0.000 000 000 000 000 000 008 529 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 529 2(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
0.000 000 000 000 000 000 008 529 2(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: 0.000 000 000 000 000 000 008 529 2.

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) 0.000 000 000 000 000 000 008 529 2 × 2 = 0 + 0.000 000 000 000 000 000 017 058 4;
  • 2) 0.000 000 000 000 000 000 017 058 4 × 2 = 0 + 0.000 000 000 000 000 000 034 116 8;
  • 3) 0.000 000 000 000 000 000 034 116 8 × 2 = 0 + 0.000 000 000 000 000 000 068 233 6;
  • 4) 0.000 000 000 000 000 000 068 233 6 × 2 = 0 + 0.000 000 000 000 000 000 136 467 2;
  • 5) 0.000 000 000 000 000 000 136 467 2 × 2 = 0 + 0.000 000 000 000 000 000 272 934 4;
  • 6) 0.000 000 000 000 000 000 272 934 4 × 2 = 0 + 0.000 000 000 000 000 000 545 868 8;
  • 7) 0.000 000 000 000 000 000 545 868 8 × 2 = 0 + 0.000 000 000 000 000 001 091 737 6;
  • 8) 0.000 000 000 000 000 001 091 737 6 × 2 = 0 + 0.000 000 000 000 000 002 183 475 2;
  • 9) 0.000 000 000 000 000 002 183 475 2 × 2 = 0 + 0.000 000 000 000 000 004 366 950 4;
  • 10) 0.000 000 000 000 000 004 366 950 4 × 2 = 0 + 0.000 000 000 000 000 008 733 900 8;
  • 11) 0.000 000 000 000 000 008 733 900 8 × 2 = 0 + 0.000 000 000 000 000 017 467 801 6;
  • 12) 0.000 000 000 000 000 017 467 801 6 × 2 = 0 + 0.000 000 000 000 000 034 935 603 2;
  • 13) 0.000 000 000 000 000 034 935 603 2 × 2 = 0 + 0.000 000 000 000 000 069 871 206 4;
  • 14) 0.000 000 000 000 000 069 871 206 4 × 2 = 0 + 0.000 000 000 000 000 139 742 412 8;
  • 15) 0.000 000 000 000 000 139 742 412 8 × 2 = 0 + 0.000 000 000 000 000 279 484 825 6;
  • 16) 0.000 000 000 000 000 279 484 825 6 × 2 = 0 + 0.000 000 000 000 000 558 969 651 2;
  • 17) 0.000 000 000 000 000 558 969 651 2 × 2 = 0 + 0.000 000 000 000 001 117 939 302 4;
  • 18) 0.000 000 000 000 001 117 939 302 4 × 2 = 0 + 0.000 000 000 000 002 235 878 604 8;
  • 19) 0.000 000 000 000 002 235 878 604 8 × 2 = 0 + 0.000 000 000 000 004 471 757 209 6;
  • 20) 0.000 000 000 000 004 471 757 209 6 × 2 = 0 + 0.000 000 000 000 008 943 514 419 2;
  • 21) 0.000 000 000 000 008 943 514 419 2 × 2 = 0 + 0.000 000 000 000 017 887 028 838 4;
  • 22) 0.000 000 000 000 017 887 028 838 4 × 2 = 0 + 0.000 000 000 000 035 774 057 676 8;
  • 23) 0.000 000 000 000 035 774 057 676 8 × 2 = 0 + 0.000 000 000 000 071 548 115 353 6;
  • 24) 0.000 000 000 000 071 548 115 353 6 × 2 = 0 + 0.000 000 000 000 143 096 230 707 2;
  • 25) 0.000 000 000 000 143 096 230 707 2 × 2 = 0 + 0.000 000 000 000 286 192 461 414 4;
  • 26) 0.000 000 000 000 286 192 461 414 4 × 2 = 0 + 0.000 000 000 000 572 384 922 828 8;
  • 27) 0.000 000 000 000 572 384 922 828 8 × 2 = 0 + 0.000 000 000 001 144 769 845 657 6;
  • 28) 0.000 000 000 001 144 769 845 657 6 × 2 = 0 + 0.000 000 000 002 289 539 691 315 2;
  • 29) 0.000 000 000 002 289 539 691 315 2 × 2 = 0 + 0.000 000 000 004 579 079 382 630 4;
  • 30) 0.000 000 000 004 579 079 382 630 4 × 2 = 0 + 0.000 000 000 009 158 158 765 260 8;
  • 31) 0.000 000 000 009 158 158 765 260 8 × 2 = 0 + 0.000 000 000 018 316 317 530 521 6;
  • 32) 0.000 000 000 018 316 317 530 521 6 × 2 = 0 + 0.000 000 000 036 632 635 061 043 2;
  • 33) 0.000 000 000 036 632 635 061 043 2 × 2 = 0 + 0.000 000 000 073 265 270 122 086 4;
  • 34) 0.000 000 000 073 265 270 122 086 4 × 2 = 0 + 0.000 000 000 146 530 540 244 172 8;
  • 35) 0.000 000 000 146 530 540 244 172 8 × 2 = 0 + 0.000 000 000 293 061 080 488 345 6;
  • 36) 0.000 000 000 293 061 080 488 345 6 × 2 = 0 + 0.000 000 000 586 122 160 976 691 2;
  • 37) 0.000 000 000 586 122 160 976 691 2 × 2 = 0 + 0.000 000 001 172 244 321 953 382 4;
  • 38) 0.000 000 001 172 244 321 953 382 4 × 2 = 0 + 0.000 000 002 344 488 643 906 764 8;
  • 39) 0.000 000 002 344 488 643 906 764 8 × 2 = 0 + 0.000 000 004 688 977 287 813 529 6;
  • 40) 0.000 000 004 688 977 287 813 529 6 × 2 = 0 + 0.000 000 009 377 954 575 627 059 2;
  • 41) 0.000 000 009 377 954 575 627 059 2 × 2 = 0 + 0.000 000 018 755 909 151 254 118 4;
  • 42) 0.000 000 018 755 909 151 254 118 4 × 2 = 0 + 0.000 000 037 511 818 302 508 236 8;
  • 43) 0.000 000 037 511 818 302 508 236 8 × 2 = 0 + 0.000 000 075 023 636 605 016 473 6;
  • 44) 0.000 000 075 023 636 605 016 473 6 × 2 = 0 + 0.000 000 150 047 273 210 032 947 2;
  • 45) 0.000 000 150 047 273 210 032 947 2 × 2 = 0 + 0.000 000 300 094 546 420 065 894 4;
  • 46) 0.000 000 300 094 546 420 065 894 4 × 2 = 0 + 0.000 000 600 189 092 840 131 788 8;
  • 47) 0.000 000 600 189 092 840 131 788 8 × 2 = 0 + 0.000 001 200 378 185 680 263 577 6;
  • 48) 0.000 001 200 378 185 680 263 577 6 × 2 = 0 + 0.000 002 400 756 371 360 527 155 2;
  • 49) 0.000 002 400 756 371 360 527 155 2 × 2 = 0 + 0.000 004 801 512 742 721 054 310 4;
  • 50) 0.000 004 801 512 742 721 054 310 4 × 2 = 0 + 0.000 009 603 025 485 442 108 620 8;
  • 51) 0.000 009 603 025 485 442 108 620 8 × 2 = 0 + 0.000 019 206 050 970 884 217 241 6;
  • 52) 0.000 019 206 050 970 884 217 241 6 × 2 = 0 + 0.000 038 412 101 941 768 434 483 2;
  • 53) 0.000 038 412 101 941 768 434 483 2 × 2 = 0 + 0.000 076 824 203 883 536 868 966 4;
  • 54) 0.000 076 824 203 883 536 868 966 4 × 2 = 0 + 0.000 153 648 407 767 073 737 932 8;
  • 55) 0.000 153 648 407 767 073 737 932 8 × 2 = 0 + 0.000 307 296 815 534 147 475 865 6;
  • 56) 0.000 307 296 815 534 147 475 865 6 × 2 = 0 + 0.000 614 593 631 068 294 951 731 2;
  • 57) 0.000 614 593 631 068 294 951 731 2 × 2 = 0 + 0.001 229 187 262 136 589 903 462 4;
  • 58) 0.001 229 187 262 136 589 903 462 4 × 2 = 0 + 0.002 458 374 524 273 179 806 924 8;
  • 59) 0.002 458 374 524 273 179 806 924 8 × 2 = 0 + 0.004 916 749 048 546 359 613 849 6;
  • 60) 0.004 916 749 048 546 359 613 849 6 × 2 = 0 + 0.009 833 498 097 092 719 227 699 2;
  • 61) 0.009 833 498 097 092 719 227 699 2 × 2 = 0 + 0.019 666 996 194 185 438 455 398 4;
  • 62) 0.019 666 996 194 185 438 455 398 4 × 2 = 0 + 0.039 333 992 388 370 876 910 796 8;
  • 63) 0.039 333 992 388 370 876 910 796 8 × 2 = 0 + 0.078 667 984 776 741 753 821 593 6;
  • 64) 0.078 667 984 776 741 753 821 593 6 × 2 = 0 + 0.157 335 969 553 483 507 643 187 2;
  • 65) 0.157 335 969 553 483 507 643 187 2 × 2 = 0 + 0.314 671 939 106 967 015 286 374 4;
  • 66) 0.314 671 939 106 967 015 286 374 4 × 2 = 0 + 0.629 343 878 213 934 030 572 748 8;
  • 67) 0.629 343 878 213 934 030 572 748 8 × 2 = 1 + 0.258 687 756 427 868 061 145 497 6;
  • 68) 0.258 687 756 427 868 061 145 497 6 × 2 = 0 + 0.517 375 512 855 736 122 290 995 2;
  • 69) 0.517 375 512 855 736 122 290 995 2 × 2 = 1 + 0.034 751 025 711 472 244 581 990 4;
  • 70) 0.034 751 025 711 472 244 581 990 4 × 2 = 0 + 0.069 502 051 422 944 489 163 980 8;
  • 71) 0.069 502 051 422 944 489 163 980 8 × 2 = 0 + 0.139 004 102 845 888 978 327 961 6;
  • 72) 0.139 004 102 845 888 978 327 961 6 × 2 = 0 + 0.278 008 205 691 777 956 655 923 2;
  • 73) 0.278 008 205 691 777 956 655 923 2 × 2 = 0 + 0.556 016 411 383 555 913 311 846 4;
  • 74) 0.556 016 411 383 555 913 311 846 4 × 2 = 1 + 0.112 032 822 767 111 826 623 692 8;
  • 75) 0.112 032 822 767 111 826 623 692 8 × 2 = 0 + 0.224 065 645 534 223 653 247 385 6;
  • 76) 0.224 065 645 534 223 653 247 385 6 × 2 = 0 + 0.448 131 291 068 447 306 494 771 2;
  • 77) 0.448 131 291 068 447 306 494 771 2 × 2 = 0 + 0.896 262 582 136 894 612 989 542 4;
  • 78) 0.896 262 582 136 894 612 989 542 4 × 2 = 1 + 0.792 525 164 273 789 225 979 084 8;
  • 79) 0.792 525 164 273 789 225 979 084 8 × 2 = 1 + 0.585 050 328 547 578 451 958 169 6;
  • 80) 0.585 050 328 547 578 451 958 169 6 × 2 = 1 + 0.170 100 657 095 156 903 916 339 2;
  • 81) 0.170 100 657 095 156 903 916 339 2 × 2 = 0 + 0.340 201 314 190 313 807 832 678 4;
  • 82) 0.340 201 314 190 313 807 832 678 4 × 2 = 0 + 0.680 402 628 380 627 615 665 356 8;
  • 83) 0.680 402 628 380 627 615 665 356 8 × 2 = 1 + 0.360 805 256 761 255 231 330 713 6;
  • 84) 0.360 805 256 761 255 231 330 713 6 × 2 = 0 + 0.721 610 513 522 510 462 661 427 2;
  • 85) 0.721 610 513 522 510 462 661 427 2 × 2 = 1 + 0.443 221 027 045 020 925 322 854 4;
  • 86) 0.443 221 027 045 020 925 322 854 4 × 2 = 0 + 0.886 442 054 090 041 850 645 708 8;
  • 87) 0.886 442 054 090 041 850 645 708 8 × 2 = 1 + 0.772 884 108 180 083 701 291 417 6;
  • 88) 0.772 884 108 180 083 701 291 417 6 × 2 = 1 + 0.545 768 216 360 167 402 582 835 2;
  • 89) 0.545 768 216 360 167 402 582 835 2 × 2 = 1 + 0.091 536 432 720 334 805 165 670 4;
  • 90) 0.091 536 432 720 334 805 165 670 4 × 2 = 0 + 0.183 072 865 440 669 610 331 340 8;
  • 91) 0.183 072 865 440 669 610 331 340 8 × 2 = 0 + 0.366 145 730 881 339 220 662 681 6;
  • 92) 0.366 145 730 881 339 220 662 681 6 × 2 = 0 + 0.732 291 461 762 678 441 325 363 2;
  • 93) 0.732 291 461 762 678 441 325 363 2 × 2 = 1 + 0.464 582 923 525 356 882 650 726 4;
  • 94) 0.464 582 923 525 356 882 650 726 4 × 2 = 0 + 0.929 165 847 050 713 765 301 452 8;
  • 95) 0.929 165 847 050 713 765 301 452 8 × 2 = 1 + 0.858 331 694 101 427 530 602 905 6;
  • 96) 0.858 331 694 101 427 530 602 905 6 × 2 = 1 + 0.716 663 388 202 855 061 205 811 2;
  • 97) 0.716 663 388 202 855 061 205 811 2 × 2 = 1 + 0.433 326 776 405 710 122 411 622 4;
  • 98) 0.433 326 776 405 710 122 411 622 4 × 2 = 0 + 0.866 653 552 811 420 244 823 244 8;
  • 99) 0.866 653 552 811 420 244 823 244 8 × 2 = 1 + 0.733 307 105 622 840 489 646 489 6;
  • 100) 0.733 307 105 622 840 489 646 489 6 × 2 = 1 + 0.466 614 211 245 680 979 292 979 2;
  • 101) 0.466 614 211 245 680 979 292 979 2 × 2 = 0 + 0.933 228 422 491 361 958 585 958 4;
  • 102) 0.933 228 422 491 361 958 585 958 4 × 2 = 1 + 0.866 456 844 982 723 917 171 916 8;
  • 103) 0.866 456 844 982 723 917 171 916 8 × 2 = 1 + 0.732 913 689 965 447 834 343 833 6;
  • 104) 0.732 913 689 965 447 834 343 833 6 × 2 = 1 + 0.465 827 379 930 895 668 687 667 2;
  • 105) 0.465 827 379 930 895 668 687 667 2 × 2 = 0 + 0.931 654 759 861 791 337 375 334 4;
  • 106) 0.931 654 759 861 791 337 375 334 4 × 2 = 1 + 0.863 309 519 723 582 674 750 668 8;
  • 107) 0.863 309 519 723 582 674 750 668 8 × 2 = 1 + 0.726 619 039 447 165 349 501 337 6;
  • 108) 0.726 619 039 447 165 349 501 337 6 × 2 = 1 + 0.453 238 078 894 330 699 002 675 2;
  • 109) 0.453 238 078 894 330 699 002 675 2 × 2 = 0 + 0.906 476 157 788 661 398 005 350 4;
  • 110) 0.906 476 157 788 661 398 005 350 4 × 2 = 1 + 0.812 952 315 577 322 796 010 700 8;
  • 111) 0.812 952 315 577 322 796 010 700 8 × 2 = 1 + 0.625 904 631 154 645 592 021 401 6;
  • 112) 0.625 904 631 154 645 592 021 401 6 × 2 = 1 + 0.251 809 262 309 291 184 042 803 2;
  • 113) 0.251 809 262 309 291 184 042 803 2 × 2 = 0 + 0.503 618 524 618 582 368 085 606 4;
  • 114) 0.503 618 524 618 582 368 085 606 4 × 2 = 1 + 0.007 237 049 237 164 736 171 212 8;
  • 115) 0.007 237 049 237 164 736 171 212 8 × 2 = 0 + 0.014 474 098 474 329 472 342 425 6;
  • 116) 0.014 474 098 474 329 472 342 425 6 × 2 = 0 + 0.028 948 196 948 658 944 684 851 2;
  • 117) 0.028 948 196 948 658 944 684 851 2 × 2 = 0 + 0.057 896 393 897 317 889 369 702 4;
  • 118) 0.057 896 393 897 317 889 369 702 4 × 2 = 0 + 0.115 792 787 794 635 778 739 404 8;
  • 119) 0.115 792 787 794 635 778 739 404 8 × 2 = 0 + 0.231 585 575 589 271 557 478 809 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:


0.000 000 000 000 000 000 008 529 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0010 1011 1000 1011 1011 0111 0111 0111 0100 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 529 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0010 1011 1000 1011 1011 0111 0111 0111 0100 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:


0.000 000 000 000 000 000 008 529 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0010 1011 1000 1011 1011 0111 0111 0111 0100 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0010 1011 1000 1011 1011 0111 0111 0111 0100 000(2) × 20 =

1.0100 0010 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 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 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 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 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 0000 =

0100 0010 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 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 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 0000


Decimal number 0.000 000 000 000 000 000 008 529 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0011 1001 0101 1100 0101 1101 1011 1011 1011 1010 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