0.000 000 000 000 000 000 008 534 61 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 61(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
0.000 000 000 000 000 000 008 534 61(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: 0.000 000 000 000 000 000 008 534 61.

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) 0.000 000 000 000 000 000 008 534 61 × 2 = 0 + 0.000 000 000 000 000 000 017 069 22;
  • 2) 0.000 000 000 000 000 000 017 069 22 × 2 = 0 + 0.000 000 000 000 000 000 034 138 44;
  • 3) 0.000 000 000 000 000 000 034 138 44 × 2 = 0 + 0.000 000 000 000 000 000 068 276 88;
  • 4) 0.000 000 000 000 000 000 068 276 88 × 2 = 0 + 0.000 000 000 000 000 000 136 553 76;
  • 5) 0.000 000 000 000 000 000 136 553 76 × 2 = 0 + 0.000 000 000 000 000 000 273 107 52;
  • 6) 0.000 000 000 000 000 000 273 107 52 × 2 = 0 + 0.000 000 000 000 000 000 546 215 04;
  • 7) 0.000 000 000 000 000 000 546 215 04 × 2 = 0 + 0.000 000 000 000 000 001 092 430 08;
  • 8) 0.000 000 000 000 000 001 092 430 08 × 2 = 0 + 0.000 000 000 000 000 002 184 860 16;
  • 9) 0.000 000 000 000 000 002 184 860 16 × 2 = 0 + 0.000 000 000 000 000 004 369 720 32;
  • 10) 0.000 000 000 000 000 004 369 720 32 × 2 = 0 + 0.000 000 000 000 000 008 739 440 64;
  • 11) 0.000 000 000 000 000 008 739 440 64 × 2 = 0 + 0.000 000 000 000 000 017 478 881 28;
  • 12) 0.000 000 000 000 000 017 478 881 28 × 2 = 0 + 0.000 000 000 000 000 034 957 762 56;
  • 13) 0.000 000 000 000 000 034 957 762 56 × 2 = 0 + 0.000 000 000 000 000 069 915 525 12;
  • 14) 0.000 000 000 000 000 069 915 525 12 × 2 = 0 + 0.000 000 000 000 000 139 831 050 24;
  • 15) 0.000 000 000 000 000 139 831 050 24 × 2 = 0 + 0.000 000 000 000 000 279 662 100 48;
  • 16) 0.000 000 000 000 000 279 662 100 48 × 2 = 0 + 0.000 000 000 000 000 559 324 200 96;
  • 17) 0.000 000 000 000 000 559 324 200 96 × 2 = 0 + 0.000 000 000 000 001 118 648 401 92;
  • 18) 0.000 000 000 000 001 118 648 401 92 × 2 = 0 + 0.000 000 000 000 002 237 296 803 84;
  • 19) 0.000 000 000 000 002 237 296 803 84 × 2 = 0 + 0.000 000 000 000 004 474 593 607 68;
  • 20) 0.000 000 000 000 004 474 593 607 68 × 2 = 0 + 0.000 000 000 000 008 949 187 215 36;
  • 21) 0.000 000 000 000 008 949 187 215 36 × 2 = 0 + 0.000 000 000 000 017 898 374 430 72;
  • 22) 0.000 000 000 000 017 898 374 430 72 × 2 = 0 + 0.000 000 000 000 035 796 748 861 44;
  • 23) 0.000 000 000 000 035 796 748 861 44 × 2 = 0 + 0.000 000 000 000 071 593 497 722 88;
  • 24) 0.000 000 000 000 071 593 497 722 88 × 2 = 0 + 0.000 000 000 000 143 186 995 445 76;
  • 25) 0.000 000 000 000 143 186 995 445 76 × 2 = 0 + 0.000 000 000 000 286 373 990 891 52;
  • 26) 0.000 000 000 000 286 373 990 891 52 × 2 = 0 + 0.000 000 000 000 572 747 981 783 04;
  • 27) 0.000 000 000 000 572 747 981 783 04 × 2 = 0 + 0.000 000 000 001 145 495 963 566 08;
  • 28) 0.000 000 000 001 145 495 963 566 08 × 2 = 0 + 0.000 000 000 002 290 991 927 132 16;
  • 29) 0.000 000 000 002 290 991 927 132 16 × 2 = 0 + 0.000 000 000 004 581 983 854 264 32;
  • 30) 0.000 000 000 004 581 983 854 264 32 × 2 = 0 + 0.000 000 000 009 163 967 708 528 64;
  • 31) 0.000 000 000 009 163 967 708 528 64 × 2 = 0 + 0.000 000 000 018 327 935 417 057 28;
  • 32) 0.000 000 000 018 327 935 417 057 28 × 2 = 0 + 0.000 000 000 036 655 870 834 114 56;
  • 33) 0.000 000 000 036 655 870 834 114 56 × 2 = 0 + 0.000 000 000 073 311 741 668 229 12;
  • 34) 0.000 000 000 073 311 741 668 229 12 × 2 = 0 + 0.000 000 000 146 623 483 336 458 24;
  • 35) 0.000 000 000 146 623 483 336 458 24 × 2 = 0 + 0.000 000 000 293 246 966 672 916 48;
  • 36) 0.000 000 000 293 246 966 672 916 48 × 2 = 0 + 0.000 000 000 586 493 933 345 832 96;
  • 37) 0.000 000 000 586 493 933 345 832 96 × 2 = 0 + 0.000 000 001 172 987 866 691 665 92;
  • 38) 0.000 000 001 172 987 866 691 665 92 × 2 = 0 + 0.000 000 002 345 975 733 383 331 84;
  • 39) 0.000 000 002 345 975 733 383 331 84 × 2 = 0 + 0.000 000 004 691 951 466 766 663 68;
  • 40) 0.000 000 004 691 951 466 766 663 68 × 2 = 0 + 0.000 000 009 383 902 933 533 327 36;
  • 41) 0.000 000 009 383 902 933 533 327 36 × 2 = 0 + 0.000 000 018 767 805 867 066 654 72;
  • 42) 0.000 000 018 767 805 867 066 654 72 × 2 = 0 + 0.000 000 037 535 611 734 133 309 44;
  • 43) 0.000 000 037 535 611 734 133 309 44 × 2 = 0 + 0.000 000 075 071 223 468 266 618 88;
  • 44) 0.000 000 075 071 223 468 266 618 88 × 2 = 0 + 0.000 000 150 142 446 936 533 237 76;
  • 45) 0.000 000 150 142 446 936 533 237 76 × 2 = 0 + 0.000 000 300 284 893 873 066 475 52;
  • 46) 0.000 000 300 284 893 873 066 475 52 × 2 = 0 + 0.000 000 600 569 787 746 132 951 04;
  • 47) 0.000 000 600 569 787 746 132 951 04 × 2 = 0 + 0.000 001 201 139 575 492 265 902 08;
  • 48) 0.000 001 201 139 575 492 265 902 08 × 2 = 0 + 0.000 002 402 279 150 984 531 804 16;
  • 49) 0.000 002 402 279 150 984 531 804 16 × 2 = 0 + 0.000 004 804 558 301 969 063 608 32;
  • 50) 0.000 004 804 558 301 969 063 608 32 × 2 = 0 + 0.000 009 609 116 603 938 127 216 64;
  • 51) 0.000 009 609 116 603 938 127 216 64 × 2 = 0 + 0.000 019 218 233 207 876 254 433 28;
  • 52) 0.000 019 218 233 207 876 254 433 28 × 2 = 0 + 0.000 038 436 466 415 752 508 866 56;
  • 53) 0.000 038 436 466 415 752 508 866 56 × 2 = 0 + 0.000 076 872 932 831 505 017 733 12;
  • 54) 0.000 076 872 932 831 505 017 733 12 × 2 = 0 + 0.000 153 745 865 663 010 035 466 24;
  • 55) 0.000 153 745 865 663 010 035 466 24 × 2 = 0 + 0.000 307 491 731 326 020 070 932 48;
  • 56) 0.000 307 491 731 326 020 070 932 48 × 2 = 0 + 0.000 614 983 462 652 040 141 864 96;
  • 57) 0.000 614 983 462 652 040 141 864 96 × 2 = 0 + 0.001 229 966 925 304 080 283 729 92;
  • 58) 0.001 229 966 925 304 080 283 729 92 × 2 = 0 + 0.002 459 933 850 608 160 567 459 84;
  • 59) 0.002 459 933 850 608 160 567 459 84 × 2 = 0 + 0.004 919 867 701 216 321 134 919 68;
  • 60) 0.004 919 867 701 216 321 134 919 68 × 2 = 0 + 0.009 839 735 402 432 642 269 839 36;
  • 61) 0.009 839 735 402 432 642 269 839 36 × 2 = 0 + 0.019 679 470 804 865 284 539 678 72;
  • 62) 0.019 679 470 804 865 284 539 678 72 × 2 = 0 + 0.039 358 941 609 730 569 079 357 44;
  • 63) 0.039 358 941 609 730 569 079 357 44 × 2 = 0 + 0.078 717 883 219 461 138 158 714 88;
  • 64) 0.078 717 883 219 461 138 158 714 88 × 2 = 0 + 0.157 435 766 438 922 276 317 429 76;
  • 65) 0.157 435 766 438 922 276 317 429 76 × 2 = 0 + 0.314 871 532 877 844 552 634 859 52;
  • 66) 0.314 871 532 877 844 552 634 859 52 × 2 = 0 + 0.629 743 065 755 689 105 269 719 04;
  • 67) 0.629 743 065 755 689 105 269 719 04 × 2 = 1 + 0.259 486 131 511 378 210 539 438 08;
  • 68) 0.259 486 131 511 378 210 539 438 08 × 2 = 0 + 0.518 972 263 022 756 421 078 876 16;
  • 69) 0.518 972 263 022 756 421 078 876 16 × 2 = 1 + 0.037 944 526 045 512 842 157 752 32;
  • 70) 0.037 944 526 045 512 842 157 752 32 × 2 = 0 + 0.075 889 052 091 025 684 315 504 64;
  • 71) 0.075 889 052 091 025 684 315 504 64 × 2 = 0 + 0.151 778 104 182 051 368 631 009 28;
  • 72) 0.151 778 104 182 051 368 631 009 28 × 2 = 0 + 0.303 556 208 364 102 737 262 018 56;
  • 73) 0.303 556 208 364 102 737 262 018 56 × 2 = 0 + 0.607 112 416 728 205 474 524 037 12;
  • 74) 0.607 112 416 728 205 474 524 037 12 × 2 = 1 + 0.214 224 833 456 410 949 048 074 24;
  • 75) 0.214 224 833 456 410 949 048 074 24 × 2 = 0 + 0.428 449 666 912 821 898 096 148 48;
  • 76) 0.428 449 666 912 821 898 096 148 48 × 2 = 0 + 0.856 899 333 825 643 796 192 296 96;
  • 77) 0.856 899 333 825 643 796 192 296 96 × 2 = 1 + 0.713 798 667 651 287 592 384 593 92;
  • 78) 0.713 798 667 651 287 592 384 593 92 × 2 = 1 + 0.427 597 335 302 575 184 769 187 84;
  • 79) 0.427 597 335 302 575 184 769 187 84 × 2 = 0 + 0.855 194 670 605 150 369 538 375 68;
  • 80) 0.855 194 670 605 150 369 538 375 68 × 2 = 1 + 0.710 389 341 210 300 739 076 751 36;
  • 81) 0.710 389 341 210 300 739 076 751 36 × 2 = 1 + 0.420 778 682 420 601 478 153 502 72;
  • 82) 0.420 778 682 420 601 478 153 502 72 × 2 = 0 + 0.841 557 364 841 202 956 307 005 44;
  • 83) 0.841 557 364 841 202 956 307 005 44 × 2 = 1 + 0.683 114 729 682 405 912 614 010 88;
  • 84) 0.683 114 729 682 405 912 614 010 88 × 2 = 1 + 0.366 229 459 364 811 825 228 021 76;
  • 85) 0.366 229 459 364 811 825 228 021 76 × 2 = 0 + 0.732 458 918 729 623 650 456 043 52;
  • 86) 0.732 458 918 729 623 650 456 043 52 × 2 = 1 + 0.464 917 837 459 247 300 912 087 04;
  • 87) 0.464 917 837 459 247 300 912 087 04 × 2 = 0 + 0.929 835 674 918 494 601 824 174 08;
  • 88) 0.929 835 674 918 494 601 824 174 08 × 2 = 1 + 0.859 671 349 836 989 203 648 348 16;
  • 89) 0.859 671 349 836 989 203 648 348 16 × 2 = 1 + 0.719 342 699 673 978 407 296 696 32;
  • 90) 0.719 342 699 673 978 407 296 696 32 × 2 = 1 + 0.438 685 399 347 956 814 593 392 64;
  • 91) 0.438 685 399 347 956 814 593 392 64 × 2 = 0 + 0.877 370 798 695 913 629 186 785 28;
  • 92) 0.877 370 798 695 913 629 186 785 28 × 2 = 1 + 0.754 741 597 391 827 258 373 570 56;
  • 93) 0.754 741 597 391 827 258 373 570 56 × 2 = 1 + 0.509 483 194 783 654 516 747 141 12;
  • 94) 0.509 483 194 783 654 516 747 141 12 × 2 = 1 + 0.018 966 389 567 309 033 494 282 24;
  • 95) 0.018 966 389 567 309 033 494 282 24 × 2 = 0 + 0.037 932 779 134 618 066 988 564 48;
  • 96) 0.037 932 779 134 618 066 988 564 48 × 2 = 0 + 0.075 865 558 269 236 133 977 128 96;
  • 97) 0.075 865 558 269 236 133 977 128 96 × 2 = 0 + 0.151 731 116 538 472 267 954 257 92;
  • 98) 0.151 731 116 538 472 267 954 257 92 × 2 = 0 + 0.303 462 233 076 944 535 908 515 84;
  • 99) 0.303 462 233 076 944 535 908 515 84 × 2 = 0 + 0.606 924 466 153 889 071 817 031 68;
  • 100) 0.606 924 466 153 889 071 817 031 68 × 2 = 1 + 0.213 848 932 307 778 143 634 063 36;
  • 101) 0.213 848 932 307 778 143 634 063 36 × 2 = 0 + 0.427 697 864 615 556 287 268 126 72;
  • 102) 0.427 697 864 615 556 287 268 126 72 × 2 = 0 + 0.855 395 729 231 112 574 536 253 44;
  • 103) 0.855 395 729 231 112 574 536 253 44 × 2 = 1 + 0.710 791 458 462 225 149 072 506 88;
  • 104) 0.710 791 458 462 225 149 072 506 88 × 2 = 1 + 0.421 582 916 924 450 298 145 013 76;
  • 105) 0.421 582 916 924 450 298 145 013 76 × 2 = 0 + 0.843 165 833 848 900 596 290 027 52;
  • 106) 0.843 165 833 848 900 596 290 027 52 × 2 = 1 + 0.686 331 667 697 801 192 580 055 04;
  • 107) 0.686 331 667 697 801 192 580 055 04 × 2 = 1 + 0.372 663 335 395 602 385 160 110 08;
  • 108) 0.372 663 335 395 602 385 160 110 08 × 2 = 0 + 0.745 326 670 791 204 770 320 220 16;
  • 109) 0.745 326 670 791 204 770 320 220 16 × 2 = 1 + 0.490 653 341 582 409 540 640 440 32;
  • 110) 0.490 653 341 582 409 540 640 440 32 × 2 = 0 + 0.981 306 683 164 819 081 280 880 64;
  • 111) 0.981 306 683 164 819 081 280 880 64 × 2 = 1 + 0.962 613 366 329 638 162 561 761 28;
  • 112) 0.962 613 366 329 638 162 561 761 28 × 2 = 1 + 0.925 226 732 659 276 325 123 522 56;
  • 113) 0.925 226 732 659 276 325 123 522 56 × 2 = 1 + 0.850 453 465 318 552 650 247 045 12;
  • 114) 0.850 453 465 318 552 650 247 045 12 × 2 = 1 + 0.700 906 930 637 105 300 494 090 24;
  • 115) 0.700 906 930 637 105 300 494 090 24 × 2 = 1 + 0.401 813 861 274 210 600 988 180 48;
  • 116) 0.401 813 861 274 210 600 988 180 48 × 2 = 0 + 0.803 627 722 548 421 201 976 360 96;
  • 117) 0.803 627 722 548 421 201 976 360 96 × 2 = 1 + 0.607 255 445 096 842 403 952 721 92;
  • 118) 0.607 255 445 096 842 403 952 721 92 × 2 = 1 + 0.214 510 890 193 684 807 905 443 84;
  • 119) 0.214 510 890 193 684 807 905 443 84 × 2 = 0 + 0.429 021 780 387 369 615 810 887 68;

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:


0.000 000 000 000 000 000 008 534 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1011 0101 1101 1100 0001 0011 0110 1011 1110 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1011 0101 1101 1100 0001 0011 0110 1011 1110 110(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:


0.000 000 000 000 000 000 008 534 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1011 0101 1101 1100 0001 0011 0110 1011 1110 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1011 0101 1101 1100 0001 0011 0110 1011 1110 110(2) × 20 =

1.0100 0010 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110(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 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110


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 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110 =

0100 0010 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110


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 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110


Decimal number 0.000 000 000 000 000 000 008 534 61 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1101 1010 1110 1110 0000 1001 1011 0101 1111 0110

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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