0.000 000 000 000 000 000 008 536 29 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 29(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 008 536 29(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 008 536 29.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 000 000 000 008 536 29 × 2 = 0 + 0.000 000 000 000 000 000 017 072 58;
  • 2) 0.000 000 000 000 000 000 017 072 58 × 2 = 0 + 0.000 000 000 000 000 000 034 145 16;
  • 3) 0.000 000 000 000 000 000 034 145 16 × 2 = 0 + 0.000 000 000 000 000 000 068 290 32;
  • 4) 0.000 000 000 000 000 000 068 290 32 × 2 = 0 + 0.000 000 000 000 000 000 136 580 64;
  • 5) 0.000 000 000 000 000 000 136 580 64 × 2 = 0 + 0.000 000 000 000 000 000 273 161 28;
  • 6) 0.000 000 000 000 000 000 273 161 28 × 2 = 0 + 0.000 000 000 000 000 000 546 322 56;
  • 7) 0.000 000 000 000 000 000 546 322 56 × 2 = 0 + 0.000 000 000 000 000 001 092 645 12;
  • 8) 0.000 000 000 000 000 001 092 645 12 × 2 = 0 + 0.000 000 000 000 000 002 185 290 24;
  • 9) 0.000 000 000 000 000 002 185 290 24 × 2 = 0 + 0.000 000 000 000 000 004 370 580 48;
  • 10) 0.000 000 000 000 000 004 370 580 48 × 2 = 0 + 0.000 000 000 000 000 008 741 160 96;
  • 11) 0.000 000 000 000 000 008 741 160 96 × 2 = 0 + 0.000 000 000 000 000 017 482 321 92;
  • 12) 0.000 000 000 000 000 017 482 321 92 × 2 = 0 + 0.000 000 000 000 000 034 964 643 84;
  • 13) 0.000 000 000 000 000 034 964 643 84 × 2 = 0 + 0.000 000 000 000 000 069 929 287 68;
  • 14) 0.000 000 000 000 000 069 929 287 68 × 2 = 0 + 0.000 000 000 000 000 139 858 575 36;
  • 15) 0.000 000 000 000 000 139 858 575 36 × 2 = 0 + 0.000 000 000 000 000 279 717 150 72;
  • 16) 0.000 000 000 000 000 279 717 150 72 × 2 = 0 + 0.000 000 000 000 000 559 434 301 44;
  • 17) 0.000 000 000 000 000 559 434 301 44 × 2 = 0 + 0.000 000 000 000 001 118 868 602 88;
  • 18) 0.000 000 000 000 001 118 868 602 88 × 2 = 0 + 0.000 000 000 000 002 237 737 205 76;
  • 19) 0.000 000 000 000 002 237 737 205 76 × 2 = 0 + 0.000 000 000 000 004 475 474 411 52;
  • 20) 0.000 000 000 000 004 475 474 411 52 × 2 = 0 + 0.000 000 000 000 008 950 948 823 04;
  • 21) 0.000 000 000 000 008 950 948 823 04 × 2 = 0 + 0.000 000 000 000 017 901 897 646 08;
  • 22) 0.000 000 000 000 017 901 897 646 08 × 2 = 0 + 0.000 000 000 000 035 803 795 292 16;
  • 23) 0.000 000 000 000 035 803 795 292 16 × 2 = 0 + 0.000 000 000 000 071 607 590 584 32;
  • 24) 0.000 000 000 000 071 607 590 584 32 × 2 = 0 + 0.000 000 000 000 143 215 181 168 64;
  • 25) 0.000 000 000 000 143 215 181 168 64 × 2 = 0 + 0.000 000 000 000 286 430 362 337 28;
  • 26) 0.000 000 000 000 286 430 362 337 28 × 2 = 0 + 0.000 000 000 000 572 860 724 674 56;
  • 27) 0.000 000 000 000 572 860 724 674 56 × 2 = 0 + 0.000 000 000 001 145 721 449 349 12;
  • 28) 0.000 000 000 001 145 721 449 349 12 × 2 = 0 + 0.000 000 000 002 291 442 898 698 24;
  • 29) 0.000 000 000 002 291 442 898 698 24 × 2 = 0 + 0.000 000 000 004 582 885 797 396 48;
  • 30) 0.000 000 000 004 582 885 797 396 48 × 2 = 0 + 0.000 000 000 009 165 771 594 792 96;
  • 31) 0.000 000 000 009 165 771 594 792 96 × 2 = 0 + 0.000 000 000 018 331 543 189 585 92;
  • 32) 0.000 000 000 018 331 543 189 585 92 × 2 = 0 + 0.000 000 000 036 663 086 379 171 84;
  • 33) 0.000 000 000 036 663 086 379 171 84 × 2 = 0 + 0.000 000 000 073 326 172 758 343 68;
  • 34) 0.000 000 000 073 326 172 758 343 68 × 2 = 0 + 0.000 000 000 146 652 345 516 687 36;
  • 35) 0.000 000 000 146 652 345 516 687 36 × 2 = 0 + 0.000 000 000 293 304 691 033 374 72;
  • 36) 0.000 000 000 293 304 691 033 374 72 × 2 = 0 + 0.000 000 000 586 609 382 066 749 44;
  • 37) 0.000 000 000 586 609 382 066 749 44 × 2 = 0 + 0.000 000 001 173 218 764 133 498 88;
  • 38) 0.000 000 001 173 218 764 133 498 88 × 2 = 0 + 0.000 000 002 346 437 528 266 997 76;
  • 39) 0.000 000 002 346 437 528 266 997 76 × 2 = 0 + 0.000 000 004 692 875 056 533 995 52;
  • 40) 0.000 000 004 692 875 056 533 995 52 × 2 = 0 + 0.000 000 009 385 750 113 067 991 04;
  • 41) 0.000 000 009 385 750 113 067 991 04 × 2 = 0 + 0.000 000 018 771 500 226 135 982 08;
  • 42) 0.000 000 018 771 500 226 135 982 08 × 2 = 0 + 0.000 000 037 543 000 452 271 964 16;
  • 43) 0.000 000 037 543 000 452 271 964 16 × 2 = 0 + 0.000 000 075 086 000 904 543 928 32;
  • 44) 0.000 000 075 086 000 904 543 928 32 × 2 = 0 + 0.000 000 150 172 001 809 087 856 64;
  • 45) 0.000 000 150 172 001 809 087 856 64 × 2 = 0 + 0.000 000 300 344 003 618 175 713 28;
  • 46) 0.000 000 300 344 003 618 175 713 28 × 2 = 0 + 0.000 000 600 688 007 236 351 426 56;
  • 47) 0.000 000 600 688 007 236 351 426 56 × 2 = 0 + 0.000 001 201 376 014 472 702 853 12;
  • 48) 0.000 001 201 376 014 472 702 853 12 × 2 = 0 + 0.000 002 402 752 028 945 405 706 24;
  • 49) 0.000 002 402 752 028 945 405 706 24 × 2 = 0 + 0.000 004 805 504 057 890 811 412 48;
  • 50) 0.000 004 805 504 057 890 811 412 48 × 2 = 0 + 0.000 009 611 008 115 781 622 824 96;
  • 51) 0.000 009 611 008 115 781 622 824 96 × 2 = 0 + 0.000 019 222 016 231 563 245 649 92;
  • 52) 0.000 019 222 016 231 563 245 649 92 × 2 = 0 + 0.000 038 444 032 463 126 491 299 84;
  • 53) 0.000 038 444 032 463 126 491 299 84 × 2 = 0 + 0.000 076 888 064 926 252 982 599 68;
  • 54) 0.000 076 888 064 926 252 982 599 68 × 2 = 0 + 0.000 153 776 129 852 505 965 199 36;
  • 55) 0.000 153 776 129 852 505 965 199 36 × 2 = 0 + 0.000 307 552 259 705 011 930 398 72;
  • 56) 0.000 307 552 259 705 011 930 398 72 × 2 = 0 + 0.000 615 104 519 410 023 860 797 44;
  • 57) 0.000 615 104 519 410 023 860 797 44 × 2 = 0 + 0.001 230 209 038 820 047 721 594 88;
  • 58) 0.001 230 209 038 820 047 721 594 88 × 2 = 0 + 0.002 460 418 077 640 095 443 189 76;
  • 59) 0.002 460 418 077 640 095 443 189 76 × 2 = 0 + 0.004 920 836 155 280 190 886 379 52;
  • 60) 0.004 920 836 155 280 190 886 379 52 × 2 = 0 + 0.009 841 672 310 560 381 772 759 04;
  • 61) 0.009 841 672 310 560 381 772 759 04 × 2 = 0 + 0.019 683 344 621 120 763 545 518 08;
  • 62) 0.019 683 344 621 120 763 545 518 08 × 2 = 0 + 0.039 366 689 242 241 527 091 036 16;
  • 63) 0.039 366 689 242 241 527 091 036 16 × 2 = 0 + 0.078 733 378 484 483 054 182 072 32;
  • 64) 0.078 733 378 484 483 054 182 072 32 × 2 = 0 + 0.157 466 756 968 966 108 364 144 64;
  • 65) 0.157 466 756 968 966 108 364 144 64 × 2 = 0 + 0.314 933 513 937 932 216 728 289 28;
  • 66) 0.314 933 513 937 932 216 728 289 28 × 2 = 0 + 0.629 867 027 875 864 433 456 578 56;
  • 67) 0.629 867 027 875 864 433 456 578 56 × 2 = 1 + 0.259 734 055 751 728 866 913 157 12;
  • 68) 0.259 734 055 751 728 866 913 157 12 × 2 = 0 + 0.519 468 111 503 457 733 826 314 24;
  • 69) 0.519 468 111 503 457 733 826 314 24 × 2 = 1 + 0.038 936 223 006 915 467 652 628 48;
  • 70) 0.038 936 223 006 915 467 652 628 48 × 2 = 0 + 0.077 872 446 013 830 935 305 256 96;
  • 71) 0.077 872 446 013 830 935 305 256 96 × 2 = 0 + 0.155 744 892 027 661 870 610 513 92;
  • 72) 0.155 744 892 027 661 870 610 513 92 × 2 = 0 + 0.311 489 784 055 323 741 221 027 84;
  • 73) 0.311 489 784 055 323 741 221 027 84 × 2 = 0 + 0.622 979 568 110 647 482 442 055 68;
  • 74) 0.622 979 568 110 647 482 442 055 68 × 2 = 1 + 0.245 959 136 221 294 964 884 111 36;
  • 75) 0.245 959 136 221 294 964 884 111 36 × 2 = 0 + 0.491 918 272 442 589 929 768 222 72;
  • 76) 0.491 918 272 442 589 929 768 222 72 × 2 = 0 + 0.983 836 544 885 179 859 536 445 44;
  • 77) 0.983 836 544 885 179 859 536 445 44 × 2 = 1 + 0.967 673 089 770 359 719 072 890 88;
  • 78) 0.967 673 089 770 359 719 072 890 88 × 2 = 1 + 0.935 346 179 540 719 438 145 781 76;
  • 79) 0.935 346 179 540 719 438 145 781 76 × 2 = 1 + 0.870 692 359 081 438 876 291 563 52;
  • 80) 0.870 692 359 081 438 876 291 563 52 × 2 = 1 + 0.741 384 718 162 877 752 583 127 04;
  • 81) 0.741 384 718 162 877 752 583 127 04 × 2 = 1 + 0.482 769 436 325 755 505 166 254 08;
  • 82) 0.482 769 436 325 755 505 166 254 08 × 2 = 0 + 0.965 538 872 651 511 010 332 508 16;
  • 83) 0.965 538 872 651 511 010 332 508 16 × 2 = 1 + 0.931 077 745 303 022 020 665 016 32;
  • 84) 0.931 077 745 303 022 020 665 016 32 × 2 = 1 + 0.862 155 490 606 044 041 330 032 64;
  • 85) 0.862 155 490 606 044 041 330 032 64 × 2 = 1 + 0.724 310 981 212 088 082 660 065 28;
  • 86) 0.724 310 981 212 088 082 660 065 28 × 2 = 1 + 0.448 621 962 424 176 165 320 130 56;
  • 87) 0.448 621 962 424 176 165 320 130 56 × 2 = 0 + 0.897 243 924 848 352 330 640 261 12;
  • 88) 0.897 243 924 848 352 330 640 261 12 × 2 = 1 + 0.794 487 849 696 704 661 280 522 24;
  • 89) 0.794 487 849 696 704 661 280 522 24 × 2 = 1 + 0.588 975 699 393 409 322 561 044 48;
  • 90) 0.588 975 699 393 409 322 561 044 48 × 2 = 1 + 0.177 951 398 786 818 645 122 088 96;
  • 91) 0.177 951 398 786 818 645 122 088 96 × 2 = 0 + 0.355 902 797 573 637 290 244 177 92;
  • 92) 0.355 902 797 573 637 290 244 177 92 × 2 = 0 + 0.711 805 595 147 274 580 488 355 84;
  • 93) 0.711 805 595 147 274 580 488 355 84 × 2 = 1 + 0.423 611 190 294 549 160 976 711 68;
  • 94) 0.423 611 190 294 549 160 976 711 68 × 2 = 0 + 0.847 222 380 589 098 321 953 423 36;
  • 95) 0.847 222 380 589 098 321 953 423 36 × 2 = 1 + 0.694 444 761 178 196 643 906 846 72;
  • 96) 0.694 444 761 178 196 643 906 846 72 × 2 = 1 + 0.388 889 522 356 393 287 813 693 44;
  • 97) 0.388 889 522 356 393 287 813 693 44 × 2 = 0 + 0.777 779 044 712 786 575 627 386 88;
  • 98) 0.777 779 044 712 786 575 627 386 88 × 2 = 1 + 0.555 558 089 425 573 151 254 773 76;
  • 99) 0.555 558 089 425 573 151 254 773 76 × 2 = 1 + 0.111 116 178 851 146 302 509 547 52;
  • 100) 0.111 116 178 851 146 302 509 547 52 × 2 = 0 + 0.222 232 357 702 292 605 019 095 04;
  • 101) 0.222 232 357 702 292 605 019 095 04 × 2 = 0 + 0.444 464 715 404 585 210 038 190 08;
  • 102) 0.444 464 715 404 585 210 038 190 08 × 2 = 0 + 0.888 929 430 809 170 420 076 380 16;
  • 103) 0.888 929 430 809 170 420 076 380 16 × 2 = 1 + 0.777 858 861 618 340 840 152 760 32;
  • 104) 0.777 858 861 618 340 840 152 760 32 × 2 = 1 + 0.555 717 723 236 681 680 305 520 64;
  • 105) 0.555 717 723 236 681 680 305 520 64 × 2 = 1 + 0.111 435 446 473 363 360 611 041 28;
  • 106) 0.111 435 446 473 363 360 611 041 28 × 2 = 0 + 0.222 870 892 946 726 721 222 082 56;
  • 107) 0.222 870 892 946 726 721 222 082 56 × 2 = 0 + 0.445 741 785 893 453 442 444 165 12;
  • 108) 0.445 741 785 893 453 442 444 165 12 × 2 = 0 + 0.891 483 571 786 906 884 888 330 24;
  • 109) 0.891 483 571 786 906 884 888 330 24 × 2 = 1 + 0.782 967 143 573 813 769 776 660 48;
  • 110) 0.782 967 143 573 813 769 776 660 48 × 2 = 1 + 0.565 934 287 147 627 539 553 320 96;
  • 111) 0.565 934 287 147 627 539 553 320 96 × 2 = 1 + 0.131 868 574 295 255 079 106 641 92;
  • 112) 0.131 868 574 295 255 079 106 641 92 × 2 = 0 + 0.263 737 148 590 510 158 213 283 84;
  • 113) 0.263 737 148 590 510 158 213 283 84 × 2 = 0 + 0.527 474 297 181 020 316 426 567 68;
  • 114) 0.527 474 297 181 020 316 426 567 68 × 2 = 1 + 0.054 948 594 362 040 632 853 135 36;
  • 115) 0.054 948 594 362 040 632 853 135 36 × 2 = 0 + 0.109 897 188 724 081 265 706 270 72;
  • 116) 0.109 897 188 724 081 265 706 270 72 × 2 = 0 + 0.219 794 377 448 162 531 412 541 44;
  • 117) 0.219 794 377 448 162 531 412 541 44 × 2 = 0 + 0.439 588 754 896 325 062 825 082 88;
  • 118) 0.439 588 754 896 325 062 825 082 88 × 2 = 0 + 0.879 177 509 792 650 125 650 165 76;
  • 119) 0.879 177 509 792 650 125 650 165 76 × 2 = 1 + 0.758 355 019 585 300 251 300 331 52;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 000 008 536 29(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1011 1101 1100 1011 0110 0011 1000 1110 0100 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 29(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1011 1101 1100 1011 0110 0011 1000 1110 0100 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 536 29(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1011 1101 1100 1011 0110 0011 1000 1110 0100 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1011 1101 1100 1011 0110 0011 1000 1110 0100 001(2) × 20 =

1.0100 0010 0111 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001(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 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001


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 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001 =

0100 0010 0111 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001


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 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001


Decimal number 0.000 000 000 000 000 000 008 536 29 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1101 1110 1110 0101 1011 0001 1100 0111 0010 0001

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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