 527 057 326 08;
  • 46) 0.328 311 708 623 732 084 142 680 775 527 057 326 08 × 2 = 0 + 0.656 623 417 247 464 168 285 361 551 054 114 652 16;
  • 47) 0.656 623 417 247 464 168 285 361 551 054 114 652 16 × 2 = 1 + 0.313 246 834 494 928 336 570 723 102 108 229 304 32;
  • 48) 0.313 246 834 494 928 336 570 723 102 108 229 304 32 × 2 = 0 + 0.626 493 668 989 856 673 141 446 204 216 458 608 64;
  • 49) 0.626 493 668 989 856 673 141 446 204 216 458 608 64 × 2 = 1 + 0.252 987 337 979 713 346 282 892 408 432 917 217 28;
  • 50) 0.252 987 337 979 713 346 282 892 408 432 917 217 28 × 2 = 0 + 0.505 974 675 959 426 692 565 784 816 865 834 434 56;
  • 51) 0.505 974 675 959 426 692 565 784 816 865 834 434 56 × 2 = 1 + 0.011 949 351 918 853 385 131 569 633 731 668 869 12;
  • 52) 0.011 949 351 918 853 385 131 569 633 731 668 869 12 × 2 = 0 + 0.023 898 703 837 706 770 263 139 267 463 337 738 24;
  • 53) 0.023 898 703 837 706 770 263 139 267 463 337 738 24 × 2 = 0 + 0.047 797 407 675 413 540 526 278 534 926 675 476 48;
  • 54) 0.047 797 407 675 413 540 526 278 534 926 675 476 48 × 2 = 0 + 0.095 594 815 350 827 081 052 557 069 853 350 952 96;
  • 55) 0.095 594 815 350 827 081 052 557 069 853 350 952 96 × 2 = 0 + 0.191 189 630 701 654 162 105 114 139 706 701 905 92;
  • 56) 0.191 189 630 701 654 162 105 114 139 706 701 905 92 × 2 = 0 + 0.382 379 261 403 308 324 210 228 279 413 403 811 84;
  • 57) 0.382 379 261 403 308 324 210 228 279 413 403 811 84 × 2 = 0 + 0.764 758 522 806 616 648 420 456 558 826 807 623 68;
  • 58) 0.764 758 522 806 616 648 420 456 558 826 807 623 68 × 2 = 1 + 0.529 517 045 613 233 296 840 913 117 653 615 247 36;
  • 59) 0.529 517 045 613 233 296 840 913 117 653 615 247 36 × 2 = 1 + 0.059 034 091 226 466 593 681 826 235 307 230 494 72;
  • 60) 0.059 034 091 226 466 593 681 826 235 307 230 494 72 × 2 = 0 + 0.118 068 182 452 933 187 363 652 470 614 460 989 44;
  • 61) 0.118 068 182 452 933 187 363 652 470 614 460 989 44 × 2 = 0 + 0.236 136 364 905 866 374 727 304 941 228 921 978 88;
  • 62) 0.236 136 364 905 866 374 727 304 941 228 921 978 88 × 2 = 0 + 0.472 272 729 811 732 749 454 609 882 457 843 957 76;
  • 63) 0.472 272 729 811 732 749 454 609 882 457 843 957 76 × 2 = 0 + 0.944 545 459 623 465 498 909 219 764 915 687 915 52;
  • 64) 0.944 545 459 623 465 498 909 219 764 915 687 915 52 × 2 = 1 + 0.889 090 919 246 930 997 818 439 529 831 375 831 04;
  • 65) 0.889 090 919 246 930 997 818 439 529 831 375 831 04 × 2 = 1 + 0.778 181 838 493 861 995 636 879 059 662 751 662 08;
  • 66) 0.778 181 838 493 861 995 636 879 059 662 751 662 08 × 2 = 1 + 0.556 363 676 987 723 991 273 758 119 325 503 324 16;
  • 67) 0.556 363 676 987 723 991 273 758 119 325 503 324 16 × 2 = 1 + 0.112 727 353 975 447 982 547 516 238 651 006 648 32;
  • 68) 0.112 727 353 975 447 982 547 516 238 651 006 648 32 × 2 = 0 + 0.225 454 707 950 895 965 095 032 477 302 013 296 64;
  • 69) 0.225 454 707 950 895 965 095 032 477 302 013 296 64 × 2 = 0 + 0.450 909 415 901 791 930 190 064 954 604 026 593 28;
  • 70) 0.450 909 415 901 791 930 190 064 954 604 026 593 28 × 2 = 0 + 0.901 818 831 803 583 860 380 129 909 208 053 186 56;
  • 71) 0.901 818 831 803 583 860 380 129 909 208 053 186 56 × 2 = 1 + 0.803 637 663 607 167 720 760 259 818 416 106 373 12;
  • 72) 0.803 637 663 607 167 720 760 259 818 416 106 373 12 × 2 = 1 + 0.607 275 327 214 335 441 520 519 636 832 212 746 24;
  • 73) 0.607 275 327 214 335 441 520 519 636 832 212 746 24 × 2 = 1 + 0.214 550 654 428 670 883 041 039 273 664 425 492 48;
  • 74) 0.214 550 654 428 670 883 041 039 273 664 425 492 48 × 2 = 0 + 0.429 101 308 857 341 766 082 078 547 328 850 984 96;
  • 75) 0.429 101 308 857 341 766 082 078 547 328 850 984 96 × 2 = 0 + 0.858 202 617 714 683 532 164 157 094 657 701 969 92;
  • 76) 0.858 202 617 714 683 532 164 157 094 657 701 969 92 × 2 = 1 + 0.716 405 235 429 367 064 328 314 189 315 403 939 84;
  • 77) 0.716 405 235 429 367 064 328 314 189 315 403 939 84 × 2 = 1 + 0.432 810 470 858 734 128 656 628 378 630 807 879 68;
  • 78) 0.432 810 470 858 734 128 656 628 378 630 807 879 68 × 2 = 0 + 0.865 620 941 717 468 257 313 256 757 261 615 759 36;
  • 79) 0.865 620 941 717 468 257 313 256 757 261 615 759 36 × 2 = 1 + 0.731 241 883 434 936 514 626 513 514 523 231 518 72;
  • 80) 0.731 241 883 434 936 514 626 513 514 523 231 518 72 × 2 = 1 + 0.462 483 766 869 873 029 253 027 029 046 463 037 44;
  • 81) 0.462 483 766 869 873 029 253 027 029 046 463 037 44 × 2 = 0 + 0.924 967 533 739 746 058 506 054 058 092 926 074 88;
  • 82) 0.924 967 533 739 746 058 506 054 058 092 926 074 88 × 2 = 1 + 0.849 935 067 479 492 117 012 108 116 185 852 149 76;
  • 83) 0.849 935 067 479 492 117 012 108 116 185 852 149 76 × 2 = 1 + 0.699 870 134 958 984 234 024 216 232 371 704 299 52;

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:


0.000 000 000 634 751 367 889 197 801 221 261 632 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

5. Positive number before normalization:

0.000 000 000 634 751 367 889 197 801 221 261 632 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 634 751 367 889 197 801 221 261 632 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) × 20 =

1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011(2) × 2-31


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): -31

Mantissa (not normalized):
1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(11-1) - 1 =

(-31 + 1 023)(10) =

992(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;
  • 992 ÷ 2 = 496 + 0;
  • 496 ÷ 2 = 248 + 0;
  • 248 ÷ 2 = 124 + 0;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

992(10) =

011 1110 0000(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. 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011 =

0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


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 1110 0000

Mantissa (52 bits) =
0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


Decimal number 0.000 000 000 634 751 367 889 197 801 221 261 632 94 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 0000 - 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011

» Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 633 3 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