0.000 000 000 000 000 000 008 535 86 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 86(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 86(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 86.

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 86 × 2 = 0 + 0.000 000 000 000 000 000 017 071 72;
  • 2) 0.000 000 000 000 000 000 017 071 72 × 2 = 0 + 0.000 000 000 000 000 000 034 143 44;
  • 3) 0.000 000 000 000 000 000 034 143 44 × 2 = 0 + 0.000 000 000 000 000 000 068 286 88;
  • 4) 0.000 000 000 000 000 000 068 286 88 × 2 = 0 + 0.000 000 000 000 000 000 136 573 76;
  • 5) 0.000 000 000 000 000 000 136 573 76 × 2 = 0 + 0.000 000 000 000 000 000 273 147 52;
  • 6) 0.000 000 000 000 000 000 273 147 52 × 2 = 0 + 0.000 000 000 000 000 000 546 295 04;
  • 7) 0.000 000 000 000 000 000 546 295 04 × 2 = 0 + 0.000 000 000 000 000 001 092 590 08;
  • 8) 0.000 000 000 000 000 001 092 590 08 × 2 = 0 + 0.000 000 000 000 000 002 185 180 16;
  • 9) 0.000 000 000 000 000 002 185 180 16 × 2 = 0 + 0.000 000 000 000 000 004 370 360 32;
  • 10) 0.000 000 000 000 000 004 370 360 32 × 2 = 0 + 0.000 000 000 000 000 008 740 720 64;
  • 11) 0.000 000 000 000 000 008 740 720 64 × 2 = 0 + 0.000 000 000 000 000 017 481 441 28;
  • 12) 0.000 000 000 000 000 017 481 441 28 × 2 = 0 + 0.000 000 000 000 000 034 962 882 56;
  • 13) 0.000 000 000 000 000 034 962 882 56 × 2 = 0 + 0.000 000 000 000 000 069 925 765 12;
  • 14) 0.000 000 000 000 000 069 925 765 12 × 2 = 0 + 0.000 000 000 000 000 139 851 530 24;
  • 15) 0.000 000 000 000 000 139 851 530 24 × 2 = 0 + 0.000 000 000 000 000 279 703 060 48;
  • 16) 0.000 000 000 000 000 279 703 060 48 × 2 = 0 + 0.000 000 000 000 000 559 406 120 96;
  • 17) 0.000 000 000 000 000 559 406 120 96 × 2 = 0 + 0.000 000 000 000 001 118 812 241 92;
  • 18) 0.000 000 000 000 001 118 812 241 92 × 2 = 0 + 0.000 000 000 000 002 237 624 483 84;
  • 19) 0.000 000 000 000 002 237 624 483 84 × 2 = 0 + 0.000 000 000 000 004 475 248 967 68;
  • 20) 0.000 000 000 000 004 475 248 967 68 × 2 = 0 + 0.000 000 000 000 008 950 497 935 36;
  • 21) 0.000 000 000 000 008 950 497 935 36 × 2 = 0 + 0.000 000 000 000 017 900 995 870 72;
  • 22) 0.000 000 000 000 017 900 995 870 72 × 2 = 0 + 0.000 000 000 000 035 801 991 741 44;
  • 23) 0.000 000 000 000 035 801 991 741 44 × 2 = 0 + 0.000 000 000 000 071 603 983 482 88;
  • 24) 0.000 000 000 000 071 603 983 482 88 × 2 = 0 + 0.000 000 000 000 143 207 966 965 76;
  • 25) 0.000 000 000 000 143 207 966 965 76 × 2 = 0 + 0.000 000 000 000 286 415 933 931 52;
  • 26) 0.000 000 000 000 286 415 933 931 52 × 2 = 0 + 0.000 000 000 000 572 831 867 863 04;
  • 27) 0.000 000 000 000 572 831 867 863 04 × 2 = 0 + 0.000 000 000 001 145 663 735 726 08;
  • 28) 0.000 000 000 001 145 663 735 726 08 × 2 = 0 + 0.000 000 000 002 291 327 471 452 16;
  • 29) 0.000 000 000 002 291 327 471 452 16 × 2 = 0 + 0.000 000 000 004 582 654 942 904 32;
  • 30) 0.000 000 000 004 582 654 942 904 32 × 2 = 0 + 0.000 000 000 009 165 309 885 808 64;
  • 31) 0.000 000 000 009 165 309 885 808 64 × 2 = 0 + 0.000 000 000 018 330 619 771 617 28;
  • 32) 0.000 000 000 018 330 619 771 617 28 × 2 = 0 + 0.000 000 000 036 661 239 543 234 56;
  • 33) 0.000 000 000 036 661 239 543 234 56 × 2 = 0 + 0.000 000 000 073 322 479 086 469 12;
  • 34) 0.000 000 000 073 322 479 086 469 12 × 2 = 0 + 0.000 000 000 146 644 958 172 938 24;
  • 35) 0.000 000 000 146 644 958 172 938 24 × 2 = 0 + 0.000 000 000 293 289 916 345 876 48;
  • 36) 0.000 000 000 293 289 916 345 876 48 × 2 = 0 + 0.000 000 000 586 579 832 691 752 96;
  • 37) 0.000 000 000 586 579 832 691 752 96 × 2 = 0 + 0.000 000 001 173 159 665 383 505 92;
  • 38) 0.000 000 001 173 159 665 383 505 92 × 2 = 0 + 0.000 000 002 346 319 330 767 011 84;
  • 39) 0.000 000 002 346 319 330 767 011 84 × 2 = 0 + 0.000 000 004 692 638 661 534 023 68;
  • 40) 0.000 000 004 692 638 661 534 023 68 × 2 = 0 + 0.000 000 009 385 277 323 068 047 36;
  • 41) 0.000 000 009 385 277 323 068 047 36 × 2 = 0 + 0.000 000 018 770 554 646 136 094 72;
  • 42) 0.000 000 018 770 554 646 136 094 72 × 2 = 0 + 0.000 000 037 541 109 292 272 189 44;
  • 43) 0.000 000 037 541 109 292 272 189 44 × 2 = 0 + 0.000 000 075 082 218 584 544 378 88;
  • 44) 0.000 000 075 082 218 584 544 378 88 × 2 = 0 + 0.000 000 150 164 437 169 088 757 76;
  • 45) 0.000 000 150 164 437 169 088 757 76 × 2 = 0 + 0.000 000 300 328 874 338 177 515 52;
  • 46) 0.000 000 300 328 874 338 177 515 52 × 2 = 0 + 0.000 000 600 657 748 676 355 031 04;
  • 47) 0.000 000 600 657 748 676 355 031 04 × 2 = 0 + 0.000 001 201 315 497 352 710 062 08;
  • 48) 0.000 001 201 315 497 352 710 062 08 × 2 = 0 + 0.000 002 402 630 994 705 420 124 16;
  • 49) 0.000 002 402 630 994 705 420 124 16 × 2 = 0 + 0.000 004 805 261 989 410 840 248 32;
  • 50) 0.000 004 805 261 989 410 840 248 32 × 2 = 0 + 0.000 009 610 523 978 821 680 496 64;
  • 51) 0.000 009 610 523 978 821 680 496 64 × 2 = 0 + 0.000 019 221 047 957 643 360 993 28;
  • 52) 0.000 019 221 047 957 643 360 993 28 × 2 = 0 + 0.000 038 442 095 915 286 721 986 56;
  • 53) 0.000 038 442 095 915 286 721 986 56 × 2 = 0 + 0.000 076 884 191 830 573 443 973 12;
  • 54) 0.000 076 884 191 830 573 443 973 12 × 2 = 0 + 0.000 153 768 383 661 146 887 946 24;
  • 55) 0.000 153 768 383 661 146 887 946 24 × 2 = 0 + 0.000 307 536 767 322 293 775 892 48;
  • 56) 0.000 307 536 767 322 293 775 892 48 × 2 = 0 + 0.000 615 073 534 644 587 551 784 96;
  • 57) 0.000 615 073 534 644 587 551 784 96 × 2 = 0 + 0.001 230 147 069 289 175 103 569 92;
  • 58) 0.001 230 147 069 289 175 103 569 92 × 2 = 0 + 0.002 460 294 138 578 350 207 139 84;
  • 59) 0.002 460 294 138 578 350 207 139 84 × 2 = 0 + 0.004 920 588 277 156 700 414 279 68;
  • 60) 0.004 920 588 277 156 700 414 279 68 × 2 = 0 + 0.009 841 176 554 313 400 828 559 36;
  • 61) 0.009 841 176 554 313 400 828 559 36 × 2 = 0 + 0.019 682 353 108 626 801 657 118 72;
  • 62) 0.019 682 353 108 626 801 657 118 72 × 2 = 0 + 0.039 364 706 217 253 603 314 237 44;
  • 63) 0.039 364 706 217 253 603 314 237 44 × 2 = 0 + 0.078 729 412 434 507 206 628 474 88;
  • 64) 0.078 729 412 434 507 206 628 474 88 × 2 = 0 + 0.157 458 824 869 014 413 256 949 76;
  • 65) 0.157 458 824 869 014 413 256 949 76 × 2 = 0 + 0.314 917 649 738 028 826 513 899 52;
  • 66) 0.314 917 649 738 028 826 513 899 52 × 2 = 0 + 0.629 835 299 476 057 653 027 799 04;
  • 67) 0.629 835 299 476 057 653 027 799 04 × 2 = 1 + 0.259 670 598 952 115 306 055 598 08;
  • 68) 0.259 670 598 952 115 306 055 598 08 × 2 = 0 + 0.519 341 197 904 230 612 111 196 16;
  • 69) 0.519 341 197 904 230 612 111 196 16 × 2 = 1 + 0.038 682 395 808 461 224 222 392 32;
  • 70) 0.038 682 395 808 461 224 222 392 32 × 2 = 0 + 0.077 364 791 616 922 448 444 784 64;
  • 71) 0.077 364 791 616 922 448 444 784 64 × 2 = 0 + 0.154 729 583 233 844 896 889 569 28;
  • 72) 0.154 729 583 233 844 896 889 569 28 × 2 = 0 + 0.309 459 166 467 689 793 779 138 56;
  • 73) 0.309 459 166 467 689 793 779 138 56 × 2 = 0 + 0.618 918 332 935 379 587 558 277 12;
  • 74) 0.618 918 332 935 379 587 558 277 12 × 2 = 1 + 0.237 836 665 870 759 175 116 554 24;
  • 75) 0.237 836 665 870 759 175 116 554 24 × 2 = 0 + 0.475 673 331 741 518 350 233 108 48;
  • 76) 0.475 673 331 741 518 350 233 108 48 × 2 = 0 + 0.951 346 663 483 036 700 466 216 96;
  • 77) 0.951 346 663 483 036 700 466 216 96 × 2 = 1 + 0.902 693 326 966 073 400 932 433 92;
  • 78) 0.902 693 326 966 073 400 932 433 92 × 2 = 1 + 0.805 386 653 932 146 801 864 867 84;
  • 79) 0.805 386 653 932 146 801 864 867 84 × 2 = 1 + 0.610 773 307 864 293 603 729 735 68;
  • 80) 0.610 773 307 864 293 603 729 735 68 × 2 = 1 + 0.221 546 615 728 587 207 459 471 36;
  • 81) 0.221 546 615 728 587 207 459 471 36 × 2 = 0 + 0.443 093 231 457 174 414 918 942 72;
  • 82) 0.443 093 231 457 174 414 918 942 72 × 2 = 0 + 0.886 186 462 914 348 829 837 885 44;
  • 83) 0.886 186 462 914 348 829 837 885 44 × 2 = 1 + 0.772 372 925 828 697 659 675 770 88;
  • 84) 0.772 372 925 828 697 659 675 770 88 × 2 = 1 + 0.544 745 851 657 395 319 351 541 76;
  • 85) 0.544 745 851 657 395 319 351 541 76 × 2 = 1 + 0.089 491 703 314 790 638 703 083 52;
  • 86) 0.089 491 703 314 790 638 703 083 52 × 2 = 0 + 0.178 983 406 629 581 277 406 167 04;
  • 87) 0.178 983 406 629 581 277 406 167 04 × 2 = 0 + 0.357 966 813 259 162 554 812 334 08;
  • 88) 0.357 966 813 259 162 554 812 334 08 × 2 = 0 + 0.715 933 626 518 325 109 624 668 16;
  • 89) 0.715 933 626 518 325 109 624 668 16 × 2 = 1 + 0.431 867 253 036 650 219 249 336 32;
  • 90) 0.431 867 253 036 650 219 249 336 32 × 2 = 0 + 0.863 734 506 073 300 438 498 672 64;
  • 91) 0.863 734 506 073 300 438 498 672 64 × 2 = 1 + 0.727 469 012 146 600 876 997 345 28;
  • 92) 0.727 469 012 146 600 876 997 345 28 × 2 = 1 + 0.454 938 024 293 201 753 994 690 56;
  • 93) 0.454 938 024 293 201 753 994 690 56 × 2 = 0 + 0.909 876 048 586 403 507 989 381 12;
  • 94) 0.909 876 048 586 403 507 989 381 12 × 2 = 1 + 0.819 752 097 172 807 015 978 762 24;
  • 95) 0.819 752 097 172 807 015 978 762 24 × 2 = 1 + 0.639 504 194 345 614 031 957 524 48;
  • 96) 0.639 504 194 345 614 031 957 524 48 × 2 = 1 + 0.279 008 388 691 228 063 915 048 96;
  • 97) 0.279 008 388 691 228 063 915 048 96 × 2 = 0 + 0.558 016 777 382 456 127 830 097 92;
  • 98) 0.558 016 777 382 456 127 830 097 92 × 2 = 1 + 0.116 033 554 764 912 255 660 195 84;
  • 99) 0.116 033 554 764 912 255 660 195 84 × 2 = 0 + 0.232 067 109 529 824 511 320 391 68;
  • 100) 0.232 067 109 529 824 511 320 391 68 × 2 = 0 + 0.464 134 219 059 649 022 640 783 36;
  • 101) 0.464 134 219 059 649 022 640 783 36 × 2 = 0 + 0.928 268 438 119 298 045 281 566 72;
  • 102) 0.928 268 438 119 298 045 281 566 72 × 2 = 1 + 0.856 536 876 238 596 090 563 133 44;
  • 103) 0.856 536 876 238 596 090 563 133 44 × 2 = 1 + 0.713 073 752 477 192 181 126 266 88;
  • 104) 0.713 073 752 477 192 181 126 266 88 × 2 = 1 + 0.426 147 504 954 384 362 252 533 76;
  • 105) 0.426 147 504 954 384 362 252 533 76 × 2 = 0 + 0.852 295 009 908 768 724 505 067 52;
  • 106) 0.852 295 009 908 768 724 505 067 52 × 2 = 1 + 0.704 590 019 817 537 449 010 135 04;
  • 107) 0.704 590 019 817 537 449 010 135 04 × 2 = 1 + 0.409 180 039 635 074 898 020 270 08;
  • 108) 0.409 180 039 635 074 898 020 270 08 × 2 = 0 + 0.818 360 079 270 149 796 040 540 16;
  • 109) 0.818 360 079 270 149 796 040 540 16 × 2 = 1 + 0.636 720 158 540 299 592 081 080 32;
  • 110) 0.636 720 158 540 299 592 081 080 32 × 2 = 1 + 0.273 440 317 080 599 184 162 160 64;
  • 111) 0.273 440 317 080 599 184 162 160 64 × 2 = 0 + 0.546 880 634 161 198 368 324 321 28;
  • 112) 0.546 880 634 161 198 368 324 321 28 × 2 = 1 + 0.093 761 268 322 396 736 648 642 56;
  • 113) 0.093 761 268 322 396 736 648 642 56 × 2 = 0 + 0.187 522 536 644 793 473 297 285 12;
  • 114) 0.187 522 536 644 793 473 297 285 12 × 2 = 0 + 0.375 045 073 289 586 946 594 570 24;
  • 115) 0.375 045 073 289 586 946 594 570 24 × 2 = 0 + 0.750 090 146 579 173 893 189 140 48;
  • 116) 0.750 090 146 579 173 893 189 140 48 × 2 = 1 + 0.500 180 293 158 347 786 378 280 96;
  • 117) 0.500 180 293 158 347 786 378 280 96 × 2 = 1 + 0.000 360 586 316 695 572 756 561 92;
  • 118) 0.000 360 586 316 695 572 756 561 92 × 2 = 0 + 0.000 721 172 633 391 145 513 123 84;
  • 119) 0.000 721 172 633 391 145 513 123 84 × 2 = 0 + 0.001 442 345 266 782 291 026 247 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:


0.000 000 000 000 000 000 008 535 86(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0011 1000 1011 0111 0100 0111 0110 1101 0001 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 86(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0011 1000 1011 0111 0100 0111 0110 1101 0001 100(2)

6. Normalize the binary representation of the number.

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


0.000 000 000 000 000 000 008 535 86(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0011 1000 1011 0111 0100 0111 0110 1101 0001 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0011 1000 1011 0111 0100 0111 0110 1101 0001 100(2) × 20 =

1.0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100(2) × 2-67


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

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


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100 =

0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100


Decimal number 0.000 000 000 000 000 000 008 535 86 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1001 1100 0101 1011 1010 0011 1011 0110 1000 1100

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100