0.000 000 000 000 000 000 008 537 23 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 23(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
0.000 000 000 000 000 000 008 537 23(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: 0.000 000 000 000 000 000 008 537 23.

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) 0.000 000 000 000 000 000 008 537 23 × 2 = 0 + 0.000 000 000 000 000 000 017 074 46;
  • 2) 0.000 000 000 000 000 000 017 074 46 × 2 = 0 + 0.000 000 000 000 000 000 034 148 92;
  • 3) 0.000 000 000 000 000 000 034 148 92 × 2 = 0 + 0.000 000 000 000 000 000 068 297 84;
  • 4) 0.000 000 000 000 000 000 068 297 84 × 2 = 0 + 0.000 000 000 000 000 000 136 595 68;
  • 5) 0.000 000 000 000 000 000 136 595 68 × 2 = 0 + 0.000 000 000 000 000 000 273 191 36;
  • 6) 0.000 000 000 000 000 000 273 191 36 × 2 = 0 + 0.000 000 000 000 000 000 546 382 72;
  • 7) 0.000 000 000 000 000 000 546 382 72 × 2 = 0 + 0.000 000 000 000 000 001 092 765 44;
  • 8) 0.000 000 000 000 000 001 092 765 44 × 2 = 0 + 0.000 000 000 000 000 002 185 530 88;
  • 9) 0.000 000 000 000 000 002 185 530 88 × 2 = 0 + 0.000 000 000 000 000 004 371 061 76;
  • 10) 0.000 000 000 000 000 004 371 061 76 × 2 = 0 + 0.000 000 000 000 000 008 742 123 52;
  • 11) 0.000 000 000 000 000 008 742 123 52 × 2 = 0 + 0.000 000 000 000 000 017 484 247 04;
  • 12) 0.000 000 000 000 000 017 484 247 04 × 2 = 0 + 0.000 000 000 000 000 034 968 494 08;
  • 13) 0.000 000 000 000 000 034 968 494 08 × 2 = 0 + 0.000 000 000 000 000 069 936 988 16;
  • 14) 0.000 000 000 000 000 069 936 988 16 × 2 = 0 + 0.000 000 000 000 000 139 873 976 32;
  • 15) 0.000 000 000 000 000 139 873 976 32 × 2 = 0 + 0.000 000 000 000 000 279 747 952 64;
  • 16) 0.000 000 000 000 000 279 747 952 64 × 2 = 0 + 0.000 000 000 000 000 559 495 905 28;
  • 17) 0.000 000 000 000 000 559 495 905 28 × 2 = 0 + 0.000 000 000 000 001 118 991 810 56;
  • 18) 0.000 000 000 000 001 118 991 810 56 × 2 = 0 + 0.000 000 000 000 002 237 983 621 12;
  • 19) 0.000 000 000 000 002 237 983 621 12 × 2 = 0 + 0.000 000 000 000 004 475 967 242 24;
  • 20) 0.000 000 000 000 004 475 967 242 24 × 2 = 0 + 0.000 000 000 000 008 951 934 484 48;
  • 21) 0.000 000 000 000 008 951 934 484 48 × 2 = 0 + 0.000 000 000 000 017 903 868 968 96;
  • 22) 0.000 000 000 000 017 903 868 968 96 × 2 = 0 + 0.000 000 000 000 035 807 737 937 92;
  • 23) 0.000 000 000 000 035 807 737 937 92 × 2 = 0 + 0.000 000 000 000 071 615 475 875 84;
  • 24) 0.000 000 000 000 071 615 475 875 84 × 2 = 0 + 0.000 000 000 000 143 230 951 751 68;
  • 25) 0.000 000 000 000 143 230 951 751 68 × 2 = 0 + 0.000 000 000 000 286 461 903 503 36;
  • 26) 0.000 000 000 000 286 461 903 503 36 × 2 = 0 + 0.000 000 000 000 572 923 807 006 72;
  • 27) 0.000 000 000 000 572 923 807 006 72 × 2 = 0 + 0.000 000 000 001 145 847 614 013 44;
  • 28) 0.000 000 000 001 145 847 614 013 44 × 2 = 0 + 0.000 000 000 002 291 695 228 026 88;
  • 29) 0.000 000 000 002 291 695 228 026 88 × 2 = 0 + 0.000 000 000 004 583 390 456 053 76;
  • 30) 0.000 000 000 004 583 390 456 053 76 × 2 = 0 + 0.000 000 000 009 166 780 912 107 52;
  • 31) 0.000 000 000 009 166 780 912 107 52 × 2 = 0 + 0.000 000 000 018 333 561 824 215 04;
  • 32) 0.000 000 000 018 333 561 824 215 04 × 2 = 0 + 0.000 000 000 036 667 123 648 430 08;
  • 33) 0.000 000 000 036 667 123 648 430 08 × 2 = 0 + 0.000 000 000 073 334 247 296 860 16;
  • 34) 0.000 000 000 073 334 247 296 860 16 × 2 = 0 + 0.000 000 000 146 668 494 593 720 32;
  • 35) 0.000 000 000 146 668 494 593 720 32 × 2 = 0 + 0.000 000 000 293 336 989 187 440 64;
  • 36) 0.000 000 000 293 336 989 187 440 64 × 2 = 0 + 0.000 000 000 586 673 978 374 881 28;
  • 37) 0.000 000 000 586 673 978 374 881 28 × 2 = 0 + 0.000 000 001 173 347 956 749 762 56;
  • 38) 0.000 000 001 173 347 956 749 762 56 × 2 = 0 + 0.000 000 002 346 695 913 499 525 12;
  • 39) 0.000 000 002 346 695 913 499 525 12 × 2 = 0 + 0.000 000 004 693 391 826 999 050 24;
  • 40) 0.000 000 004 693 391 826 999 050 24 × 2 = 0 + 0.000 000 009 386 783 653 998 100 48;
  • 41) 0.000 000 009 386 783 653 998 100 48 × 2 = 0 + 0.000 000 018 773 567 307 996 200 96;
  • 42) 0.000 000 018 773 567 307 996 200 96 × 2 = 0 + 0.000 000 037 547 134 615 992 401 92;
  • 43) 0.000 000 037 547 134 615 992 401 92 × 2 = 0 + 0.000 000 075 094 269 231 984 803 84;
  • 44) 0.000 000 075 094 269 231 984 803 84 × 2 = 0 + 0.000 000 150 188 538 463 969 607 68;
  • 45) 0.000 000 150 188 538 463 969 607 68 × 2 = 0 + 0.000 000 300 377 076 927 939 215 36;
  • 46) 0.000 000 300 377 076 927 939 215 36 × 2 = 0 + 0.000 000 600 754 153 855 878 430 72;
  • 47) 0.000 000 600 754 153 855 878 430 72 × 2 = 0 + 0.000 001 201 508 307 711 756 861 44;
  • 48) 0.000 001 201 508 307 711 756 861 44 × 2 = 0 + 0.000 002 403 016 615 423 513 722 88;
  • 49) 0.000 002 403 016 615 423 513 722 88 × 2 = 0 + 0.000 004 806 033 230 847 027 445 76;
  • 50) 0.000 004 806 033 230 847 027 445 76 × 2 = 0 + 0.000 009 612 066 461 694 054 891 52;
  • 51) 0.000 009 612 066 461 694 054 891 52 × 2 = 0 + 0.000 019 224 132 923 388 109 783 04;
  • 52) 0.000 019 224 132 923 388 109 783 04 × 2 = 0 + 0.000 038 448 265 846 776 219 566 08;
  • 53) 0.000 038 448 265 846 776 219 566 08 × 2 = 0 + 0.000 076 896 531 693 552 439 132 16;
  • 54) 0.000 076 896 531 693 552 439 132 16 × 2 = 0 + 0.000 153 793 063 387 104 878 264 32;
  • 55) 0.000 153 793 063 387 104 878 264 32 × 2 = 0 + 0.000 307 586 126 774 209 756 528 64;
  • 56) 0.000 307 586 126 774 209 756 528 64 × 2 = 0 + 0.000 615 172 253 548 419 513 057 28;
  • 57) 0.000 615 172 253 548 419 513 057 28 × 2 = 0 + 0.001 230 344 507 096 839 026 114 56;
  • 58) 0.001 230 344 507 096 839 026 114 56 × 2 = 0 + 0.002 460 689 014 193 678 052 229 12;
  • 59) 0.002 460 689 014 193 678 052 229 12 × 2 = 0 + 0.004 921 378 028 387 356 104 458 24;
  • 60) 0.004 921 378 028 387 356 104 458 24 × 2 = 0 + 0.009 842 756 056 774 712 208 916 48;
  • 61) 0.009 842 756 056 774 712 208 916 48 × 2 = 0 + 0.019 685 512 113 549 424 417 832 96;
  • 62) 0.019 685 512 113 549 424 417 832 96 × 2 = 0 + 0.039 371 024 227 098 848 835 665 92;
  • 63) 0.039 371 024 227 098 848 835 665 92 × 2 = 0 + 0.078 742 048 454 197 697 671 331 84;
  • 64) 0.078 742 048 454 197 697 671 331 84 × 2 = 0 + 0.157 484 096 908 395 395 342 663 68;
  • 65) 0.157 484 096 908 395 395 342 663 68 × 2 = 0 + 0.314 968 193 816 790 790 685 327 36;
  • 66) 0.314 968 193 816 790 790 685 327 36 × 2 = 0 + 0.629 936 387 633 581 581 370 654 72;
  • 67) 0.629 936 387 633 581 581 370 654 72 × 2 = 1 + 0.259 872 775 267 163 162 741 309 44;
  • 68) 0.259 872 775 267 163 162 741 309 44 × 2 = 0 + 0.519 745 550 534 326 325 482 618 88;
  • 69) 0.519 745 550 534 326 325 482 618 88 × 2 = 1 + 0.039 491 101 068 652 650 965 237 76;
  • 70) 0.039 491 101 068 652 650 965 237 76 × 2 = 0 + 0.078 982 202 137 305 301 930 475 52;
  • 71) 0.078 982 202 137 305 301 930 475 52 × 2 = 0 + 0.157 964 404 274 610 603 860 951 04;
  • 72) 0.157 964 404 274 610 603 860 951 04 × 2 = 0 + 0.315 928 808 549 221 207 721 902 08;
  • 73) 0.315 928 808 549 221 207 721 902 08 × 2 = 0 + 0.631 857 617 098 442 415 443 804 16;
  • 74) 0.631 857 617 098 442 415 443 804 16 × 2 = 1 + 0.263 715 234 196 884 830 887 608 32;
  • 75) 0.263 715 234 196 884 830 887 608 32 × 2 = 0 + 0.527 430 468 393 769 661 775 216 64;
  • 76) 0.527 430 468 393 769 661 775 216 64 × 2 = 1 + 0.054 860 936 787 539 323 550 433 28;
  • 77) 0.054 860 936 787 539 323 550 433 28 × 2 = 0 + 0.109 721 873 575 078 647 100 866 56;
  • 78) 0.109 721 873 575 078 647 100 866 56 × 2 = 0 + 0.219 443 747 150 157 294 201 733 12;
  • 79) 0.219 443 747 150 157 294 201 733 12 × 2 = 0 + 0.438 887 494 300 314 588 403 466 24;
  • 80) 0.438 887 494 300 314 588 403 466 24 × 2 = 0 + 0.877 774 988 600 629 176 806 932 48;
  • 81) 0.877 774 988 600 629 176 806 932 48 × 2 = 1 + 0.755 549 977 201 258 353 613 864 96;
  • 82) 0.755 549 977 201 258 353 613 864 96 × 2 = 1 + 0.511 099 954 402 516 707 227 729 92;
  • 83) 0.511 099 954 402 516 707 227 729 92 × 2 = 1 + 0.022 199 908 805 033 414 455 459 84;
  • 84) 0.022 199 908 805 033 414 455 459 84 × 2 = 0 + 0.044 399 817 610 066 828 910 919 68;
  • 85) 0.044 399 817 610 066 828 910 919 68 × 2 = 0 + 0.088 799 635 220 133 657 821 839 36;
  • 86) 0.088 799 635 220 133 657 821 839 36 × 2 = 0 + 0.177 599 270 440 267 315 643 678 72;
  • 87) 0.177 599 270 440 267 315 643 678 72 × 2 = 0 + 0.355 198 540 880 534 631 287 357 44;
  • 88) 0.355 198 540 880 534 631 287 357 44 × 2 = 0 + 0.710 397 081 761 069 262 574 714 88;
  • 89) 0.710 397 081 761 069 262 574 714 88 × 2 = 1 + 0.420 794 163 522 138 525 149 429 76;
  • 90) 0.420 794 163 522 138 525 149 429 76 × 2 = 0 + 0.841 588 327 044 277 050 298 859 52;
  • 91) 0.841 588 327 044 277 050 298 859 52 × 2 = 1 + 0.683 176 654 088 554 100 597 719 04;
  • 92) 0.683 176 654 088 554 100 597 719 04 × 2 = 1 + 0.366 353 308 177 108 201 195 438 08;
  • 93) 0.366 353 308 177 108 201 195 438 08 × 2 = 0 + 0.732 706 616 354 216 402 390 876 16;
  • 94) 0.732 706 616 354 216 402 390 876 16 × 2 = 1 + 0.465 413 232 708 432 804 781 752 32;
  • 95) 0.465 413 232 708 432 804 781 752 32 × 2 = 0 + 0.930 826 465 416 865 609 563 504 64;
  • 96) 0.930 826 465 416 865 609 563 504 64 × 2 = 1 + 0.861 652 930 833 731 219 127 009 28;
  • 97) 0.861 652 930 833 731 219 127 009 28 × 2 = 1 + 0.723 305 861 667 462 438 254 018 56;
  • 98) 0.723 305 861 667 462 438 254 018 56 × 2 = 1 + 0.446 611 723 334 924 876 508 037 12;
  • 99) 0.446 611 723 334 924 876 508 037 12 × 2 = 0 + 0.893 223 446 669 849 753 016 074 24;
  • 100) 0.893 223 446 669 849 753 016 074 24 × 2 = 1 + 0.786 446 893 339 699 506 032 148 48;
  • 101) 0.786 446 893 339 699 506 032 148 48 × 2 = 1 + 0.572 893 786 679 399 012 064 296 96;
  • 102) 0.572 893 786 679 399 012 064 296 96 × 2 = 1 + 0.145 787 573 358 798 024 128 593 92;
  • 103) 0.145 787 573 358 798 024 128 593 92 × 2 = 0 + 0.291 575 146 717 596 048 257 187 84;
  • 104) 0.291 575 146 717 596 048 257 187 84 × 2 = 0 + 0.583 150 293 435 192 096 514 375 68;
  • 105) 0.583 150 293 435 192 096 514 375 68 × 2 = 1 + 0.166 300 586 870 384 193 028 751 36;
  • 106) 0.166 300 586 870 384 193 028 751 36 × 2 = 0 + 0.332 601 173 740 768 386 057 502 72;
  • 107) 0.332 601 173 740 768 386 057 502 72 × 2 = 0 + 0.665 202 347 481 536 772 115 005 44;
  • 108) 0.665 202 347 481 536 772 115 005 44 × 2 = 1 + 0.330 404 694 963 073 544 230 010 88;
  • 109) 0.330 404 694 963 073 544 230 010 88 × 2 = 0 + 0.660 809 389 926 147 088 460 021 76;
  • 110) 0.660 809 389 926 147 088 460 021 76 × 2 = 1 + 0.321 618 779 852 294 176 920 043 52;
  • 111) 0.321 618 779 852 294 176 920 043 52 × 2 = 0 + 0.643 237 559 704 588 353 840 087 04;
  • 112) 0.643 237 559 704 588 353 840 087 04 × 2 = 1 + 0.286 475 119 409 176 707 680 174 08;
  • 113) 0.286 475 119 409 176 707 680 174 08 × 2 = 0 + 0.572 950 238 818 353 415 360 348 16;
  • 114) 0.572 950 238 818 353 415 360 348 16 × 2 = 1 + 0.145 900 477 636 706 830 720 696 32;
  • 115) 0.145 900 477 636 706 830 720 696 32 × 2 = 0 + 0.291 800 955 273 413 661 441 392 64;
  • 116) 0.291 800 955 273 413 661 441 392 64 × 2 = 0 + 0.583 601 910 546 827 322 882 785 28;
  • 117) 0.583 601 910 546 827 322 882 785 28 × 2 = 1 + 0.167 203 821 093 654 645 765 570 56;
  • 118) 0.167 203 821 093 654 645 765 570 56 × 2 = 0 + 0.334 407 642 187 309 291 531 141 12;
  • 119) 0.334 407 642 187 309 291 531 141 12 × 2 = 0 + 0.668 815 284 374 618 583 062 282 24;

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:


0.000 000 000 000 000 000 008 537 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1110 0000 1011 0101 1101 1100 1001 0101 0100 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1110 0000 1011 0101 1101 1100 1001 0101 0100 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:


0.000 000 000 000 000 000 008 537 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1110 0000 1011 0101 1101 1100 1001 0101 0100 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1110 0000 1011 0101 1101 1100 1001 0101 0100 100(2) × 20 =

1.0100 0010 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100(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 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100


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 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100 =

0100 0010 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100


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 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100


Decimal number 0.000 000 000 000 000 000 008 537 23 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0111 0000 0101 1010 1110 1110 0100 1010 1010 0100

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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