0.000 000 000 000 000 000 008 543 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 543 7(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
0.000 000 000 000 000 000 008 543 7(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: 0.000 000 000 000 000 000 008 543 7.

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) 0.000 000 000 000 000 000 008 543 7 × 2 = 0 + 0.000 000 000 000 000 000 017 087 4;
  • 2) 0.000 000 000 000 000 000 017 087 4 × 2 = 0 + 0.000 000 000 000 000 000 034 174 8;
  • 3) 0.000 000 000 000 000 000 034 174 8 × 2 = 0 + 0.000 000 000 000 000 000 068 349 6;
  • 4) 0.000 000 000 000 000 000 068 349 6 × 2 = 0 + 0.000 000 000 000 000 000 136 699 2;
  • 5) 0.000 000 000 000 000 000 136 699 2 × 2 = 0 + 0.000 000 000 000 000 000 273 398 4;
  • 6) 0.000 000 000 000 000 000 273 398 4 × 2 = 0 + 0.000 000 000 000 000 000 546 796 8;
  • 7) 0.000 000 000 000 000 000 546 796 8 × 2 = 0 + 0.000 000 000 000 000 001 093 593 6;
  • 8) 0.000 000 000 000 000 001 093 593 6 × 2 = 0 + 0.000 000 000 000 000 002 187 187 2;
  • 9) 0.000 000 000 000 000 002 187 187 2 × 2 = 0 + 0.000 000 000 000 000 004 374 374 4;
  • 10) 0.000 000 000 000 000 004 374 374 4 × 2 = 0 + 0.000 000 000 000 000 008 748 748 8;
  • 11) 0.000 000 000 000 000 008 748 748 8 × 2 = 0 + 0.000 000 000 000 000 017 497 497 6;
  • 12) 0.000 000 000 000 000 017 497 497 6 × 2 = 0 + 0.000 000 000 000 000 034 994 995 2;
  • 13) 0.000 000 000 000 000 034 994 995 2 × 2 = 0 + 0.000 000 000 000 000 069 989 990 4;
  • 14) 0.000 000 000 000 000 069 989 990 4 × 2 = 0 + 0.000 000 000 000 000 139 979 980 8;
  • 15) 0.000 000 000 000 000 139 979 980 8 × 2 = 0 + 0.000 000 000 000 000 279 959 961 6;
  • 16) 0.000 000 000 000 000 279 959 961 6 × 2 = 0 + 0.000 000 000 000 000 559 919 923 2;
  • 17) 0.000 000 000 000 000 559 919 923 2 × 2 = 0 + 0.000 000 000 000 001 119 839 846 4;
  • 18) 0.000 000 000 000 001 119 839 846 4 × 2 = 0 + 0.000 000 000 000 002 239 679 692 8;
  • 19) 0.000 000 000 000 002 239 679 692 8 × 2 = 0 + 0.000 000 000 000 004 479 359 385 6;
  • 20) 0.000 000 000 000 004 479 359 385 6 × 2 = 0 + 0.000 000 000 000 008 958 718 771 2;
  • 21) 0.000 000 000 000 008 958 718 771 2 × 2 = 0 + 0.000 000 000 000 017 917 437 542 4;
  • 22) 0.000 000 000 000 017 917 437 542 4 × 2 = 0 + 0.000 000 000 000 035 834 875 084 8;
  • 23) 0.000 000 000 000 035 834 875 084 8 × 2 = 0 + 0.000 000 000 000 071 669 750 169 6;
  • 24) 0.000 000 000 000 071 669 750 169 6 × 2 = 0 + 0.000 000 000 000 143 339 500 339 2;
  • 25) 0.000 000 000 000 143 339 500 339 2 × 2 = 0 + 0.000 000 000 000 286 679 000 678 4;
  • 26) 0.000 000 000 000 286 679 000 678 4 × 2 = 0 + 0.000 000 000 000 573 358 001 356 8;
  • 27) 0.000 000 000 000 573 358 001 356 8 × 2 = 0 + 0.000 000 000 001 146 716 002 713 6;
  • 28) 0.000 000 000 001 146 716 002 713 6 × 2 = 0 + 0.000 000 000 002 293 432 005 427 2;
  • 29) 0.000 000 000 002 293 432 005 427 2 × 2 = 0 + 0.000 000 000 004 586 864 010 854 4;
  • 30) 0.000 000 000 004 586 864 010 854 4 × 2 = 0 + 0.000 000 000 009 173 728 021 708 8;
  • 31) 0.000 000 000 009 173 728 021 708 8 × 2 = 0 + 0.000 000 000 018 347 456 043 417 6;
  • 32) 0.000 000 000 018 347 456 043 417 6 × 2 = 0 + 0.000 000 000 036 694 912 086 835 2;
  • 33) 0.000 000 000 036 694 912 086 835 2 × 2 = 0 + 0.000 000 000 073 389 824 173 670 4;
  • 34) 0.000 000 000 073 389 824 173 670 4 × 2 = 0 + 0.000 000 000 146 779 648 347 340 8;
  • 35) 0.000 000 000 146 779 648 347 340 8 × 2 = 0 + 0.000 000 000 293 559 296 694 681 6;
  • 36) 0.000 000 000 293 559 296 694 681 6 × 2 = 0 + 0.000 000 000 587 118 593 389 363 2;
  • 37) 0.000 000 000 587 118 593 389 363 2 × 2 = 0 + 0.000 000 001 174 237 186 778 726 4;
  • 38) 0.000 000 001 174 237 186 778 726 4 × 2 = 0 + 0.000 000 002 348 474 373 557 452 8;
  • 39) 0.000 000 002 348 474 373 557 452 8 × 2 = 0 + 0.000 000 004 696 948 747 114 905 6;
  • 40) 0.000 000 004 696 948 747 114 905 6 × 2 = 0 + 0.000 000 009 393 897 494 229 811 2;
  • 41) 0.000 000 009 393 897 494 229 811 2 × 2 = 0 + 0.000 000 018 787 794 988 459 622 4;
  • 42) 0.000 000 018 787 794 988 459 622 4 × 2 = 0 + 0.000 000 037 575 589 976 919 244 8;
  • 43) 0.000 000 037 575 589 976 919 244 8 × 2 = 0 + 0.000 000 075 151 179 953 838 489 6;
  • 44) 0.000 000 075 151 179 953 838 489 6 × 2 = 0 + 0.000 000 150 302 359 907 676 979 2;
  • 45) 0.000 000 150 302 359 907 676 979 2 × 2 = 0 + 0.000 000 300 604 719 815 353 958 4;
  • 46) 0.000 000 300 604 719 815 353 958 4 × 2 = 0 + 0.000 000 601 209 439 630 707 916 8;
  • 47) 0.000 000 601 209 439 630 707 916 8 × 2 = 0 + 0.000 001 202 418 879 261 415 833 6;
  • 48) 0.000 001 202 418 879 261 415 833 6 × 2 = 0 + 0.000 002 404 837 758 522 831 667 2;
  • 49) 0.000 002 404 837 758 522 831 667 2 × 2 = 0 + 0.000 004 809 675 517 045 663 334 4;
  • 50) 0.000 004 809 675 517 045 663 334 4 × 2 = 0 + 0.000 009 619 351 034 091 326 668 8;
  • 51) 0.000 009 619 351 034 091 326 668 8 × 2 = 0 + 0.000 019 238 702 068 182 653 337 6;
  • 52) 0.000 019 238 702 068 182 653 337 6 × 2 = 0 + 0.000 038 477 404 136 365 306 675 2;
  • 53) 0.000 038 477 404 136 365 306 675 2 × 2 = 0 + 0.000 076 954 808 272 730 613 350 4;
  • 54) 0.000 076 954 808 272 730 613 350 4 × 2 = 0 + 0.000 153 909 616 545 461 226 700 8;
  • 55) 0.000 153 909 616 545 461 226 700 8 × 2 = 0 + 0.000 307 819 233 090 922 453 401 6;
  • 56) 0.000 307 819 233 090 922 453 401 6 × 2 = 0 + 0.000 615 638 466 181 844 906 803 2;
  • 57) 0.000 615 638 466 181 844 906 803 2 × 2 = 0 + 0.001 231 276 932 363 689 813 606 4;
  • 58) 0.001 231 276 932 363 689 813 606 4 × 2 = 0 + 0.002 462 553 864 727 379 627 212 8;
  • 59) 0.002 462 553 864 727 379 627 212 8 × 2 = 0 + 0.004 925 107 729 454 759 254 425 6;
  • 60) 0.004 925 107 729 454 759 254 425 6 × 2 = 0 + 0.009 850 215 458 909 518 508 851 2;
  • 61) 0.009 850 215 458 909 518 508 851 2 × 2 = 0 + 0.019 700 430 917 819 037 017 702 4;
  • 62) 0.019 700 430 917 819 037 017 702 4 × 2 = 0 + 0.039 400 861 835 638 074 035 404 8;
  • 63) 0.039 400 861 835 638 074 035 404 8 × 2 = 0 + 0.078 801 723 671 276 148 070 809 6;
  • 64) 0.078 801 723 671 276 148 070 809 6 × 2 = 0 + 0.157 603 447 342 552 296 141 619 2;
  • 65) 0.157 603 447 342 552 296 141 619 2 × 2 = 0 + 0.315 206 894 685 104 592 283 238 4;
  • 66) 0.315 206 894 685 104 592 283 238 4 × 2 = 0 + 0.630 413 789 370 209 184 566 476 8;
  • 67) 0.630 413 789 370 209 184 566 476 8 × 2 = 1 + 0.260 827 578 740 418 369 132 953 6;
  • 68) 0.260 827 578 740 418 369 132 953 6 × 2 = 0 + 0.521 655 157 480 836 738 265 907 2;
  • 69) 0.521 655 157 480 836 738 265 907 2 × 2 = 1 + 0.043 310 314 961 673 476 531 814 4;
  • 70) 0.043 310 314 961 673 476 531 814 4 × 2 = 0 + 0.086 620 629 923 346 953 063 628 8;
  • 71) 0.086 620 629 923 346 953 063 628 8 × 2 = 0 + 0.173 241 259 846 693 906 127 257 6;
  • 72) 0.173 241 259 846 693 906 127 257 6 × 2 = 0 + 0.346 482 519 693 387 812 254 515 2;
  • 73) 0.346 482 519 693 387 812 254 515 2 × 2 = 0 + 0.692 965 039 386 775 624 509 030 4;
  • 74) 0.692 965 039 386 775 624 509 030 4 × 2 = 1 + 0.385 930 078 773 551 249 018 060 8;
  • 75) 0.385 930 078 773 551 249 018 060 8 × 2 = 0 + 0.771 860 157 547 102 498 036 121 6;
  • 76) 0.771 860 157 547 102 498 036 121 6 × 2 = 1 + 0.543 720 315 094 204 996 072 243 2;
  • 77) 0.543 720 315 094 204 996 072 243 2 × 2 = 1 + 0.087 440 630 188 409 992 144 486 4;
  • 78) 0.087 440 630 188 409 992 144 486 4 × 2 = 0 + 0.174 881 260 376 819 984 288 972 8;
  • 79) 0.174 881 260 376 819 984 288 972 8 × 2 = 0 + 0.349 762 520 753 639 968 577 945 6;
  • 80) 0.349 762 520 753 639 968 577 945 6 × 2 = 0 + 0.699 525 041 507 279 937 155 891 2;
  • 81) 0.699 525 041 507 279 937 155 891 2 × 2 = 1 + 0.399 050 083 014 559 874 311 782 4;
  • 82) 0.399 050 083 014 559 874 311 782 4 × 2 = 0 + 0.798 100 166 029 119 748 623 564 8;
  • 83) 0.798 100 166 029 119 748 623 564 8 × 2 = 1 + 0.596 200 332 058 239 497 247 129 6;
  • 84) 0.596 200 332 058 239 497 247 129 6 × 2 = 1 + 0.192 400 664 116 478 994 494 259 2;
  • 85) 0.192 400 664 116 478 994 494 259 2 × 2 = 0 + 0.384 801 328 232 957 988 988 518 4;
  • 86) 0.384 801 328 232 957 988 988 518 4 × 2 = 0 + 0.769 602 656 465 915 977 977 036 8;
  • 87) 0.769 602 656 465 915 977 977 036 8 × 2 = 1 + 0.539 205 312 931 831 955 954 073 6;
  • 88) 0.539 205 312 931 831 955 954 073 6 × 2 = 1 + 0.078 410 625 863 663 911 908 147 2;
  • 89) 0.078 410 625 863 663 911 908 147 2 × 2 = 0 + 0.156 821 251 727 327 823 816 294 4;
  • 90) 0.156 821 251 727 327 823 816 294 4 × 2 = 0 + 0.313 642 503 454 655 647 632 588 8;
  • 91) 0.313 642 503 454 655 647 632 588 8 × 2 = 0 + 0.627 285 006 909 311 295 265 177 6;
  • 92) 0.627 285 006 909 311 295 265 177 6 × 2 = 1 + 0.254 570 013 818 622 590 530 355 2;
  • 93) 0.254 570 013 818 622 590 530 355 2 × 2 = 0 + 0.509 140 027 637 245 181 060 710 4;
  • 94) 0.509 140 027 637 245 181 060 710 4 × 2 = 1 + 0.018 280 055 274 490 362 121 420 8;
  • 95) 0.018 280 055 274 490 362 121 420 8 × 2 = 0 + 0.036 560 110 548 980 724 242 841 6;
  • 96) 0.036 560 110 548 980 724 242 841 6 × 2 = 0 + 0.073 120 221 097 961 448 485 683 2;
  • 97) 0.073 120 221 097 961 448 485 683 2 × 2 = 0 + 0.146 240 442 195 922 896 971 366 4;
  • 98) 0.146 240 442 195 922 896 971 366 4 × 2 = 0 + 0.292 480 884 391 845 793 942 732 8;
  • 99) 0.292 480 884 391 845 793 942 732 8 × 2 = 0 + 0.584 961 768 783 691 587 885 465 6;
  • 100) 0.584 961 768 783 691 587 885 465 6 × 2 = 1 + 0.169 923 537 567 383 175 770 931 2;
  • 101) 0.169 923 537 567 383 175 770 931 2 × 2 = 0 + 0.339 847 075 134 766 351 541 862 4;
  • 102) 0.339 847 075 134 766 351 541 862 4 × 2 = 0 + 0.679 694 150 269 532 703 083 724 8;
  • 103) 0.679 694 150 269 532 703 083 724 8 × 2 = 1 + 0.359 388 300 539 065 406 167 449 6;
  • 104) 0.359 388 300 539 065 406 167 449 6 × 2 = 0 + 0.718 776 601 078 130 812 334 899 2;
  • 105) 0.718 776 601 078 130 812 334 899 2 × 2 = 1 + 0.437 553 202 156 261 624 669 798 4;
  • 106) 0.437 553 202 156 261 624 669 798 4 × 2 = 0 + 0.875 106 404 312 523 249 339 596 8;
  • 107) 0.875 106 404 312 523 249 339 596 8 × 2 = 1 + 0.750 212 808 625 046 498 679 193 6;
  • 108) 0.750 212 808 625 046 498 679 193 6 × 2 = 1 + 0.500 425 617 250 092 997 358 387 2;
  • 109) 0.500 425 617 250 092 997 358 387 2 × 2 = 1 + 0.000 851 234 500 185 994 716 774 4;
  • 110) 0.000 851 234 500 185 994 716 774 4 × 2 = 0 + 0.001 702 469 000 371 989 433 548 8;
  • 111) 0.001 702 469 000 371 989 433 548 8 × 2 = 0 + 0.003 404 938 000 743 978 867 097 6;
  • 112) 0.003 404 938 000 743 978 867 097 6 × 2 = 0 + 0.006 809 876 001 487 957 734 195 2;
  • 113) 0.006 809 876 001 487 957 734 195 2 × 2 = 0 + 0.013 619 752 002 975 915 468 390 4;
  • 114) 0.013 619 752 002 975 915 468 390 4 × 2 = 0 + 0.027 239 504 005 951 830 936 780 8;
  • 115) 0.027 239 504 005 951 830 936 780 8 × 2 = 0 + 0.054 479 008 011 903 661 873 561 6;
  • 116) 0.054 479 008 011 903 661 873 561 6 × 2 = 0 + 0.108 958 016 023 807 323 747 123 2;
  • 117) 0.108 958 016 023 807 323 747 123 2 × 2 = 0 + 0.217 916 032 047 614 647 494 246 4;
  • 118) 0.217 916 032 047 614 647 494 246 4 × 2 = 0 + 0.435 832 064 095 229 294 988 492 8;
  • 119) 0.435 832 064 095 229 294 988 492 8 × 2 = 0 + 0.871 664 128 190 458 589 976 985 6;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 000 008 543 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 1011 0011 0001 0100 0001 0010 1011 1000 0000 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 543 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 1011 0011 0001 0100 0001 0010 1011 1000 0000 000(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 543 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 1011 0011 0001 0100 0001 0010 1011 1000 0000 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 1011 0011 0001 0100 0001 0010 1011 1000 0000 000(2) × 20 =

1.0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

956(10) =

011 1011 1100(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. 0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000 =

0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000


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 1011 1100

Mantissa (52 bits) =
0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000


Decimal number 0.000 000 000 000 000 000 008 543 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1100 0101 1001 1000 1010 0000 1001 0101 1100 0000 0000

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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