0.000 000 000 000 000 000 008 537 94 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 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
0.000 000 000 000 000 000 008 537 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: 0.000 000 000 000 000 000 008 537 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) 0.000 000 000 000 000 000 008 537 94 × 2 = 0 + 0.000 000 000 000 000 000 017 075 88;
  • 2) 0.000 000 000 000 000 000 017 075 88 × 2 = 0 + 0.000 000 000 000 000 000 034 151 76;
  • 3) 0.000 000 000 000 000 000 034 151 76 × 2 = 0 + 0.000 000 000 000 000 000 068 303 52;
  • 4) 0.000 000 000 000 000 000 068 303 52 × 2 = 0 + 0.000 000 000 000 000 000 136 607 04;
  • 5) 0.000 000 000 000 000 000 136 607 04 × 2 = 0 + 0.000 000 000 000 000 000 273 214 08;
  • 6) 0.000 000 000 000 000 000 273 214 08 × 2 = 0 + 0.000 000 000 000 000 000 546 428 16;
  • 7) 0.000 000 000 000 000 000 546 428 16 × 2 = 0 + 0.000 000 000 000 000 001 092 856 32;
  • 8) 0.000 000 000 000 000 001 092 856 32 × 2 = 0 + 0.000 000 000 000 000 002 185 712 64;
  • 9) 0.000 000 000 000 000 002 185 712 64 × 2 = 0 + 0.000 000 000 000 000 004 371 425 28;
  • 10) 0.000 000 000 000 000 004 371 425 28 × 2 = 0 + 0.000 000 000 000 000 008 742 850 56;
  • 11) 0.000 000 000 000 000 008 742 850 56 × 2 = 0 + 0.000 000 000 000 000 017 485 701 12;
  • 12) 0.000 000 000 000 000 017 485 701 12 × 2 = 0 + 0.000 000 000 000 000 034 971 402 24;
  • 13) 0.000 000 000 000 000 034 971 402 24 × 2 = 0 + 0.000 000 000 000 000 069 942 804 48;
  • 14) 0.000 000 000 000 000 069 942 804 48 × 2 = 0 + 0.000 000 000 000 000 139 885 608 96;
  • 15) 0.000 000 000 000 000 139 885 608 96 × 2 = 0 + 0.000 000 000 000 000 279 771 217 92;
  • 16) 0.000 000 000 000 000 279 771 217 92 × 2 = 0 + 0.000 000 000 000 000 559 542 435 84;
  • 17) 0.000 000 000 000 000 559 542 435 84 × 2 = 0 + 0.000 000 000 000 001 119 084 871 68;
  • 18) 0.000 000 000 000 001 119 084 871 68 × 2 = 0 + 0.000 000 000 000 002 238 169 743 36;
  • 19) 0.000 000 000 000 002 238 169 743 36 × 2 = 0 + 0.000 000 000 000 004 476 339 486 72;
  • 20) 0.000 000 000 000 004 476 339 486 72 × 2 = 0 + 0.000 000 000 000 008 952 678 973 44;
  • 21) 0.000 000 000 000 008 952 678 973 44 × 2 = 0 + 0.000 000 000 000 017 905 357 946 88;
  • 22) 0.000 000 000 000 017 905 357 946 88 × 2 = 0 + 0.000 000 000 000 035 810 715 893 76;
  • 23) 0.000 000 000 000 035 810 715 893 76 × 2 = 0 + 0.000 000 000 000 071 621 431 787 52;
  • 24) 0.000 000 000 000 071 621 431 787 52 × 2 = 0 + 0.000 000 000 000 143 242 863 575 04;
  • 25) 0.000 000 000 000 143 242 863 575 04 × 2 = 0 + 0.000 000 000 000 286 485 727 150 08;
  • 26) 0.000 000 000 000 286 485 727 150 08 × 2 = 0 + 0.000 000 000 000 572 971 454 300 16;
  • 27) 0.000 000 000 000 572 971 454 300 16 × 2 = 0 + 0.000 000 000 001 145 942 908 600 32;
  • 28) 0.000 000 000 001 145 942 908 600 32 × 2 = 0 + 0.000 000 000 002 291 885 817 200 64;
  • 29) 0.000 000 000 002 291 885 817 200 64 × 2 = 0 + 0.000 000 000 004 583 771 634 401 28;
  • 30) 0.000 000 000 004 583 771 634 401 28 × 2 = 0 + 0.000 000 000 009 167 543 268 802 56;
  • 31) 0.000 000 000 009 167 543 268 802 56 × 2 = 0 + 0.000 000 000 018 335 086 537 605 12;
  • 32) 0.000 000 000 018 335 086 537 605 12 × 2 = 0 + 0.000 000 000 036 670 173 075 210 24;
  • 33) 0.000 000 000 036 670 173 075 210 24 × 2 = 0 + 0.000 000 000 073 340 346 150 420 48;
  • 34) 0.000 000 000 073 340 346 150 420 48 × 2 = 0 + 0.000 000 000 146 680 692 300 840 96;
  • 35) 0.000 000 000 146 680 692 300 840 96 × 2 = 0 + 0.000 000 000 293 361 384 601 681 92;
  • 36) 0.000 000 000 293 361 384 601 681 92 × 2 = 0 + 0.000 000 000 586 722 769 203 363 84;
  • 37) 0.000 000 000 586 722 769 203 363 84 × 2 = 0 + 0.000 000 001 173 445 538 406 727 68;
  • 38) 0.000 000 001 173 445 538 406 727 68 × 2 = 0 + 0.000 000 002 346 891 076 813 455 36;
  • 39) 0.000 000 002 346 891 076 813 455 36 × 2 = 0 + 0.000 000 004 693 782 153 626 910 72;
  • 40) 0.000 000 004 693 782 153 626 910 72 × 2 = 0 + 0.000 000 009 387 564 307 253 821 44;
  • 41) 0.000 000 009 387 564 307 253 821 44 × 2 = 0 + 0.000 000 018 775 128 614 507 642 88;
  • 42) 0.000 000 018 775 128 614 507 642 88 × 2 = 0 + 0.000 000 037 550 257 229 015 285 76;
  • 43) 0.000 000 037 550 257 229 015 285 76 × 2 = 0 + 0.000 000 075 100 514 458 030 571 52;
  • 44) 0.000 000 075 100 514 458 030 571 52 × 2 = 0 + 0.000 000 150 201 028 916 061 143 04;
  • 45) 0.000 000 150 201 028 916 061 143 04 × 2 = 0 + 0.000 000 300 402 057 832 122 286 08;
  • 46) 0.000 000 300 402 057 832 122 286 08 × 2 = 0 + 0.000 000 600 804 115 664 244 572 16;
  • 47) 0.000 000 600 804 115 664 244 572 16 × 2 = 0 + 0.000 001 201 608 231 328 489 144 32;
  • 48) 0.000 001 201 608 231 328 489 144 32 × 2 = 0 + 0.000 002 403 216 462 656 978 288 64;
  • 49) 0.000 002 403 216 462 656 978 288 64 × 2 = 0 + 0.000 004 806 432 925 313 956 577 28;
  • 50) 0.000 004 806 432 925 313 956 577 28 × 2 = 0 + 0.000 009 612 865 850 627 913 154 56;
  • 51) 0.000 009 612 865 850 627 913 154 56 × 2 = 0 + 0.000 019 225 731 701 255 826 309 12;
  • 52) 0.000 019 225 731 701 255 826 309 12 × 2 = 0 + 0.000 038 451 463 402 511 652 618 24;
  • 53) 0.000 038 451 463 402 511 652 618 24 × 2 = 0 + 0.000 076 902 926 805 023 305 236 48;
  • 54) 0.000 076 902 926 805 023 305 236 48 × 2 = 0 + 0.000 153 805 853 610 046 610 472 96;
  • 55) 0.000 153 805 853 610 046 610 472 96 × 2 = 0 + 0.000 307 611 707 220 093 220 945 92;
  • 56) 0.000 307 611 707 220 093 220 945 92 × 2 = 0 + 0.000 615 223 414 440 186 441 891 84;
  • 57) 0.000 615 223 414 440 186 441 891 84 × 2 = 0 + 0.001 230 446 828 880 372 883 783 68;
  • 58) 0.001 230 446 828 880 372 883 783 68 × 2 = 0 + 0.002 460 893 657 760 745 767 567 36;
  • 59) 0.002 460 893 657 760 745 767 567 36 × 2 = 0 + 0.004 921 787 315 521 491 535 134 72;
  • 60) 0.004 921 787 315 521 491 535 134 72 × 2 = 0 + 0.009 843 574 631 042 983 070 269 44;
  • 61) 0.009 843 574 631 042 983 070 269 44 × 2 = 0 + 0.019 687 149 262 085 966 140 538 88;
  • 62) 0.019 687 149 262 085 966 140 538 88 × 2 = 0 + 0.039 374 298 524 171 932 281 077 76;
  • 63) 0.039 374 298 524 171 932 281 077 76 × 2 = 0 + 0.078 748 597 048 343 864 562 155 52;
  • 64) 0.078 748 597 048 343 864 562 155 52 × 2 = 0 + 0.157 497 194 096 687 729 124 311 04;
  • 65) 0.157 497 194 096 687 729 124 311 04 × 2 = 0 + 0.314 994 388 193 375 458 248 622 08;
  • 66) 0.314 994 388 193 375 458 248 622 08 × 2 = 0 + 0.629 988 776 386 750 916 497 244 16;
  • 67) 0.629 988 776 386 750 916 497 244 16 × 2 = 1 + 0.259 977 552 773 501 832 994 488 32;
  • 68) 0.259 977 552 773 501 832 994 488 32 × 2 = 0 + 0.519 955 105 547 003 665 988 976 64;
  • 69) 0.519 955 105 547 003 665 988 976 64 × 2 = 1 + 0.039 910 211 094 007 331 977 953 28;
  • 70) 0.039 910 211 094 007 331 977 953 28 × 2 = 0 + 0.079 820 422 188 014 663 955 906 56;
  • 71) 0.079 820 422 188 014 663 955 906 56 × 2 = 0 + 0.159 640 844 376 029 327 911 813 12;
  • 72) 0.159 640 844 376 029 327 911 813 12 × 2 = 0 + 0.319 281 688 752 058 655 823 626 24;
  • 73) 0.319 281 688 752 058 655 823 626 24 × 2 = 0 + 0.638 563 377 504 117 311 647 252 48;
  • 74) 0.638 563 377 504 117 311 647 252 48 × 2 = 1 + 0.277 126 755 008 234 623 294 504 96;
  • 75) 0.277 126 755 008 234 623 294 504 96 × 2 = 0 + 0.554 253 510 016 469 246 589 009 92;
  • 76) 0.554 253 510 016 469 246 589 009 92 × 2 = 1 + 0.108 507 020 032 938 493 178 019 84;
  • 77) 0.108 507 020 032 938 493 178 019 84 × 2 = 0 + 0.217 014 040 065 876 986 356 039 68;
  • 78) 0.217 014 040 065 876 986 356 039 68 × 2 = 0 + 0.434 028 080 131 753 972 712 079 36;
  • 79) 0.434 028 080 131 753 972 712 079 36 × 2 = 0 + 0.868 056 160 263 507 945 424 158 72;
  • 80) 0.868 056 160 263 507 945 424 158 72 × 2 = 1 + 0.736 112 320 527 015 890 848 317 44;
  • 81) 0.736 112 320 527 015 890 848 317 44 × 2 = 1 + 0.472 224 641 054 031 781 696 634 88;
  • 82) 0.472 224 641 054 031 781 696 634 88 × 2 = 0 + 0.944 449 282 108 063 563 393 269 76;
  • 83) 0.944 449 282 108 063 563 393 269 76 × 2 = 1 + 0.888 898 564 216 127 126 786 539 52;
  • 84) 0.888 898 564 216 127 126 786 539 52 × 2 = 1 + 0.777 797 128 432 254 253 573 079 04;
  • 85) 0.777 797 128 432 254 253 573 079 04 × 2 = 1 + 0.555 594 256 864 508 507 146 158 08;
  • 86) 0.555 594 256 864 508 507 146 158 08 × 2 = 1 + 0.111 188 513 729 017 014 292 316 16;
  • 87) 0.111 188 513 729 017 014 292 316 16 × 2 = 0 + 0.222 377 027 458 034 028 584 632 32;
  • 88) 0.222 377 027 458 034 028 584 632 32 × 2 = 0 + 0.444 754 054 916 068 057 169 264 64;
  • 89) 0.444 754 054 916 068 057 169 264 64 × 2 = 0 + 0.889 508 109 832 136 114 338 529 28;
  • 90) 0.889 508 109 832 136 114 338 529 28 × 2 = 1 + 0.779 016 219 664 272 228 677 058 56;
  • 91) 0.779 016 219 664 272 228 677 058 56 × 2 = 1 + 0.558 032 439 328 544 457 354 117 12;
  • 92) 0.558 032 439 328 544 457 354 117 12 × 2 = 1 + 0.116 064 878 657 088 914 708 234 24;
  • 93) 0.116 064 878 657 088 914 708 234 24 × 2 = 0 + 0.232 129 757 314 177 829 416 468 48;
  • 94) 0.232 129 757 314 177 829 416 468 48 × 2 = 0 + 0.464 259 514 628 355 658 832 936 96;
  • 95) 0.464 259 514 628 355 658 832 936 96 × 2 = 0 + 0.928 519 029 256 711 317 665 873 92;
  • 96) 0.928 519 029 256 711 317 665 873 92 × 2 = 1 + 0.857 038 058 513 422 635 331 747 84;
  • 97) 0.857 038 058 513 422 635 331 747 84 × 2 = 1 + 0.714 076 117 026 845 270 663 495 68;
  • 98) 0.714 076 117 026 845 270 663 495 68 × 2 = 1 + 0.428 152 234 053 690 541 326 991 36;
  • 99) 0.428 152 234 053 690 541 326 991 36 × 2 = 0 + 0.856 304 468 107 381 082 653 982 72;
  • 100) 0.856 304 468 107 381 082 653 982 72 × 2 = 1 + 0.712 608 936 214 762 165 307 965 44;
  • 101) 0.712 608 936 214 762 165 307 965 44 × 2 = 1 + 0.425 217 872 429 524 330 615 930 88;
  • 102) 0.425 217 872 429 524 330 615 930 88 × 2 = 0 + 0.850 435 744 859 048 661 231 861 76;
  • 103) 0.850 435 744 859 048 661 231 861 76 × 2 = 1 + 0.700 871 489 718 097 322 463 723 52;
  • 104) 0.700 871 489 718 097 322 463 723 52 × 2 = 1 + 0.401 742 979 436 194 644 927 447 04;
  • 105) 0.401 742 979 436 194 644 927 447 04 × 2 = 0 + 0.803 485 958 872 389 289 854 894 08;
  • 106) 0.803 485 958 872 389 289 854 894 08 × 2 = 1 + 0.606 971 917 744 778 579 709 788 16;
  • 107) 0.606 971 917 744 778 579 709 788 16 × 2 = 1 + 0.213 943 835 489 557 159 419 576 32;
  • 108) 0.213 943 835 489 557 159 419 576 32 × 2 = 0 + 0.427 887 670 979 114 318 839 152 64;
  • 109) 0.427 887 670 979 114 318 839 152 64 × 2 = 0 + 0.855 775 341 958 228 637 678 305 28;
  • 110) 0.855 775 341 958 228 637 678 305 28 × 2 = 1 + 0.711 550 683 916 457 275 356 610 56;
  • 111) 0.711 550 683 916 457 275 356 610 56 × 2 = 1 + 0.423 101 367 832 914 550 713 221 12;
  • 112) 0.423 101 367 832 914 550 713 221 12 × 2 = 0 + 0.846 202 735 665 829 101 426 442 24;
  • 113) 0.846 202 735 665 829 101 426 442 24 × 2 = 1 + 0.692 405 471 331 658 202 852 884 48;
  • 114) 0.692 405 471 331 658 202 852 884 48 × 2 = 1 + 0.384 810 942 663 316 405 705 768 96;
  • 115) 0.384 810 942 663 316 405 705 768 96 × 2 = 0 + 0.769 621 885 326 632 811 411 537 92;
  • 116) 0.769 621 885 326 632 811 411 537 92 × 2 = 1 + 0.539 243 770 653 265 622 823 075 84;
  • 117) 0.539 243 770 653 265 622 823 075 84 × 2 = 1 + 0.078 487 541 306 531 245 646 151 68;
  • 118) 0.078 487 541 306 531 245 646 151 68 × 2 = 0 + 0.156 975 082 613 062 491 292 303 36;
  • 119) 0.156 975 082 613 062 491 292 303 36 × 2 = 0 + 0.313 950 165 226 124 982 584 606 72;

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:


0.000 000 000 000 000 000 008 537 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1011 1100 0111 0001 1101 1011 0110 0110 1101 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1011 1100 0111 0001 1101 1011 0110 0110 1101 100(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:


0.000 000 000 000 000 000 008 537 94(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1011 1100 0111 0001 1101 1011 0110 0110 1101 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1011 1100 0111 0001 1101 1011 0110 0110 1101 100(2) × 20 =

1.0100 0010 1000 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100(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 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100


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 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100 =

0100 0010 1000 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100


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 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100


Decimal number 0.000 000 000 000 000 000 008 537 94 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1101 1110 0011 1000 1110 1101 1011 0011 0110 1100

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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