0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419(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
0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419(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: 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419.

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) 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419 × 2 = 0 + 0.062 831 853 071 795 864 769 252 867 665 590 057 683 943 387 987 502 116 838;
  • 2) 0.062 831 853 071 795 864 769 252 867 665 590 057 683 943 387 987 502 116 838 × 2 = 0 + 0.125 663 706 143 591 729 538 505 735 331 180 115 367 886 775 975 004 233 676;
  • 3) 0.125 663 706 143 591 729 538 505 735 331 180 115 367 886 775 975 004 233 676 × 2 = 0 + 0.251 327 412 287 183 459 077 011 470 662 360 230 735 773 551 950 008 467 352;
  • 4) 0.251 327 412 287 183 459 077 011 470 662 360 230 735 773 551 950 008 467 352 × 2 = 0 + 0.502 654 824 574 366 918 154 022 941 324 720 461 471 547 103 900 016 934 704;
  • 5) 0.502 654 824 574 366 918 154 022 941 324 720 461 471 547 103 900 016 934 704 × 2 = 1 + 0.005 309 649 148 733 836 308 045 882 649 440 922 943 094 207 800 033 869 408;
  • 6) 0.005 309 649 148 733 836 308 045 882 649 440 922 943 094 207 800 033 869 408 × 2 = 0 + 0.010 619 298 297 467 672 616 091 765 298 881 845 886 188 415 600 067 738 816;
  • 7) 0.010 619 298 297 467 672 616 091 765 298 881 845 886 188 415 600 067 738 816 × 2 = 0 + 0.021 238 596 594 935 345 232 183 530 597 763 691 772 376 831 200 135 477 632;
  • 8) 0.021 238 596 594 935 345 232 183 530 597 763 691 772 376 831 200 135 477 632 × 2 = 0 + 0.042 477 193 189 870 690 464 367 061 195 527 383 544 753 662 400 270 955 264;
  • 9) 0.042 477 193 189 870 690 464 367 061 195 527 383 544 753 662 400 270 955 264 × 2 = 0 + 0.084 954 386 379 741 380 928 734 122 391 054 767 089 507 324 800 541 910 528;
  • 10) 0.084 954 386 379 741 380 928 734 122 391 054 767 089 507 324 800 541 910 528 × 2 = 0 + 0.169 908 772 759 482 761 857 468 244 782 109 534 179 014 649 601 083 821 056;
  • 11) 0.169 908 772 759 482 761 857 468 244 782 109 534 179 014 649 601 083 821 056 × 2 = 0 + 0.339 817 545 518 965 523 714 936 489 564 219 068 358 029 299 202 167 642 112;
  • 12) 0.339 817 545 518 965 523 714 936 489 564 219 068 358 029 299 202 167 642 112 × 2 = 0 + 0.679 635 091 037 931 047 429 872 979 128 438 136 716 058 598 404 335 284 224;
  • 13) 0.679 635 091 037 931 047 429 872 979 128 438 136 716 058 598 404 335 284 224 × 2 = 1 + 0.359 270 182 075 862 094 859 745 958 256 876 273 432 117 196 808 670 568 448;
  • 14) 0.359 270 182 075 862 094 859 745 958 256 876 273 432 117 196 808 670 568 448 × 2 = 0 + 0.718 540 364 151 724 189 719 491 916 513 752 546 864 234 393 617 341 136 896;
  • 15) 0.718 540 364 151 724 189 719 491 916 513 752 546 864 234 393 617 341 136 896 × 2 = 1 + 0.437 080 728 303 448 379 438 983 833 027 505 093 728 468 787 234 682 273 792;
  • 16) 0.437 080 728 303 448 379 438 983 833 027 505 093 728 468 787 234 682 273 792 × 2 = 0 + 0.874 161 456 606 896 758 877 967 666 055 010 187 456 937 574 469 364 547 584;
  • 17) 0.874 161 456 606 896 758 877 967 666 055 010 187 456 937 574 469 364 547 584 × 2 = 1 + 0.748 322 913 213 793 517 755 935 332 110 020 374 913 875 148 938 729 095 168;
  • 18) 0.748 322 913 213 793 517 755 935 332 110 020 374 913 875 148 938 729 095 168 × 2 = 1 + 0.496 645 826 427 587 035 511 870 664 220 040 749 827 750 297 877 458 190 336;
  • 19) 0.496 645 826 427 587 035 511 870 664 220 040 749 827 750 297 877 458 190 336 × 2 = 0 + 0.993 291 652 855 174 071 023 741 328 440 081 499 655 500 595 754 916 380 672;
  • 20) 0.993 291 652 855 174 071 023 741 328 440 081 499 655 500 595 754 916 380 672 × 2 = 1 + 0.986 583 305 710 348 142 047 482 656 880 162 999 311 001 191 509 832 761 344;
  • 21) 0.986 583 305 710 348 142 047 482 656 880 162 999 311 001 191 509 832 761 344 × 2 = 1 + 0.973 166 611 420 696 284 094 965 313 760 325 998 622 002 383 019 665 522 688;
  • 22) 0.973 166 611 420 696 284 094 965 313 760 325 998 622 002 383 019 665 522 688 × 2 = 1 + 0.946 333 222 841 392 568 189 930 627 520 651 997 244 004 766 039 331 045 376;
  • 23) 0.946 333 222 841 392 568 189 930 627 520 651 997 244 004 766 039 331 045 376 × 2 = 1 + 0.892 666 445 682 785 136 379 861 255 041 303 994 488 009 532 078 662 090 752;
  • 24) 0.892 666 445 682 785 136 379 861 255 041 303 994 488 009 532 078 662 090 752 × 2 = 1 + 0.785 332 891 365 570 272 759 722 510 082 607 988 976 019 064 157 324 181 504;
  • 25) 0.785 332 891 365 570 272 759 722 510 082 607 988 976 019 064 157 324 181 504 × 2 = 1 + 0.570 665 782 731 140 545 519 445 020 165 215 977 952 038 128 314 648 363 008;
  • 26) 0.570 665 782 731 140 545 519 445 020 165 215 977 952 038 128 314 648 363 008 × 2 = 1 + 0.141 331 565 462 281 091 038 890 040 330 431 955 904 076 256 629 296 726 016;
  • 27) 0.141 331 565 462 281 091 038 890 040 330 431 955 904 076 256 629 296 726 016 × 2 = 0 + 0.282 663 130 924 562 182 077 780 080 660 863 911 808 152 513 258 593 452 032;
  • 28) 0.282 663 130 924 562 182 077 780 080 660 863 911 808 152 513 258 593 452 032 × 2 = 0 + 0.565 326 261 849 124 364 155 560 161 321 727 823 616 305 026 517 186 904 064;
  • 29) 0.565 326 261 849 124 364 155 560 161 321 727 823 616 305 026 517 186 904 064 × 2 = 1 + 0.130 652 523 698 248 728 311 120 322 643 455 647 232 610 053 034 373 808 128;
  • 30) 0.130 652 523 698 248 728 311 120 322 643 455 647 232 610 053 034 373 808 128 × 2 = 0 + 0.261 305 047 396 497 456 622 240 645 286 911 294 465 220 106 068 747 616 256;
  • 31) 0.261 305 047 396 497 456 622 240 645 286 911 294 465 220 106 068 747 616 256 × 2 = 0 + 0.522 610 094 792 994 913 244 481 290 573 822 588 930 440 212 137 495 232 512;
  • 32) 0.522 610 094 792 994 913 244 481 290 573 822 588 930 440 212 137 495 232 512 × 2 = 1 + 0.045 220 189 585 989 826 488 962 581 147 645 177 860 880 424 274 990 465 024;
  • 33) 0.045 220 189 585 989 826 488 962 581 147 645 177 860 880 424 274 990 465 024 × 2 = 0 + 0.090 440 379 171 979 652 977 925 162 295 290 355 721 760 848 549 980 930 048;
  • 34) 0.090 440 379 171 979 652 977 925 162 295 290 355 721 760 848 549 980 930 048 × 2 = 0 + 0.180 880 758 343 959 305 955 850 324 590 580 711 443 521 697 099 961 860 096;
  • 35) 0.180 880 758 343 959 305 955 850 324 590 580 711 443 521 697 099 961 860 096 × 2 = 0 + 0.361 761 516 687 918 611 911 700 649 181 161 422 887 043 394 199 923 720 192;
  • 36) 0.361 761 516 687 918 611 911 700 649 181 161 422 887 043 394 199 923 720 192 × 2 = 0 + 0.723 523 033 375 837 223 823 401 298 362 322 845 774 086 788 399 847 440 384;
  • 37) 0.723 523 033 375 837 223 823 401 298 362 322 845 774 086 788 399 847 440 384 × 2 = 1 + 0.447 046 066 751 674 447 646 802 596 724 645 691 548 173 576 799 694 880 768;
  • 38) 0.447 046 066 751 674 447 646 802 596 724 645 691 548 173 576 799 694 880 768 × 2 = 0 + 0.894 092 133 503 348 895 293 605 193 449 291 383 096 347 153 599 389 761 536;
  • 39) 0.894 092 133 503 348 895 293 605 193 449 291 383 096 347 153 599 389 761 536 × 2 = 1 + 0.788 184 267 006 697 790 587 210 386 898 582 766 192 694 307 198 779 523 072;
  • 40) 0.788 184 267 006 697 790 587 210 386 898 582 766 192 694 307 198 779 523 072 × 2 = 1 + 0.576 368 534 013 395 581 174 420 773 797 165 532 385 388 614 397 559 046 144;
  • 41) 0.576 368 534 013 395 581 174 420 773 797 165 532 385 388 614 397 559 046 144 × 2 = 1 + 0.152 737 068 026 791 162 348 841 547 594 331 064 770 777 228 795 118 092 288;
  • 42) 0.152 737 068 026 791 162 348 841 547 594 331 064 770 777 228 795 118 092 288 × 2 = 0 + 0.305 474 136 053 582 324 697 683 095 188 662 129 541 554 457 590 236 184 576;
  • 43) 0.305 474 136 053 582 324 697 683 095 188 662 129 541 554 457 590 236 184 576 × 2 = 0 + 0.610 948 272 107 164 649 395 366 190 377 324 259 083 108 915 180 472 369 152;
  • 44) 0.610 948 272 107 164 649 395 366 190 377 324 259 083 108 915 180 472 369 152 × 2 = 1 + 0.221 896 544 214 329 298 790 732 380 754 648 518 166 217 830 360 944 738 304;
  • 45) 0.221 896 544 214 329 298 790 732 380 754 648 518 166 217 830 360 944 738 304 × 2 = 0 + 0.443 793 088 428 658 597 581 464 761 509 297 036 332 435 660 721 889 476 608;
  • 46) 0.443 793 088 428 658 597 581 464 761 509 297 036 332 435 660 721 889 476 608 × 2 = 0 + 0.887 586 176 857 317 195 162 929 523 018 594 072 664 871 321 443 778 953 216;
  • 47) 0.887 586 176 857 317 195 162 929 523 018 594 072 664 871 321 443 778 953 216 × 2 = 1 + 0.775 172 353 714 634 390 325 859 046 037 188 145 329 742 642 887 557 906 432;
  • 48) 0.775 172 353 714 634 390 325 859 046 037 188 145 329 742 642 887 557 906 432 × 2 = 1 + 0.550 344 707 429 268 780 651 718 092 074 376 290 659 485 285 775 115 812 864;
  • 49) 0.550 344 707 429 268 780 651 718 092 074 376 290 659 485 285 775 115 812 864 × 2 = 1 + 0.100 689 414 858 537 561 303 436 184 148 752 581 318 970 571 550 231 625 728;
  • 50) 0.100 689 414 858 537 561 303 436 184 148 752 581 318 970 571 550 231 625 728 × 2 = 0 + 0.201 378 829 717 075 122 606 872 368 297 505 162 637 941 143 100 463 251 456;
  • 51) 0.201 378 829 717 075 122 606 872 368 297 505 162 637 941 143 100 463 251 456 × 2 = 0 + 0.402 757 659 434 150 245 213 744 736 595 010 325 275 882 286 200 926 502 912;
  • 52) 0.402 757 659 434 150 245 213 744 736 595 010 325 275 882 286 200 926 502 912 × 2 = 0 + 0.805 515 318 868 300 490 427 489 473 190 020 650 551 764 572 401 853 005 824;
  • 53) 0.805 515 318 868 300 490 427 489 473 190 020 650 551 764 572 401 853 005 824 × 2 = 1 + 0.611 030 637 736 600 980 854 978 946 380 041 301 103 529 144 803 706 011 648;
  • 54) 0.611 030 637 736 600 980 854 978 946 380 041 301 103 529 144 803 706 011 648 × 2 = 1 + 0.222 061 275 473 201 961 709 957 892 760 082 602 207 058 289 607 412 023 296;
  • 55) 0.222 061 275 473 201 961 709 957 892 760 082 602 207 058 289 607 412 023 296 × 2 = 0 + 0.444 122 550 946 403 923 419 915 785 520 165 204 414 116 579 214 824 046 592;
  • 56) 0.444 122 550 946 403 923 419 915 785 520 165 204 414 116 579 214 824 046 592 × 2 = 0 + 0.888 245 101 892 807 846 839 831 571 040 330 408 828 233 158 429 648 093 184;
  • 57) 0.888 245 101 892 807 846 839 831 571 040 330 408 828 233 158 429 648 093 184 × 2 = 1 + 0.776 490 203 785 615 693 679 663 142 080 660 817 656 466 316 859 296 186 368;

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:


0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2)

5. Positive number before normalization:

0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2) × 20 =

1.0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001(2) × 2-5


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): -5

Mantissa (not normalized):
1.0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 509 ÷ 2 = 254 + 1;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1018(10) =

011 1111 1010(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. 0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001 =

0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


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 1111 1010

Mantissa (52 bits) =
0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


Decimal number 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 751 058 419 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1010 - 0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001

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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