0.000 000 000 000 000 000 008 535 63 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 63(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
0.000 000 000 000 000 000 008 535 63(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: 0.000 000 000 000 000 000 008 535 63.

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) 0.000 000 000 000 000 000 008 535 63 × 2 = 0 + 0.000 000 000 000 000 000 017 071 26;
  • 2) 0.000 000 000 000 000 000 017 071 26 × 2 = 0 + 0.000 000 000 000 000 000 034 142 52;
  • 3) 0.000 000 000 000 000 000 034 142 52 × 2 = 0 + 0.000 000 000 000 000 000 068 285 04;
  • 4) 0.000 000 000 000 000 000 068 285 04 × 2 = 0 + 0.000 000 000 000 000 000 136 570 08;
  • 5) 0.000 000 000 000 000 000 136 570 08 × 2 = 0 + 0.000 000 000 000 000 000 273 140 16;
  • 6) 0.000 000 000 000 000 000 273 140 16 × 2 = 0 + 0.000 000 000 000 000 000 546 280 32;
  • 7) 0.000 000 000 000 000 000 546 280 32 × 2 = 0 + 0.000 000 000 000 000 001 092 560 64;
  • 8) 0.000 000 000 000 000 001 092 560 64 × 2 = 0 + 0.000 000 000 000 000 002 185 121 28;
  • 9) 0.000 000 000 000 000 002 185 121 28 × 2 = 0 + 0.000 000 000 000 000 004 370 242 56;
  • 10) 0.000 000 000 000 000 004 370 242 56 × 2 = 0 + 0.000 000 000 000 000 008 740 485 12;
  • 11) 0.000 000 000 000 000 008 740 485 12 × 2 = 0 + 0.000 000 000 000 000 017 480 970 24;
  • 12) 0.000 000 000 000 000 017 480 970 24 × 2 = 0 + 0.000 000 000 000 000 034 961 940 48;
  • 13) 0.000 000 000 000 000 034 961 940 48 × 2 = 0 + 0.000 000 000 000 000 069 923 880 96;
  • 14) 0.000 000 000 000 000 069 923 880 96 × 2 = 0 + 0.000 000 000 000 000 139 847 761 92;
  • 15) 0.000 000 000 000 000 139 847 761 92 × 2 = 0 + 0.000 000 000 000 000 279 695 523 84;
  • 16) 0.000 000 000 000 000 279 695 523 84 × 2 = 0 + 0.000 000 000 000 000 559 391 047 68;
  • 17) 0.000 000 000 000 000 559 391 047 68 × 2 = 0 + 0.000 000 000 000 001 118 782 095 36;
  • 18) 0.000 000 000 000 001 118 782 095 36 × 2 = 0 + 0.000 000 000 000 002 237 564 190 72;
  • 19) 0.000 000 000 000 002 237 564 190 72 × 2 = 0 + 0.000 000 000 000 004 475 128 381 44;
  • 20) 0.000 000 000 000 004 475 128 381 44 × 2 = 0 + 0.000 000 000 000 008 950 256 762 88;
  • 21) 0.000 000 000 000 008 950 256 762 88 × 2 = 0 + 0.000 000 000 000 017 900 513 525 76;
  • 22) 0.000 000 000 000 017 900 513 525 76 × 2 = 0 + 0.000 000 000 000 035 801 027 051 52;
  • 23) 0.000 000 000 000 035 801 027 051 52 × 2 = 0 + 0.000 000 000 000 071 602 054 103 04;
  • 24) 0.000 000 000 000 071 602 054 103 04 × 2 = 0 + 0.000 000 000 000 143 204 108 206 08;
  • 25) 0.000 000 000 000 143 204 108 206 08 × 2 = 0 + 0.000 000 000 000 286 408 216 412 16;
  • 26) 0.000 000 000 000 286 408 216 412 16 × 2 = 0 + 0.000 000 000 000 572 816 432 824 32;
  • 27) 0.000 000 000 000 572 816 432 824 32 × 2 = 0 + 0.000 000 000 001 145 632 865 648 64;
  • 28) 0.000 000 000 001 145 632 865 648 64 × 2 = 0 + 0.000 000 000 002 291 265 731 297 28;
  • 29) 0.000 000 000 002 291 265 731 297 28 × 2 = 0 + 0.000 000 000 004 582 531 462 594 56;
  • 30) 0.000 000 000 004 582 531 462 594 56 × 2 = 0 + 0.000 000 000 009 165 062 925 189 12;
  • 31) 0.000 000 000 009 165 062 925 189 12 × 2 = 0 + 0.000 000 000 018 330 125 850 378 24;
  • 32) 0.000 000 000 018 330 125 850 378 24 × 2 = 0 + 0.000 000 000 036 660 251 700 756 48;
  • 33) 0.000 000 000 036 660 251 700 756 48 × 2 = 0 + 0.000 000 000 073 320 503 401 512 96;
  • 34) 0.000 000 000 073 320 503 401 512 96 × 2 = 0 + 0.000 000 000 146 641 006 803 025 92;
  • 35) 0.000 000 000 146 641 006 803 025 92 × 2 = 0 + 0.000 000 000 293 282 013 606 051 84;
  • 36) 0.000 000 000 293 282 013 606 051 84 × 2 = 0 + 0.000 000 000 586 564 027 212 103 68;
  • 37) 0.000 000 000 586 564 027 212 103 68 × 2 = 0 + 0.000 000 001 173 128 054 424 207 36;
  • 38) 0.000 000 001 173 128 054 424 207 36 × 2 = 0 + 0.000 000 002 346 256 108 848 414 72;
  • 39) 0.000 000 002 346 256 108 848 414 72 × 2 = 0 + 0.000 000 004 692 512 217 696 829 44;
  • 40) 0.000 000 004 692 512 217 696 829 44 × 2 = 0 + 0.000 000 009 385 024 435 393 658 88;
  • 41) 0.000 000 009 385 024 435 393 658 88 × 2 = 0 + 0.000 000 018 770 048 870 787 317 76;
  • 42) 0.000 000 018 770 048 870 787 317 76 × 2 = 0 + 0.000 000 037 540 097 741 574 635 52;
  • 43) 0.000 000 037 540 097 741 574 635 52 × 2 = 0 + 0.000 000 075 080 195 483 149 271 04;
  • 44) 0.000 000 075 080 195 483 149 271 04 × 2 = 0 + 0.000 000 150 160 390 966 298 542 08;
  • 45) 0.000 000 150 160 390 966 298 542 08 × 2 = 0 + 0.000 000 300 320 781 932 597 084 16;
  • 46) 0.000 000 300 320 781 932 597 084 16 × 2 = 0 + 0.000 000 600 641 563 865 194 168 32;
  • 47) 0.000 000 600 641 563 865 194 168 32 × 2 = 0 + 0.000 001 201 283 127 730 388 336 64;
  • 48) 0.000 001 201 283 127 730 388 336 64 × 2 = 0 + 0.000 002 402 566 255 460 776 673 28;
  • 49) 0.000 002 402 566 255 460 776 673 28 × 2 = 0 + 0.000 004 805 132 510 921 553 346 56;
  • 50) 0.000 004 805 132 510 921 553 346 56 × 2 = 0 + 0.000 009 610 265 021 843 106 693 12;
  • 51) 0.000 009 610 265 021 843 106 693 12 × 2 = 0 + 0.000 019 220 530 043 686 213 386 24;
  • 52) 0.000 019 220 530 043 686 213 386 24 × 2 = 0 + 0.000 038 441 060 087 372 426 772 48;
  • 53) 0.000 038 441 060 087 372 426 772 48 × 2 = 0 + 0.000 076 882 120 174 744 853 544 96;
  • 54) 0.000 076 882 120 174 744 853 544 96 × 2 = 0 + 0.000 153 764 240 349 489 707 089 92;
  • 55) 0.000 153 764 240 349 489 707 089 92 × 2 = 0 + 0.000 307 528 480 698 979 414 179 84;
  • 56) 0.000 307 528 480 698 979 414 179 84 × 2 = 0 + 0.000 615 056 961 397 958 828 359 68;
  • 57) 0.000 615 056 961 397 958 828 359 68 × 2 = 0 + 0.001 230 113 922 795 917 656 719 36;
  • 58) 0.001 230 113 922 795 917 656 719 36 × 2 = 0 + 0.002 460 227 845 591 835 313 438 72;
  • 59) 0.002 460 227 845 591 835 313 438 72 × 2 = 0 + 0.004 920 455 691 183 670 626 877 44;
  • 60) 0.004 920 455 691 183 670 626 877 44 × 2 = 0 + 0.009 840 911 382 367 341 253 754 88;
  • 61) 0.009 840 911 382 367 341 253 754 88 × 2 = 0 + 0.019 681 822 764 734 682 507 509 76;
  • 62) 0.019 681 822 764 734 682 507 509 76 × 2 = 0 + 0.039 363 645 529 469 365 015 019 52;
  • 63) 0.039 363 645 529 469 365 015 019 52 × 2 = 0 + 0.078 727 291 058 938 730 030 039 04;
  • 64) 0.078 727 291 058 938 730 030 039 04 × 2 = 0 + 0.157 454 582 117 877 460 060 078 08;
  • 65) 0.157 454 582 117 877 460 060 078 08 × 2 = 0 + 0.314 909 164 235 754 920 120 156 16;
  • 66) 0.314 909 164 235 754 920 120 156 16 × 2 = 0 + 0.629 818 328 471 509 840 240 312 32;
  • 67) 0.629 818 328 471 509 840 240 312 32 × 2 = 1 + 0.259 636 656 943 019 680 480 624 64;
  • 68) 0.259 636 656 943 019 680 480 624 64 × 2 = 0 + 0.519 273 313 886 039 360 961 249 28;
  • 69) 0.519 273 313 886 039 360 961 249 28 × 2 = 1 + 0.038 546 627 772 078 721 922 498 56;
  • 70) 0.038 546 627 772 078 721 922 498 56 × 2 = 0 + 0.077 093 255 544 157 443 844 997 12;
  • 71) 0.077 093 255 544 157 443 844 997 12 × 2 = 0 + 0.154 186 511 088 314 887 689 994 24;
  • 72) 0.154 186 511 088 314 887 689 994 24 × 2 = 0 + 0.308 373 022 176 629 775 379 988 48;
  • 73) 0.308 373 022 176 629 775 379 988 48 × 2 = 0 + 0.616 746 044 353 259 550 759 976 96;
  • 74) 0.616 746 044 353 259 550 759 976 96 × 2 = 1 + 0.233 492 088 706 519 101 519 953 92;
  • 75) 0.233 492 088 706 519 101 519 953 92 × 2 = 0 + 0.466 984 177 413 038 203 039 907 84;
  • 76) 0.466 984 177 413 038 203 039 907 84 × 2 = 0 + 0.933 968 354 826 076 406 079 815 68;
  • 77) 0.933 968 354 826 076 406 079 815 68 × 2 = 1 + 0.867 936 709 652 152 812 159 631 36;
  • 78) 0.867 936 709 652 152 812 159 631 36 × 2 = 1 + 0.735 873 419 304 305 624 319 262 72;
  • 79) 0.735 873 419 304 305 624 319 262 72 × 2 = 1 + 0.471 746 838 608 611 248 638 525 44;
  • 80) 0.471 746 838 608 611 248 638 525 44 × 2 = 0 + 0.943 493 677 217 222 497 277 050 88;
  • 81) 0.943 493 677 217 222 497 277 050 88 × 2 = 1 + 0.886 987 354 434 444 994 554 101 76;
  • 82) 0.886 987 354 434 444 994 554 101 76 × 2 = 1 + 0.773 974 708 868 889 989 108 203 52;
  • 83) 0.773 974 708 868 889 989 108 203 52 × 2 = 1 + 0.547 949 417 737 779 978 216 407 04;
  • 84) 0.547 949 417 737 779 978 216 407 04 × 2 = 1 + 0.095 898 835 475 559 956 432 814 08;
  • 85) 0.095 898 835 475 559 956 432 814 08 × 2 = 0 + 0.191 797 670 951 119 912 865 628 16;
  • 86) 0.191 797 670 951 119 912 865 628 16 × 2 = 0 + 0.383 595 341 902 239 825 731 256 32;
  • 87) 0.383 595 341 902 239 825 731 256 32 × 2 = 0 + 0.767 190 683 804 479 651 462 512 64;
  • 88) 0.767 190 683 804 479 651 462 512 64 × 2 = 1 + 0.534 381 367 608 959 302 925 025 28;
  • 89) 0.534 381 367 608 959 302 925 025 28 × 2 = 1 + 0.068 762 735 217 918 605 850 050 56;
  • 90) 0.068 762 735 217 918 605 850 050 56 × 2 = 0 + 0.137 525 470 435 837 211 700 101 12;
  • 91) 0.137 525 470 435 837 211 700 101 12 × 2 = 0 + 0.275 050 940 871 674 423 400 202 24;
  • 92) 0.275 050 940 871 674 423 400 202 24 × 2 = 0 + 0.550 101 881 743 348 846 800 404 48;
  • 93) 0.550 101 881 743 348 846 800 404 48 × 2 = 1 + 0.100 203 763 486 697 693 600 808 96;
  • 94) 0.100 203 763 486 697 693 600 808 96 × 2 = 0 + 0.200 407 526 973 395 387 201 617 92;
  • 95) 0.200 407 526 973 395 387 201 617 92 × 2 = 0 + 0.400 815 053 946 790 774 403 235 84;
  • 96) 0.400 815 053 946 790 774 403 235 84 × 2 = 0 + 0.801 630 107 893 581 548 806 471 68;
  • 97) 0.801 630 107 893 581 548 806 471 68 × 2 = 1 + 0.603 260 215 787 163 097 612 943 36;
  • 98) 0.603 260 215 787 163 097 612 943 36 × 2 = 1 + 0.206 520 431 574 326 195 225 886 72;
  • 99) 0.206 520 431 574 326 195 225 886 72 × 2 = 0 + 0.413 040 863 148 652 390 451 773 44;
  • 100) 0.413 040 863 148 652 390 451 773 44 × 2 = 0 + 0.826 081 726 297 304 780 903 546 88;
  • 101) 0.826 081 726 297 304 780 903 546 88 × 2 = 1 + 0.652 163 452 594 609 561 807 093 76;
  • 102) 0.652 163 452 594 609 561 807 093 76 × 2 = 1 + 0.304 326 905 189 219 123 614 187 52;
  • 103) 0.304 326 905 189 219 123 614 187 52 × 2 = 0 + 0.608 653 810 378 438 247 228 375 04;
  • 104) 0.608 653 810 378 438 247 228 375 04 × 2 = 1 + 0.217 307 620 756 876 494 456 750 08;
  • 105) 0.217 307 620 756 876 494 456 750 08 × 2 = 0 + 0.434 615 241 513 752 988 913 500 16;
  • 106) 0.434 615 241 513 752 988 913 500 16 × 2 = 0 + 0.869 230 483 027 505 977 827 000 32;
  • 107) 0.869 230 483 027 505 977 827 000 32 × 2 = 1 + 0.738 460 966 055 011 955 654 000 64;
  • 108) 0.738 460 966 055 011 955 654 000 64 × 2 = 1 + 0.476 921 932 110 023 911 308 001 28;
  • 109) 0.476 921 932 110 023 911 308 001 28 × 2 = 0 + 0.953 843 864 220 047 822 616 002 56;
  • 110) 0.953 843 864 220 047 822 616 002 56 × 2 = 1 + 0.907 687 728 440 095 645 232 005 12;
  • 111) 0.907 687 728 440 095 645 232 005 12 × 2 = 1 + 0.815 375 456 880 191 290 464 010 24;
  • 112) 0.815 375 456 880 191 290 464 010 24 × 2 = 1 + 0.630 750 913 760 382 580 928 020 48;
  • 113) 0.630 750 913 760 382 580 928 020 48 × 2 = 1 + 0.261 501 827 520 765 161 856 040 96;
  • 114) 0.261 501 827 520 765 161 856 040 96 × 2 = 0 + 0.523 003 655 041 530 323 712 081 92;
  • 115) 0.523 003 655 041 530 323 712 081 92 × 2 = 1 + 0.046 007 310 083 060 647 424 163 84;
  • 116) 0.046 007 310 083 060 647 424 163 84 × 2 = 0 + 0.092 014 620 166 121 294 848 327 68;
  • 117) 0.092 014 620 166 121 294 848 327 68 × 2 = 0 + 0.184 029 240 332 242 589 696 655 36;
  • 118) 0.184 029 240 332 242 589 696 655 36 × 2 = 0 + 0.368 058 480 664 485 179 393 310 72;
  • 119) 0.368 058 480 664 485 179 393 310 72 × 2 = 0 + 0.736 116 961 328 970 358 786 621 44;

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:


0.000 000 000 000 000 000 008 535 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1111 0001 1000 1000 1100 1101 0011 0111 1010 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1111 0001 1000 1000 1100 1101 0011 0111 1010 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:


0.000 000 000 000 000 000 008 535 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1111 0001 1000 1000 1100 1101 0011 0111 1010 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1111 0001 1000 1000 1100 1101 0011 0111 1010 000(2) × 20 =

1.0100 0010 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 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 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 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 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 0000 =

0100 0010 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 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 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 0000


Decimal number 0.000 000 000 000 000 000 008 535 63 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0111 1000 1100 0100 0110 0110 1001 1011 1101 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