0.000 000 000 000 000 000 008 536 81 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 81(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
0.000 000 000 000 000 000 008 536 81(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: 0.000 000 000 000 000 000 008 536 81.

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) 0.000 000 000 000 000 000 008 536 81 × 2 = 0 + 0.000 000 000 000 000 000 017 073 62;
  • 2) 0.000 000 000 000 000 000 017 073 62 × 2 = 0 + 0.000 000 000 000 000 000 034 147 24;
  • 3) 0.000 000 000 000 000 000 034 147 24 × 2 = 0 + 0.000 000 000 000 000 000 068 294 48;
  • 4) 0.000 000 000 000 000 000 068 294 48 × 2 = 0 + 0.000 000 000 000 000 000 136 588 96;
  • 5) 0.000 000 000 000 000 000 136 588 96 × 2 = 0 + 0.000 000 000 000 000 000 273 177 92;
  • 6) 0.000 000 000 000 000 000 273 177 92 × 2 = 0 + 0.000 000 000 000 000 000 546 355 84;
  • 7) 0.000 000 000 000 000 000 546 355 84 × 2 = 0 + 0.000 000 000 000 000 001 092 711 68;
  • 8) 0.000 000 000 000 000 001 092 711 68 × 2 = 0 + 0.000 000 000 000 000 002 185 423 36;
  • 9) 0.000 000 000 000 000 002 185 423 36 × 2 = 0 + 0.000 000 000 000 000 004 370 846 72;
  • 10) 0.000 000 000 000 000 004 370 846 72 × 2 = 0 + 0.000 000 000 000 000 008 741 693 44;
  • 11) 0.000 000 000 000 000 008 741 693 44 × 2 = 0 + 0.000 000 000 000 000 017 483 386 88;
  • 12) 0.000 000 000 000 000 017 483 386 88 × 2 = 0 + 0.000 000 000 000 000 034 966 773 76;
  • 13) 0.000 000 000 000 000 034 966 773 76 × 2 = 0 + 0.000 000 000 000 000 069 933 547 52;
  • 14) 0.000 000 000 000 000 069 933 547 52 × 2 = 0 + 0.000 000 000 000 000 139 867 095 04;
  • 15) 0.000 000 000 000 000 139 867 095 04 × 2 = 0 + 0.000 000 000 000 000 279 734 190 08;
  • 16) 0.000 000 000 000 000 279 734 190 08 × 2 = 0 + 0.000 000 000 000 000 559 468 380 16;
  • 17) 0.000 000 000 000 000 559 468 380 16 × 2 = 0 + 0.000 000 000 000 001 118 936 760 32;
  • 18) 0.000 000 000 000 001 118 936 760 32 × 2 = 0 + 0.000 000 000 000 002 237 873 520 64;
  • 19) 0.000 000 000 000 002 237 873 520 64 × 2 = 0 + 0.000 000 000 000 004 475 747 041 28;
  • 20) 0.000 000 000 000 004 475 747 041 28 × 2 = 0 + 0.000 000 000 000 008 951 494 082 56;
  • 21) 0.000 000 000 000 008 951 494 082 56 × 2 = 0 + 0.000 000 000 000 017 902 988 165 12;
  • 22) 0.000 000 000 000 017 902 988 165 12 × 2 = 0 + 0.000 000 000 000 035 805 976 330 24;
  • 23) 0.000 000 000 000 035 805 976 330 24 × 2 = 0 + 0.000 000 000 000 071 611 952 660 48;
  • 24) 0.000 000 000 000 071 611 952 660 48 × 2 = 0 + 0.000 000 000 000 143 223 905 320 96;
  • 25) 0.000 000 000 000 143 223 905 320 96 × 2 = 0 + 0.000 000 000 000 286 447 810 641 92;
  • 26) 0.000 000 000 000 286 447 810 641 92 × 2 = 0 + 0.000 000 000 000 572 895 621 283 84;
  • 27) 0.000 000 000 000 572 895 621 283 84 × 2 = 0 + 0.000 000 000 001 145 791 242 567 68;
  • 28) 0.000 000 000 001 145 791 242 567 68 × 2 = 0 + 0.000 000 000 002 291 582 485 135 36;
  • 29) 0.000 000 000 002 291 582 485 135 36 × 2 = 0 + 0.000 000 000 004 583 164 970 270 72;
  • 30) 0.000 000 000 004 583 164 970 270 72 × 2 = 0 + 0.000 000 000 009 166 329 940 541 44;
  • 31) 0.000 000 000 009 166 329 940 541 44 × 2 = 0 + 0.000 000 000 018 332 659 881 082 88;
  • 32) 0.000 000 000 018 332 659 881 082 88 × 2 = 0 + 0.000 000 000 036 665 319 762 165 76;
  • 33) 0.000 000 000 036 665 319 762 165 76 × 2 = 0 + 0.000 000 000 073 330 639 524 331 52;
  • 34) 0.000 000 000 073 330 639 524 331 52 × 2 = 0 + 0.000 000 000 146 661 279 048 663 04;
  • 35) 0.000 000 000 146 661 279 048 663 04 × 2 = 0 + 0.000 000 000 293 322 558 097 326 08;
  • 36) 0.000 000 000 293 322 558 097 326 08 × 2 = 0 + 0.000 000 000 586 645 116 194 652 16;
  • 37) 0.000 000 000 586 645 116 194 652 16 × 2 = 0 + 0.000 000 001 173 290 232 389 304 32;
  • 38) 0.000 000 001 173 290 232 389 304 32 × 2 = 0 + 0.000 000 002 346 580 464 778 608 64;
  • 39) 0.000 000 002 346 580 464 778 608 64 × 2 = 0 + 0.000 000 004 693 160 929 557 217 28;
  • 40) 0.000 000 004 693 160 929 557 217 28 × 2 = 0 + 0.000 000 009 386 321 859 114 434 56;
  • 41) 0.000 000 009 386 321 859 114 434 56 × 2 = 0 + 0.000 000 018 772 643 718 228 869 12;
  • 42) 0.000 000 018 772 643 718 228 869 12 × 2 = 0 + 0.000 000 037 545 287 436 457 738 24;
  • 43) 0.000 000 037 545 287 436 457 738 24 × 2 = 0 + 0.000 000 075 090 574 872 915 476 48;
  • 44) 0.000 000 075 090 574 872 915 476 48 × 2 = 0 + 0.000 000 150 181 149 745 830 952 96;
  • 45) 0.000 000 150 181 149 745 830 952 96 × 2 = 0 + 0.000 000 300 362 299 491 661 905 92;
  • 46) 0.000 000 300 362 299 491 661 905 92 × 2 = 0 + 0.000 000 600 724 598 983 323 811 84;
  • 47) 0.000 000 600 724 598 983 323 811 84 × 2 = 0 + 0.000 001 201 449 197 966 647 623 68;
  • 48) 0.000 001 201 449 197 966 647 623 68 × 2 = 0 + 0.000 002 402 898 395 933 295 247 36;
  • 49) 0.000 002 402 898 395 933 295 247 36 × 2 = 0 + 0.000 004 805 796 791 866 590 494 72;
  • 50) 0.000 004 805 796 791 866 590 494 72 × 2 = 0 + 0.000 009 611 593 583 733 180 989 44;
  • 51) 0.000 009 611 593 583 733 180 989 44 × 2 = 0 + 0.000 019 223 187 167 466 361 978 88;
  • 52) 0.000 019 223 187 167 466 361 978 88 × 2 = 0 + 0.000 038 446 374 334 932 723 957 76;
  • 53) 0.000 038 446 374 334 932 723 957 76 × 2 = 0 + 0.000 076 892 748 669 865 447 915 52;
  • 54) 0.000 076 892 748 669 865 447 915 52 × 2 = 0 + 0.000 153 785 497 339 730 895 831 04;
  • 55) 0.000 153 785 497 339 730 895 831 04 × 2 = 0 + 0.000 307 570 994 679 461 791 662 08;
  • 56) 0.000 307 570 994 679 461 791 662 08 × 2 = 0 + 0.000 615 141 989 358 923 583 324 16;
  • 57) 0.000 615 141 989 358 923 583 324 16 × 2 = 0 + 0.001 230 283 978 717 847 166 648 32;
  • 58) 0.001 230 283 978 717 847 166 648 32 × 2 = 0 + 0.002 460 567 957 435 694 333 296 64;
  • 59) 0.002 460 567 957 435 694 333 296 64 × 2 = 0 + 0.004 921 135 914 871 388 666 593 28;
  • 60) 0.004 921 135 914 871 388 666 593 28 × 2 = 0 + 0.009 842 271 829 742 777 333 186 56;
  • 61) 0.009 842 271 829 742 777 333 186 56 × 2 = 0 + 0.019 684 543 659 485 554 666 373 12;
  • 62) 0.019 684 543 659 485 554 666 373 12 × 2 = 0 + 0.039 369 087 318 971 109 332 746 24;
  • 63) 0.039 369 087 318 971 109 332 746 24 × 2 = 0 + 0.078 738 174 637 942 218 665 492 48;
  • 64) 0.078 738 174 637 942 218 665 492 48 × 2 = 0 + 0.157 476 349 275 884 437 330 984 96;
  • 65) 0.157 476 349 275 884 437 330 984 96 × 2 = 0 + 0.314 952 698 551 768 874 661 969 92;
  • 66) 0.314 952 698 551 768 874 661 969 92 × 2 = 0 + 0.629 905 397 103 537 749 323 939 84;
  • 67) 0.629 905 397 103 537 749 323 939 84 × 2 = 1 + 0.259 810 794 207 075 498 647 879 68;
  • 68) 0.259 810 794 207 075 498 647 879 68 × 2 = 0 + 0.519 621 588 414 150 997 295 759 36;
  • 69) 0.519 621 588 414 150 997 295 759 36 × 2 = 1 + 0.039 243 176 828 301 994 591 518 72;
  • 70) 0.039 243 176 828 301 994 591 518 72 × 2 = 0 + 0.078 486 353 656 603 989 183 037 44;
  • 71) 0.078 486 353 656 603 989 183 037 44 × 2 = 0 + 0.156 972 707 313 207 978 366 074 88;
  • 72) 0.156 972 707 313 207 978 366 074 88 × 2 = 0 + 0.313 945 414 626 415 956 732 149 76;
  • 73) 0.313 945 414 626 415 956 732 149 76 × 2 = 0 + 0.627 890 829 252 831 913 464 299 52;
  • 74) 0.627 890 829 252 831 913 464 299 52 × 2 = 1 + 0.255 781 658 505 663 826 928 599 04;
  • 75) 0.255 781 658 505 663 826 928 599 04 × 2 = 0 + 0.511 563 317 011 327 653 857 198 08;
  • 76) 0.511 563 317 011 327 653 857 198 08 × 2 = 1 + 0.023 126 634 022 655 307 714 396 16;
  • 77) 0.023 126 634 022 655 307 714 396 16 × 2 = 0 + 0.046 253 268 045 310 615 428 792 32;
  • 78) 0.046 253 268 045 310 615 428 792 32 × 2 = 0 + 0.092 506 536 090 621 230 857 584 64;
  • 79) 0.092 506 536 090 621 230 857 584 64 × 2 = 0 + 0.185 013 072 181 242 461 715 169 28;
  • 80) 0.185 013 072 181 242 461 715 169 28 × 2 = 0 + 0.370 026 144 362 484 923 430 338 56;
  • 81) 0.370 026 144 362 484 923 430 338 56 × 2 = 0 + 0.740 052 288 724 969 846 860 677 12;
  • 82) 0.740 052 288 724 969 846 860 677 12 × 2 = 1 + 0.480 104 577 449 939 693 721 354 24;
  • 83) 0.480 104 577 449 939 693 721 354 24 × 2 = 0 + 0.960 209 154 899 879 387 442 708 48;
  • 84) 0.960 209 154 899 879 387 442 708 48 × 2 = 1 + 0.920 418 309 799 758 774 885 416 96;
  • 85) 0.920 418 309 799 758 774 885 416 96 × 2 = 1 + 0.840 836 619 599 517 549 770 833 92;
  • 86) 0.840 836 619 599 517 549 770 833 92 × 2 = 1 + 0.681 673 239 199 035 099 541 667 84;
  • 87) 0.681 673 239 199 035 099 541 667 84 × 2 = 1 + 0.363 346 478 398 070 199 083 335 68;
  • 88) 0.363 346 478 398 070 199 083 335 68 × 2 = 0 + 0.726 692 956 796 140 398 166 671 36;
  • 89) 0.726 692 956 796 140 398 166 671 36 × 2 = 1 + 0.453 385 913 592 280 796 333 342 72;
  • 90) 0.453 385 913 592 280 796 333 342 72 × 2 = 0 + 0.906 771 827 184 561 592 666 685 44;
  • 91) 0.906 771 827 184 561 592 666 685 44 × 2 = 1 + 0.813 543 654 369 123 185 333 370 88;
  • 92) 0.813 543 654 369 123 185 333 370 88 × 2 = 1 + 0.627 087 308 738 246 370 666 741 76;
  • 93) 0.627 087 308 738 246 370 666 741 76 × 2 = 1 + 0.254 174 617 476 492 741 333 483 52;
  • 94) 0.254 174 617 476 492 741 333 483 52 × 2 = 0 + 0.508 349 234 952 985 482 666 967 04;
  • 95) 0.508 349 234 952 985 482 666 967 04 × 2 = 1 + 0.016 698 469 905 970 965 333 934 08;
  • 96) 0.016 698 469 905 970 965 333 934 08 × 2 = 0 + 0.033 396 939 811 941 930 667 868 16;
  • 97) 0.033 396 939 811 941 930 667 868 16 × 2 = 0 + 0.066 793 879 623 883 861 335 736 32;
  • 98) 0.066 793 879 623 883 861 335 736 32 × 2 = 0 + 0.133 587 759 247 767 722 671 472 64;
  • 99) 0.133 587 759 247 767 722 671 472 64 × 2 = 0 + 0.267 175 518 495 535 445 342 945 28;
  • 100) 0.267 175 518 495 535 445 342 945 28 × 2 = 0 + 0.534 351 036 991 070 890 685 890 56;
  • 101) 0.534 351 036 991 070 890 685 890 56 × 2 = 1 + 0.068 702 073 982 141 781 371 781 12;
  • 102) 0.068 702 073 982 141 781 371 781 12 × 2 = 0 + 0.137 404 147 964 283 562 743 562 24;
  • 103) 0.137 404 147 964 283 562 743 562 24 × 2 = 0 + 0.274 808 295 928 567 125 487 124 48;
  • 104) 0.274 808 295 928 567 125 487 124 48 × 2 = 0 + 0.549 616 591 857 134 250 974 248 96;
  • 105) 0.549 616 591 857 134 250 974 248 96 × 2 = 1 + 0.099 233 183 714 268 501 948 497 92;
  • 106) 0.099 233 183 714 268 501 948 497 92 × 2 = 0 + 0.198 466 367 428 537 003 896 995 84;
  • 107) 0.198 466 367 428 537 003 896 995 84 × 2 = 0 + 0.396 932 734 857 074 007 793 991 68;
  • 108) 0.396 932 734 857 074 007 793 991 68 × 2 = 0 + 0.793 865 469 714 148 015 587 983 36;
  • 109) 0.793 865 469 714 148 015 587 983 36 × 2 = 1 + 0.587 730 939 428 296 031 175 966 72;
  • 110) 0.587 730 939 428 296 031 175 966 72 × 2 = 1 + 0.175 461 878 856 592 062 351 933 44;
  • 111) 0.175 461 878 856 592 062 351 933 44 × 2 = 0 + 0.350 923 757 713 184 124 703 866 88;
  • 112) 0.350 923 757 713 184 124 703 866 88 × 2 = 0 + 0.701 847 515 426 368 249 407 733 76;
  • 113) 0.701 847 515 426 368 249 407 733 76 × 2 = 1 + 0.403 695 030 852 736 498 815 467 52;
  • 114) 0.403 695 030 852 736 498 815 467 52 × 2 = 0 + 0.807 390 061 705 472 997 630 935 04;
  • 115) 0.807 390 061 705 472 997 630 935 04 × 2 = 1 + 0.614 780 123 410 945 995 261 870 08;
  • 116) 0.614 780 123 410 945 995 261 870 08 × 2 = 1 + 0.229 560 246 821 891 990 523 740 16;
  • 117) 0.229 560 246 821 891 990 523 740 16 × 2 = 0 + 0.459 120 493 643 783 981 047 480 32;
  • 118) 0.459 120 493 643 783 981 047 480 32 × 2 = 0 + 0.918 240 987 287 567 962 094 960 64;
  • 119) 0.918 240 987 287 567 962 094 960 64 × 2 = 1 + 0.836 481 974 575 135 924 189 921 28;

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:


0.000 000 000 000 000 000 008 536 81(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 1110 1011 1010 0000 1000 1000 1100 1011 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 81(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 1110 1011 1010 0000 1000 1000 1100 1011 001(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:


0.000 000 000 000 000 000 008 536 81(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 1110 1011 1010 0000 1000 1000 1100 1011 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 1110 1011 1010 0000 1000 1000 1100 1011 001(2) × 20 =

1.0100 0010 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 1001(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 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 1001


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 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 1001 =

0100 0010 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 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 1011 1100

Mantissa (52 bits) =
0100 0010 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 1001


Decimal number 0.000 000 000 000 000 000 008 536 81 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0010 1111 0101 1101 0000 0100 0100 0110 0101 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