0.000 000 000 000 000 000 008 535 03 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 03(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 03(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 03.

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 03 × 2 = 0 + 0.000 000 000 000 000 000 017 070 06;
  • 2) 0.000 000 000 000 000 000 017 070 06 × 2 = 0 + 0.000 000 000 000 000 000 034 140 12;
  • 3) 0.000 000 000 000 000 000 034 140 12 × 2 = 0 + 0.000 000 000 000 000 000 068 280 24;
  • 4) 0.000 000 000 000 000 000 068 280 24 × 2 = 0 + 0.000 000 000 000 000 000 136 560 48;
  • 5) 0.000 000 000 000 000 000 136 560 48 × 2 = 0 + 0.000 000 000 000 000 000 273 120 96;
  • 6) 0.000 000 000 000 000 000 273 120 96 × 2 = 0 + 0.000 000 000 000 000 000 546 241 92;
  • 7) 0.000 000 000 000 000 000 546 241 92 × 2 = 0 + 0.000 000 000 000 000 001 092 483 84;
  • 8) 0.000 000 000 000 000 001 092 483 84 × 2 = 0 + 0.000 000 000 000 000 002 184 967 68;
  • 9) 0.000 000 000 000 000 002 184 967 68 × 2 = 0 + 0.000 000 000 000 000 004 369 935 36;
  • 10) 0.000 000 000 000 000 004 369 935 36 × 2 = 0 + 0.000 000 000 000 000 008 739 870 72;
  • 11) 0.000 000 000 000 000 008 739 870 72 × 2 = 0 + 0.000 000 000 000 000 017 479 741 44;
  • 12) 0.000 000 000 000 000 017 479 741 44 × 2 = 0 + 0.000 000 000 000 000 034 959 482 88;
  • 13) 0.000 000 000 000 000 034 959 482 88 × 2 = 0 + 0.000 000 000 000 000 069 918 965 76;
  • 14) 0.000 000 000 000 000 069 918 965 76 × 2 = 0 + 0.000 000 000 000 000 139 837 931 52;
  • 15) 0.000 000 000 000 000 139 837 931 52 × 2 = 0 + 0.000 000 000 000 000 279 675 863 04;
  • 16) 0.000 000 000 000 000 279 675 863 04 × 2 = 0 + 0.000 000 000 000 000 559 351 726 08;
  • 17) 0.000 000 000 000 000 559 351 726 08 × 2 = 0 + 0.000 000 000 000 001 118 703 452 16;
  • 18) 0.000 000 000 000 001 118 703 452 16 × 2 = 0 + 0.000 000 000 000 002 237 406 904 32;
  • 19) 0.000 000 000 000 002 237 406 904 32 × 2 = 0 + 0.000 000 000 000 004 474 813 808 64;
  • 20) 0.000 000 000 000 004 474 813 808 64 × 2 = 0 + 0.000 000 000 000 008 949 627 617 28;
  • 21) 0.000 000 000 000 008 949 627 617 28 × 2 = 0 + 0.000 000 000 000 017 899 255 234 56;
  • 22) 0.000 000 000 000 017 899 255 234 56 × 2 = 0 + 0.000 000 000 000 035 798 510 469 12;
  • 23) 0.000 000 000 000 035 798 510 469 12 × 2 = 0 + 0.000 000 000 000 071 597 020 938 24;
  • 24) 0.000 000 000 000 071 597 020 938 24 × 2 = 0 + 0.000 000 000 000 143 194 041 876 48;
  • 25) 0.000 000 000 000 143 194 041 876 48 × 2 = 0 + 0.000 000 000 000 286 388 083 752 96;
  • 26) 0.000 000 000 000 286 388 083 752 96 × 2 = 0 + 0.000 000 000 000 572 776 167 505 92;
  • 27) 0.000 000 000 000 572 776 167 505 92 × 2 = 0 + 0.000 000 000 001 145 552 335 011 84;
  • 28) 0.000 000 000 001 145 552 335 011 84 × 2 = 0 + 0.000 000 000 002 291 104 670 023 68;
  • 29) 0.000 000 000 002 291 104 670 023 68 × 2 = 0 + 0.000 000 000 004 582 209 340 047 36;
  • 30) 0.000 000 000 004 582 209 340 047 36 × 2 = 0 + 0.000 000 000 009 164 418 680 094 72;
  • 31) 0.000 000 000 009 164 418 680 094 72 × 2 = 0 + 0.000 000 000 018 328 837 360 189 44;
  • 32) 0.000 000 000 018 328 837 360 189 44 × 2 = 0 + 0.000 000 000 036 657 674 720 378 88;
  • 33) 0.000 000 000 036 657 674 720 378 88 × 2 = 0 + 0.000 000 000 073 315 349 440 757 76;
  • 34) 0.000 000 000 073 315 349 440 757 76 × 2 = 0 + 0.000 000 000 146 630 698 881 515 52;
  • 35) 0.000 000 000 146 630 698 881 515 52 × 2 = 0 + 0.000 000 000 293 261 397 763 031 04;
  • 36) 0.000 000 000 293 261 397 763 031 04 × 2 = 0 + 0.000 000 000 586 522 795 526 062 08;
  • 37) 0.000 000 000 586 522 795 526 062 08 × 2 = 0 + 0.000 000 001 173 045 591 052 124 16;
  • 38) 0.000 000 001 173 045 591 052 124 16 × 2 = 0 + 0.000 000 002 346 091 182 104 248 32;
  • 39) 0.000 000 002 346 091 182 104 248 32 × 2 = 0 + 0.000 000 004 692 182 364 208 496 64;
  • 40) 0.000 000 004 692 182 364 208 496 64 × 2 = 0 + 0.000 000 009 384 364 728 416 993 28;
  • 41) 0.000 000 009 384 364 728 416 993 28 × 2 = 0 + 0.000 000 018 768 729 456 833 986 56;
  • 42) 0.000 000 018 768 729 456 833 986 56 × 2 = 0 + 0.000 000 037 537 458 913 667 973 12;
  • 43) 0.000 000 037 537 458 913 667 973 12 × 2 = 0 + 0.000 000 075 074 917 827 335 946 24;
  • 44) 0.000 000 075 074 917 827 335 946 24 × 2 = 0 + 0.000 000 150 149 835 654 671 892 48;
  • 45) 0.000 000 150 149 835 654 671 892 48 × 2 = 0 + 0.000 000 300 299 671 309 343 784 96;
  • 46) 0.000 000 300 299 671 309 343 784 96 × 2 = 0 + 0.000 000 600 599 342 618 687 569 92;
  • 47) 0.000 000 600 599 342 618 687 569 92 × 2 = 0 + 0.000 001 201 198 685 237 375 139 84;
  • 48) 0.000 001 201 198 685 237 375 139 84 × 2 = 0 + 0.000 002 402 397 370 474 750 279 68;
  • 49) 0.000 002 402 397 370 474 750 279 68 × 2 = 0 + 0.000 004 804 794 740 949 500 559 36;
  • 50) 0.000 004 804 794 740 949 500 559 36 × 2 = 0 + 0.000 009 609 589 481 899 001 118 72;
  • 51) 0.000 009 609 589 481 899 001 118 72 × 2 = 0 + 0.000 019 219 178 963 798 002 237 44;
  • 52) 0.000 019 219 178 963 798 002 237 44 × 2 = 0 + 0.000 038 438 357 927 596 004 474 88;
  • 53) 0.000 038 438 357 927 596 004 474 88 × 2 = 0 + 0.000 076 876 715 855 192 008 949 76;
  • 54) 0.000 076 876 715 855 192 008 949 76 × 2 = 0 + 0.000 153 753 431 710 384 017 899 52;
  • 55) 0.000 153 753 431 710 384 017 899 52 × 2 = 0 + 0.000 307 506 863 420 768 035 799 04;
  • 56) 0.000 307 506 863 420 768 035 799 04 × 2 = 0 + 0.000 615 013 726 841 536 071 598 08;
  • 57) 0.000 615 013 726 841 536 071 598 08 × 2 = 0 + 0.001 230 027 453 683 072 143 196 16;
  • 58) 0.001 230 027 453 683 072 143 196 16 × 2 = 0 + 0.002 460 054 907 366 144 286 392 32;
  • 59) 0.002 460 054 907 366 144 286 392 32 × 2 = 0 + 0.004 920 109 814 732 288 572 784 64;
  • 60) 0.004 920 109 814 732 288 572 784 64 × 2 = 0 + 0.009 840 219 629 464 577 145 569 28;
  • 61) 0.009 840 219 629 464 577 145 569 28 × 2 = 0 + 0.019 680 439 258 929 154 291 138 56;
  • 62) 0.019 680 439 258 929 154 291 138 56 × 2 = 0 + 0.039 360 878 517 858 308 582 277 12;
  • 63) 0.039 360 878 517 858 308 582 277 12 × 2 = 0 + 0.078 721 757 035 716 617 164 554 24;
  • 64) 0.078 721 757 035 716 617 164 554 24 × 2 = 0 + 0.157 443 514 071 433 234 329 108 48;
  • 65) 0.157 443 514 071 433 234 329 108 48 × 2 = 0 + 0.314 887 028 142 866 468 658 216 96;
  • 66) 0.314 887 028 142 866 468 658 216 96 × 2 = 0 + 0.629 774 056 285 732 937 316 433 92;
  • 67) 0.629 774 056 285 732 937 316 433 92 × 2 = 1 + 0.259 548 112 571 465 874 632 867 84;
  • 68) 0.259 548 112 571 465 874 632 867 84 × 2 = 0 + 0.519 096 225 142 931 749 265 735 68;
  • 69) 0.519 096 225 142 931 749 265 735 68 × 2 = 1 + 0.038 192 450 285 863 498 531 471 36;
  • 70) 0.038 192 450 285 863 498 531 471 36 × 2 = 0 + 0.076 384 900 571 726 997 062 942 72;
  • 71) 0.076 384 900 571 726 997 062 942 72 × 2 = 0 + 0.152 769 801 143 453 994 125 885 44;
  • 72) 0.152 769 801 143 453 994 125 885 44 × 2 = 0 + 0.305 539 602 286 907 988 251 770 88;
  • 73) 0.305 539 602 286 907 988 251 770 88 × 2 = 0 + 0.611 079 204 573 815 976 503 541 76;
  • 74) 0.611 079 204 573 815 976 503 541 76 × 2 = 1 + 0.222 158 409 147 631 953 007 083 52;
  • 75) 0.222 158 409 147 631 953 007 083 52 × 2 = 0 + 0.444 316 818 295 263 906 014 167 04;
  • 76) 0.444 316 818 295 263 906 014 167 04 × 2 = 0 + 0.888 633 636 590 527 812 028 334 08;
  • 77) 0.888 633 636 590 527 812 028 334 08 × 2 = 1 + 0.777 267 273 181 055 624 056 668 16;
  • 78) 0.777 267 273 181 055 624 056 668 16 × 2 = 1 + 0.554 534 546 362 111 248 113 336 32;
  • 79) 0.554 534 546 362 111 248 113 336 32 × 2 = 1 + 0.109 069 092 724 222 496 226 672 64;
  • 80) 0.109 069 092 724 222 496 226 672 64 × 2 = 0 + 0.218 138 185 448 444 992 453 345 28;
  • 81) 0.218 138 185 448 444 992 453 345 28 × 2 = 0 + 0.436 276 370 896 889 984 906 690 56;
  • 82) 0.436 276 370 896 889 984 906 690 56 × 2 = 0 + 0.872 552 741 793 779 969 813 381 12;
  • 83) 0.872 552 741 793 779 969 813 381 12 × 2 = 1 + 0.745 105 483 587 559 939 626 762 24;
  • 84) 0.745 105 483 587 559 939 626 762 24 × 2 = 1 + 0.490 210 967 175 119 879 253 524 48;
  • 85) 0.490 210 967 175 119 879 253 524 48 × 2 = 0 + 0.980 421 934 350 239 758 507 048 96;
  • 86) 0.980 421 934 350 239 758 507 048 96 × 2 = 1 + 0.960 843 868 700 479 517 014 097 92;
  • 87) 0.960 843 868 700 479 517 014 097 92 × 2 = 1 + 0.921 687 737 400 959 034 028 195 84;
  • 88) 0.921 687 737 400 959 034 028 195 84 × 2 = 1 + 0.843 375 474 801 918 068 056 391 68;
  • 89) 0.843 375 474 801 918 068 056 391 68 × 2 = 1 + 0.686 750 949 603 836 136 112 783 36;
  • 90) 0.686 750 949 603 836 136 112 783 36 × 2 = 1 + 0.373 501 899 207 672 272 225 566 72;
  • 91) 0.373 501 899 207 672 272 225 566 72 × 2 = 0 + 0.747 003 798 415 344 544 451 133 44;
  • 92) 0.747 003 798 415 344 544 451 133 44 × 2 = 1 + 0.494 007 596 830 689 088 902 266 88;
  • 93) 0.494 007 596 830 689 088 902 266 88 × 2 = 0 + 0.988 015 193 661 378 177 804 533 76;
  • 94) 0.988 015 193 661 378 177 804 533 76 × 2 = 1 + 0.976 030 387 322 756 355 609 067 52;
  • 95) 0.976 030 387 322 756 355 609 067 52 × 2 = 1 + 0.952 060 774 645 512 711 218 135 04;
  • 96) 0.952 060 774 645 512 711 218 135 04 × 2 = 1 + 0.904 121 549 291 025 422 436 270 08;
  • 97) 0.904 121 549 291 025 422 436 270 08 × 2 = 1 + 0.808 243 098 582 050 844 872 540 16;
  • 98) 0.808 243 098 582 050 844 872 540 16 × 2 = 1 + 0.616 486 197 164 101 689 745 080 32;
  • 99) 0.616 486 197 164 101 689 745 080 32 × 2 = 1 + 0.232 972 394 328 203 379 490 160 64;
  • 100) 0.232 972 394 328 203 379 490 160 64 × 2 = 0 + 0.465 944 788 656 406 758 980 321 28;
  • 101) 0.465 944 788 656 406 758 980 321 28 × 2 = 0 + 0.931 889 577 312 813 517 960 642 56;
  • 102) 0.931 889 577 312 813 517 960 642 56 × 2 = 1 + 0.863 779 154 625 627 035 921 285 12;
  • 103) 0.863 779 154 625 627 035 921 285 12 × 2 = 1 + 0.727 558 309 251 254 071 842 570 24;
  • 104) 0.727 558 309 251 254 071 842 570 24 × 2 = 1 + 0.455 116 618 502 508 143 685 140 48;
  • 105) 0.455 116 618 502 508 143 685 140 48 × 2 = 0 + 0.910 233 237 005 016 287 370 280 96;
  • 106) 0.910 233 237 005 016 287 370 280 96 × 2 = 1 + 0.820 466 474 010 032 574 740 561 92;
  • 107) 0.820 466 474 010 032 574 740 561 92 × 2 = 1 + 0.640 932 948 020 065 149 481 123 84;
  • 108) 0.640 932 948 020 065 149 481 123 84 × 2 = 1 + 0.281 865 896 040 130 298 962 247 68;
  • 109) 0.281 865 896 040 130 298 962 247 68 × 2 = 0 + 0.563 731 792 080 260 597 924 495 36;
  • 110) 0.563 731 792 080 260 597 924 495 36 × 2 = 1 + 0.127 463 584 160 521 195 848 990 72;
  • 111) 0.127 463 584 160 521 195 848 990 72 × 2 = 0 + 0.254 927 168 321 042 391 697 981 44;
  • 112) 0.254 927 168 321 042 391 697 981 44 × 2 = 0 + 0.509 854 336 642 084 783 395 962 88;
  • 113) 0.509 854 336 642 084 783 395 962 88 × 2 = 1 + 0.019 708 673 284 169 566 791 925 76;
  • 114) 0.019 708 673 284 169 566 791 925 76 × 2 = 0 + 0.039 417 346 568 339 133 583 851 52;
  • 115) 0.039 417 346 568 339 133 583 851 52 × 2 = 0 + 0.078 834 693 136 678 267 167 703 04;
  • 116) 0.078 834 693 136 678 267 167 703 04 × 2 = 0 + 0.157 669 386 273 356 534 335 406 08;
  • 117) 0.157 669 386 273 356 534 335 406 08 × 2 = 0 + 0.315 338 772 546 713 068 670 812 16;
  • 118) 0.315 338 772 546 713 068 670 812 16 × 2 = 0 + 0.630 677 545 093 426 137 341 624 32;
  • 119) 0.630 677 545 093 426 137 341 624 32 × 2 = 1 + 0.261 355 090 186 852 274 683 248 64;

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 03(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0011 0111 1101 0111 1110 0111 0111 0100 1000 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 03(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0011 0111 1101 0111 1110 0111 0111 0100 1000 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:


0.000 000 000 000 000 000 008 535 03(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0011 0111 1101 0111 1110 0111 0111 0100 1000 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0011 0111 1101 0111 1110 0111 0111 0100 1000 001(2) × 20 =

1.0100 0010 0111 0001 1011 1110 1011 1111 0011 1011 1010 0100 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 0001 1011 1110 1011 1111 0011 1011 1010 0100 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 0001 1011 1110 1011 1111 0011 1011 1010 0100 0001 =

0100 0010 0111 0001 1011 1110 1011 1111 0011 1011 1010 0100 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 0001 1011 1110 1011 1111 0011 1011 1010 0100 0001


Decimal number 0.000 000 000 000 000 000 008 535 03 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0001 1011 1110 1011 1111 0011 1011 1010 0100 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