0.000 000 000 000 000 000 008 535 51 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 51(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 51(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 51.

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 51 × 2 = 0 + 0.000 000 000 000 000 000 017 071 02;
  • 2) 0.000 000 000 000 000 000 017 071 02 × 2 = 0 + 0.000 000 000 000 000 000 034 142 04;
  • 3) 0.000 000 000 000 000 000 034 142 04 × 2 = 0 + 0.000 000 000 000 000 000 068 284 08;
  • 4) 0.000 000 000 000 000 000 068 284 08 × 2 = 0 + 0.000 000 000 000 000 000 136 568 16;
  • 5) 0.000 000 000 000 000 000 136 568 16 × 2 = 0 + 0.000 000 000 000 000 000 273 136 32;
  • 6) 0.000 000 000 000 000 000 273 136 32 × 2 = 0 + 0.000 000 000 000 000 000 546 272 64;
  • 7) 0.000 000 000 000 000 000 546 272 64 × 2 = 0 + 0.000 000 000 000 000 001 092 545 28;
  • 8) 0.000 000 000 000 000 001 092 545 28 × 2 = 0 + 0.000 000 000 000 000 002 185 090 56;
  • 9) 0.000 000 000 000 000 002 185 090 56 × 2 = 0 + 0.000 000 000 000 000 004 370 181 12;
  • 10) 0.000 000 000 000 000 004 370 181 12 × 2 = 0 + 0.000 000 000 000 000 008 740 362 24;
  • 11) 0.000 000 000 000 000 008 740 362 24 × 2 = 0 + 0.000 000 000 000 000 017 480 724 48;
  • 12) 0.000 000 000 000 000 017 480 724 48 × 2 = 0 + 0.000 000 000 000 000 034 961 448 96;
  • 13) 0.000 000 000 000 000 034 961 448 96 × 2 = 0 + 0.000 000 000 000 000 069 922 897 92;
  • 14) 0.000 000 000 000 000 069 922 897 92 × 2 = 0 + 0.000 000 000 000 000 139 845 795 84;
  • 15) 0.000 000 000 000 000 139 845 795 84 × 2 = 0 + 0.000 000 000 000 000 279 691 591 68;
  • 16) 0.000 000 000 000 000 279 691 591 68 × 2 = 0 + 0.000 000 000 000 000 559 383 183 36;
  • 17) 0.000 000 000 000 000 559 383 183 36 × 2 = 0 + 0.000 000 000 000 001 118 766 366 72;
  • 18) 0.000 000 000 000 001 118 766 366 72 × 2 = 0 + 0.000 000 000 000 002 237 532 733 44;
  • 19) 0.000 000 000 000 002 237 532 733 44 × 2 = 0 + 0.000 000 000 000 004 475 065 466 88;
  • 20) 0.000 000 000 000 004 475 065 466 88 × 2 = 0 + 0.000 000 000 000 008 950 130 933 76;
  • 21) 0.000 000 000 000 008 950 130 933 76 × 2 = 0 + 0.000 000 000 000 017 900 261 867 52;
  • 22) 0.000 000 000 000 017 900 261 867 52 × 2 = 0 + 0.000 000 000 000 035 800 523 735 04;
  • 23) 0.000 000 000 000 035 800 523 735 04 × 2 = 0 + 0.000 000 000 000 071 601 047 470 08;
  • 24) 0.000 000 000 000 071 601 047 470 08 × 2 = 0 + 0.000 000 000 000 143 202 094 940 16;
  • 25) 0.000 000 000 000 143 202 094 940 16 × 2 = 0 + 0.000 000 000 000 286 404 189 880 32;
  • 26) 0.000 000 000 000 286 404 189 880 32 × 2 = 0 + 0.000 000 000 000 572 808 379 760 64;
  • 27) 0.000 000 000 000 572 808 379 760 64 × 2 = 0 + 0.000 000 000 001 145 616 759 521 28;
  • 28) 0.000 000 000 001 145 616 759 521 28 × 2 = 0 + 0.000 000 000 002 291 233 519 042 56;
  • 29) 0.000 000 000 002 291 233 519 042 56 × 2 = 0 + 0.000 000 000 004 582 467 038 085 12;
  • 30) 0.000 000 000 004 582 467 038 085 12 × 2 = 0 + 0.000 000 000 009 164 934 076 170 24;
  • 31) 0.000 000 000 009 164 934 076 170 24 × 2 = 0 + 0.000 000 000 018 329 868 152 340 48;
  • 32) 0.000 000 000 018 329 868 152 340 48 × 2 = 0 + 0.000 000 000 036 659 736 304 680 96;
  • 33) 0.000 000 000 036 659 736 304 680 96 × 2 = 0 + 0.000 000 000 073 319 472 609 361 92;
  • 34) 0.000 000 000 073 319 472 609 361 92 × 2 = 0 + 0.000 000 000 146 638 945 218 723 84;
  • 35) 0.000 000 000 146 638 945 218 723 84 × 2 = 0 + 0.000 000 000 293 277 890 437 447 68;
  • 36) 0.000 000 000 293 277 890 437 447 68 × 2 = 0 + 0.000 000 000 586 555 780 874 895 36;
  • 37) 0.000 000 000 586 555 780 874 895 36 × 2 = 0 + 0.000 000 001 173 111 561 749 790 72;
  • 38) 0.000 000 001 173 111 561 749 790 72 × 2 = 0 + 0.000 000 002 346 223 123 499 581 44;
  • 39) 0.000 000 002 346 223 123 499 581 44 × 2 = 0 + 0.000 000 004 692 446 246 999 162 88;
  • 40) 0.000 000 004 692 446 246 999 162 88 × 2 = 0 + 0.000 000 009 384 892 493 998 325 76;
  • 41) 0.000 000 009 384 892 493 998 325 76 × 2 = 0 + 0.000 000 018 769 784 987 996 651 52;
  • 42) 0.000 000 018 769 784 987 996 651 52 × 2 = 0 + 0.000 000 037 539 569 975 993 303 04;
  • 43) 0.000 000 037 539 569 975 993 303 04 × 2 = 0 + 0.000 000 075 079 139 951 986 606 08;
  • 44) 0.000 000 075 079 139 951 986 606 08 × 2 = 0 + 0.000 000 150 158 279 903 973 212 16;
  • 45) 0.000 000 150 158 279 903 973 212 16 × 2 = 0 + 0.000 000 300 316 559 807 946 424 32;
  • 46) 0.000 000 300 316 559 807 946 424 32 × 2 = 0 + 0.000 000 600 633 119 615 892 848 64;
  • 47) 0.000 000 600 633 119 615 892 848 64 × 2 = 0 + 0.000 001 201 266 239 231 785 697 28;
  • 48) 0.000 001 201 266 239 231 785 697 28 × 2 = 0 + 0.000 002 402 532 478 463 571 394 56;
  • 49) 0.000 002 402 532 478 463 571 394 56 × 2 = 0 + 0.000 004 805 064 956 927 142 789 12;
  • 50) 0.000 004 805 064 956 927 142 789 12 × 2 = 0 + 0.000 009 610 129 913 854 285 578 24;
  • 51) 0.000 009 610 129 913 854 285 578 24 × 2 = 0 + 0.000 019 220 259 827 708 571 156 48;
  • 52) 0.000 019 220 259 827 708 571 156 48 × 2 = 0 + 0.000 038 440 519 655 417 142 312 96;
  • 53) 0.000 038 440 519 655 417 142 312 96 × 2 = 0 + 0.000 076 881 039 310 834 284 625 92;
  • 54) 0.000 076 881 039 310 834 284 625 92 × 2 = 0 + 0.000 153 762 078 621 668 569 251 84;
  • 55) 0.000 153 762 078 621 668 569 251 84 × 2 = 0 + 0.000 307 524 157 243 337 138 503 68;
  • 56) 0.000 307 524 157 243 337 138 503 68 × 2 = 0 + 0.000 615 048 314 486 674 277 007 36;
  • 57) 0.000 615 048 314 486 674 277 007 36 × 2 = 0 + 0.001 230 096 628 973 348 554 014 72;
  • 58) 0.001 230 096 628 973 348 554 014 72 × 2 = 0 + 0.002 460 193 257 946 697 108 029 44;
  • 59) 0.002 460 193 257 946 697 108 029 44 × 2 = 0 + 0.004 920 386 515 893 394 216 058 88;
  • 60) 0.004 920 386 515 893 394 216 058 88 × 2 = 0 + 0.009 840 773 031 786 788 432 117 76;
  • 61) 0.009 840 773 031 786 788 432 117 76 × 2 = 0 + 0.019 681 546 063 573 576 864 235 52;
  • 62) 0.019 681 546 063 573 576 864 235 52 × 2 = 0 + 0.039 363 092 127 147 153 728 471 04;
  • 63) 0.039 363 092 127 147 153 728 471 04 × 2 = 0 + 0.078 726 184 254 294 307 456 942 08;
  • 64) 0.078 726 184 254 294 307 456 942 08 × 2 = 0 + 0.157 452 368 508 588 614 913 884 16;
  • 65) 0.157 452 368 508 588 614 913 884 16 × 2 = 0 + 0.314 904 737 017 177 229 827 768 32;
  • 66) 0.314 904 737 017 177 229 827 768 32 × 2 = 0 + 0.629 809 474 034 354 459 655 536 64;
  • 67) 0.629 809 474 034 354 459 655 536 64 × 2 = 1 + 0.259 618 948 068 708 919 311 073 28;
  • 68) 0.259 618 948 068 708 919 311 073 28 × 2 = 0 + 0.519 237 896 137 417 838 622 146 56;
  • 69) 0.519 237 896 137 417 838 622 146 56 × 2 = 1 + 0.038 475 792 274 835 677 244 293 12;
  • 70) 0.038 475 792 274 835 677 244 293 12 × 2 = 0 + 0.076 951 584 549 671 354 488 586 24;
  • 71) 0.076 951 584 549 671 354 488 586 24 × 2 = 0 + 0.153 903 169 099 342 708 977 172 48;
  • 72) 0.153 903 169 099 342 708 977 172 48 × 2 = 0 + 0.307 806 338 198 685 417 954 344 96;
  • 73) 0.307 806 338 198 685 417 954 344 96 × 2 = 0 + 0.615 612 676 397 370 835 908 689 92;
  • 74) 0.615 612 676 397 370 835 908 689 92 × 2 = 1 + 0.231 225 352 794 741 671 817 379 84;
  • 75) 0.231 225 352 794 741 671 817 379 84 × 2 = 0 + 0.462 450 705 589 483 343 634 759 68;
  • 76) 0.462 450 705 589 483 343 634 759 68 × 2 = 0 + 0.924 901 411 178 966 687 269 519 36;
  • 77) 0.924 901 411 178 966 687 269 519 36 × 2 = 1 + 0.849 802 822 357 933 374 539 038 72;
  • 78) 0.849 802 822 357 933 374 539 038 72 × 2 = 1 + 0.699 605 644 715 866 749 078 077 44;
  • 79) 0.699 605 644 715 866 749 078 077 44 × 2 = 1 + 0.399 211 289 431 733 498 156 154 88;
  • 80) 0.399 211 289 431 733 498 156 154 88 × 2 = 0 + 0.798 422 578 863 466 996 312 309 76;
  • 81) 0.798 422 578 863 466 996 312 309 76 × 2 = 1 + 0.596 845 157 726 933 992 624 619 52;
  • 82) 0.596 845 157 726 933 992 624 619 52 × 2 = 1 + 0.193 690 315 453 867 985 249 239 04;
  • 83) 0.193 690 315 453 867 985 249 239 04 × 2 = 0 + 0.387 380 630 907 735 970 498 478 08;
  • 84) 0.387 380 630 907 735 970 498 478 08 × 2 = 0 + 0.774 761 261 815 471 940 996 956 16;
  • 85) 0.774 761 261 815 471 940 996 956 16 × 2 = 1 + 0.549 522 523 630 943 881 993 912 32;
  • 86) 0.549 522 523 630 943 881 993 912 32 × 2 = 1 + 0.099 045 047 261 887 763 987 824 64;
  • 87) 0.099 045 047 261 887 763 987 824 64 × 2 = 0 + 0.198 090 094 523 775 527 975 649 28;
  • 88) 0.198 090 094 523 775 527 975 649 28 × 2 = 0 + 0.396 180 189 047 551 055 951 298 56;
  • 89) 0.396 180 189 047 551 055 951 298 56 × 2 = 0 + 0.792 360 378 095 102 111 902 597 12;
  • 90) 0.792 360 378 095 102 111 902 597 12 × 2 = 1 + 0.584 720 756 190 204 223 805 194 24;
  • 91) 0.584 720 756 190 204 223 805 194 24 × 2 = 1 + 0.169 441 512 380 408 447 610 388 48;
  • 92) 0.169 441 512 380 408 447 610 388 48 × 2 = 0 + 0.338 883 024 760 816 895 220 776 96;
  • 93) 0.338 883 024 760 816 895 220 776 96 × 2 = 0 + 0.677 766 049 521 633 790 441 553 92;
  • 94) 0.677 766 049 521 633 790 441 553 92 × 2 = 1 + 0.355 532 099 043 267 580 883 107 84;
  • 95) 0.355 532 099 043 267 580 883 107 84 × 2 = 0 + 0.711 064 198 086 535 161 766 215 68;
  • 96) 0.711 064 198 086 535 161 766 215 68 × 2 = 1 + 0.422 128 396 173 070 323 532 431 36;
  • 97) 0.422 128 396 173 070 323 532 431 36 × 2 = 0 + 0.844 256 792 346 140 647 064 862 72;
  • 98) 0.844 256 792 346 140 647 064 862 72 × 2 = 1 + 0.688 513 584 692 281 294 129 725 44;
  • 99) 0.688 513 584 692 281 294 129 725 44 × 2 = 1 + 0.377 027 169 384 562 588 259 450 88;
  • 100) 0.377 027 169 384 562 588 259 450 88 × 2 = 0 + 0.754 054 338 769 125 176 518 901 76;
  • 101) 0.754 054 338 769 125 176 518 901 76 × 2 = 1 + 0.508 108 677 538 250 353 037 803 52;
  • 102) 0.508 108 677 538 250 353 037 803 52 × 2 = 1 + 0.016 217 355 076 500 706 075 607 04;
  • 103) 0.016 217 355 076 500 706 075 607 04 × 2 = 0 + 0.032 434 710 153 001 412 151 214 08;
  • 104) 0.032 434 710 153 001 412 151 214 08 × 2 = 0 + 0.064 869 420 306 002 824 302 428 16;
  • 105) 0.064 869 420 306 002 824 302 428 16 × 2 = 0 + 0.129 738 840 612 005 648 604 856 32;
  • 106) 0.129 738 840 612 005 648 604 856 32 × 2 = 0 + 0.259 477 681 224 011 297 209 712 64;
  • 107) 0.259 477 681 224 011 297 209 712 64 × 2 = 0 + 0.518 955 362 448 022 594 419 425 28;
  • 108) 0.518 955 362 448 022 594 419 425 28 × 2 = 1 + 0.037 910 724 896 045 188 838 850 56;
  • 109) 0.037 910 724 896 045 188 838 850 56 × 2 = 0 + 0.075 821 449 792 090 377 677 701 12;
  • 110) 0.075 821 449 792 090 377 677 701 12 × 2 = 0 + 0.151 642 899 584 180 755 355 402 24;
  • 111) 0.151 642 899 584 180 755 355 402 24 × 2 = 0 + 0.303 285 799 168 361 510 710 804 48;
  • 112) 0.303 285 799 168 361 510 710 804 48 × 2 = 0 + 0.606 571 598 336 723 021 421 608 96;
  • 113) 0.606 571 598 336 723 021 421 608 96 × 2 = 1 + 0.213 143 196 673 446 042 843 217 92;
  • 114) 0.213 143 196 673 446 042 843 217 92 × 2 = 0 + 0.426 286 393 346 892 085 686 435 84;
  • 115) 0.426 286 393 346 892 085 686 435 84 × 2 = 0 + 0.852 572 786 693 784 171 372 871 68;
  • 116) 0.852 572 786 693 784 171 372 871 68 × 2 = 1 + 0.705 145 573 387 568 342 745 743 36;
  • 117) 0.705 145 573 387 568 342 745 743 36 × 2 = 1 + 0.410 291 146 775 136 685 491 486 72;
  • 118) 0.410 291 146 775 136 685 491 486 72 × 2 = 0 + 0.820 582 293 550 273 370 982 973 44;
  • 119) 0.820 582 293 550 273 370 982 973 44 × 2 = 1 + 0.641 164 587 100 546 741 965 946 88;

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 51(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1100 0110 0101 0110 1100 0001 0000 1001 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 51(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1100 0110 0101 0110 1100 0001 0000 1001 101(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 51(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1100 0110 0101 0110 1100 0001 0000 1001 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1100 0110 0101 0110 1100 0001 0000 1001 101(2) × 20 =

1.0100 0010 0111 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101(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 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101


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 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101 =

0100 0010 0111 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101


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 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101


Decimal number 0.000 000 000 000 000 000 008 535 51 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0110 0110 0011 0010 1011 0110 0000 1000 0100 1101

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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