0.000 000 000 000 000 000 008 534 13 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 13(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
0.000 000 000 000 000 000 008 534 13(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: 0.000 000 000 000 000 000 008 534 13.

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) 0.000 000 000 000 000 000 008 534 13 × 2 = 0 + 0.000 000 000 000 000 000 017 068 26;
  • 2) 0.000 000 000 000 000 000 017 068 26 × 2 = 0 + 0.000 000 000 000 000 000 034 136 52;
  • 3) 0.000 000 000 000 000 000 034 136 52 × 2 = 0 + 0.000 000 000 000 000 000 068 273 04;
  • 4) 0.000 000 000 000 000 000 068 273 04 × 2 = 0 + 0.000 000 000 000 000 000 136 546 08;
  • 5) 0.000 000 000 000 000 000 136 546 08 × 2 = 0 + 0.000 000 000 000 000 000 273 092 16;
  • 6) 0.000 000 000 000 000 000 273 092 16 × 2 = 0 + 0.000 000 000 000 000 000 546 184 32;
  • 7) 0.000 000 000 000 000 000 546 184 32 × 2 = 0 + 0.000 000 000 000 000 001 092 368 64;
  • 8) 0.000 000 000 000 000 001 092 368 64 × 2 = 0 + 0.000 000 000 000 000 002 184 737 28;
  • 9) 0.000 000 000 000 000 002 184 737 28 × 2 = 0 + 0.000 000 000 000 000 004 369 474 56;
  • 10) 0.000 000 000 000 000 004 369 474 56 × 2 = 0 + 0.000 000 000 000 000 008 738 949 12;
  • 11) 0.000 000 000 000 000 008 738 949 12 × 2 = 0 + 0.000 000 000 000 000 017 477 898 24;
  • 12) 0.000 000 000 000 000 017 477 898 24 × 2 = 0 + 0.000 000 000 000 000 034 955 796 48;
  • 13) 0.000 000 000 000 000 034 955 796 48 × 2 = 0 + 0.000 000 000 000 000 069 911 592 96;
  • 14) 0.000 000 000 000 000 069 911 592 96 × 2 = 0 + 0.000 000 000 000 000 139 823 185 92;
  • 15) 0.000 000 000 000 000 139 823 185 92 × 2 = 0 + 0.000 000 000 000 000 279 646 371 84;
  • 16) 0.000 000 000 000 000 279 646 371 84 × 2 = 0 + 0.000 000 000 000 000 559 292 743 68;
  • 17) 0.000 000 000 000 000 559 292 743 68 × 2 = 0 + 0.000 000 000 000 001 118 585 487 36;
  • 18) 0.000 000 000 000 001 118 585 487 36 × 2 = 0 + 0.000 000 000 000 002 237 170 974 72;
  • 19) 0.000 000 000 000 002 237 170 974 72 × 2 = 0 + 0.000 000 000 000 004 474 341 949 44;
  • 20) 0.000 000 000 000 004 474 341 949 44 × 2 = 0 + 0.000 000 000 000 008 948 683 898 88;
  • 21) 0.000 000 000 000 008 948 683 898 88 × 2 = 0 + 0.000 000 000 000 017 897 367 797 76;
  • 22) 0.000 000 000 000 017 897 367 797 76 × 2 = 0 + 0.000 000 000 000 035 794 735 595 52;
  • 23) 0.000 000 000 000 035 794 735 595 52 × 2 = 0 + 0.000 000 000 000 071 589 471 191 04;
  • 24) 0.000 000 000 000 071 589 471 191 04 × 2 = 0 + 0.000 000 000 000 143 178 942 382 08;
  • 25) 0.000 000 000 000 143 178 942 382 08 × 2 = 0 + 0.000 000 000 000 286 357 884 764 16;
  • 26) 0.000 000 000 000 286 357 884 764 16 × 2 = 0 + 0.000 000 000 000 572 715 769 528 32;
  • 27) 0.000 000 000 000 572 715 769 528 32 × 2 = 0 + 0.000 000 000 001 145 431 539 056 64;
  • 28) 0.000 000 000 001 145 431 539 056 64 × 2 = 0 + 0.000 000 000 002 290 863 078 113 28;
  • 29) 0.000 000 000 002 290 863 078 113 28 × 2 = 0 + 0.000 000 000 004 581 726 156 226 56;
  • 30) 0.000 000 000 004 581 726 156 226 56 × 2 = 0 + 0.000 000 000 009 163 452 312 453 12;
  • 31) 0.000 000 000 009 163 452 312 453 12 × 2 = 0 + 0.000 000 000 018 326 904 624 906 24;
  • 32) 0.000 000 000 018 326 904 624 906 24 × 2 = 0 + 0.000 000 000 036 653 809 249 812 48;
  • 33) 0.000 000 000 036 653 809 249 812 48 × 2 = 0 + 0.000 000 000 073 307 618 499 624 96;
  • 34) 0.000 000 000 073 307 618 499 624 96 × 2 = 0 + 0.000 000 000 146 615 236 999 249 92;
  • 35) 0.000 000 000 146 615 236 999 249 92 × 2 = 0 + 0.000 000 000 293 230 473 998 499 84;
  • 36) 0.000 000 000 293 230 473 998 499 84 × 2 = 0 + 0.000 000 000 586 460 947 996 999 68;
  • 37) 0.000 000 000 586 460 947 996 999 68 × 2 = 0 + 0.000 000 001 172 921 895 993 999 36;
  • 38) 0.000 000 001 172 921 895 993 999 36 × 2 = 0 + 0.000 000 002 345 843 791 987 998 72;
  • 39) 0.000 000 002 345 843 791 987 998 72 × 2 = 0 + 0.000 000 004 691 687 583 975 997 44;
  • 40) 0.000 000 004 691 687 583 975 997 44 × 2 = 0 + 0.000 000 009 383 375 167 951 994 88;
  • 41) 0.000 000 009 383 375 167 951 994 88 × 2 = 0 + 0.000 000 018 766 750 335 903 989 76;
  • 42) 0.000 000 018 766 750 335 903 989 76 × 2 = 0 + 0.000 000 037 533 500 671 807 979 52;
  • 43) 0.000 000 037 533 500 671 807 979 52 × 2 = 0 + 0.000 000 075 067 001 343 615 959 04;
  • 44) 0.000 000 075 067 001 343 615 959 04 × 2 = 0 + 0.000 000 150 134 002 687 231 918 08;
  • 45) 0.000 000 150 134 002 687 231 918 08 × 2 = 0 + 0.000 000 300 268 005 374 463 836 16;
  • 46) 0.000 000 300 268 005 374 463 836 16 × 2 = 0 + 0.000 000 600 536 010 748 927 672 32;
  • 47) 0.000 000 600 536 010 748 927 672 32 × 2 = 0 + 0.000 001 201 072 021 497 855 344 64;
  • 48) 0.000 001 201 072 021 497 855 344 64 × 2 = 0 + 0.000 002 402 144 042 995 710 689 28;
  • 49) 0.000 002 402 144 042 995 710 689 28 × 2 = 0 + 0.000 004 804 288 085 991 421 378 56;
  • 50) 0.000 004 804 288 085 991 421 378 56 × 2 = 0 + 0.000 009 608 576 171 982 842 757 12;
  • 51) 0.000 009 608 576 171 982 842 757 12 × 2 = 0 + 0.000 019 217 152 343 965 685 514 24;
  • 52) 0.000 019 217 152 343 965 685 514 24 × 2 = 0 + 0.000 038 434 304 687 931 371 028 48;
  • 53) 0.000 038 434 304 687 931 371 028 48 × 2 = 0 + 0.000 076 868 609 375 862 742 056 96;
  • 54) 0.000 076 868 609 375 862 742 056 96 × 2 = 0 + 0.000 153 737 218 751 725 484 113 92;
  • 55) 0.000 153 737 218 751 725 484 113 92 × 2 = 0 + 0.000 307 474 437 503 450 968 227 84;
  • 56) 0.000 307 474 437 503 450 968 227 84 × 2 = 0 + 0.000 614 948 875 006 901 936 455 68;
  • 57) 0.000 614 948 875 006 901 936 455 68 × 2 = 0 + 0.001 229 897 750 013 803 872 911 36;
  • 58) 0.001 229 897 750 013 803 872 911 36 × 2 = 0 + 0.002 459 795 500 027 607 745 822 72;
  • 59) 0.002 459 795 500 027 607 745 822 72 × 2 = 0 + 0.004 919 591 000 055 215 491 645 44;
  • 60) 0.004 919 591 000 055 215 491 645 44 × 2 = 0 + 0.009 839 182 000 110 430 983 290 88;
  • 61) 0.009 839 182 000 110 430 983 290 88 × 2 = 0 + 0.019 678 364 000 220 861 966 581 76;
  • 62) 0.019 678 364 000 220 861 966 581 76 × 2 = 0 + 0.039 356 728 000 441 723 933 163 52;
  • 63) 0.039 356 728 000 441 723 933 163 52 × 2 = 0 + 0.078 713 456 000 883 447 866 327 04;
  • 64) 0.078 713 456 000 883 447 866 327 04 × 2 = 0 + 0.157 426 912 001 766 895 732 654 08;
  • 65) 0.157 426 912 001 766 895 732 654 08 × 2 = 0 + 0.314 853 824 003 533 791 465 308 16;
  • 66) 0.314 853 824 003 533 791 465 308 16 × 2 = 0 + 0.629 707 648 007 067 582 930 616 32;
  • 67) 0.629 707 648 007 067 582 930 616 32 × 2 = 1 + 0.259 415 296 014 135 165 861 232 64;
  • 68) 0.259 415 296 014 135 165 861 232 64 × 2 = 0 + 0.518 830 592 028 270 331 722 465 28;
  • 69) 0.518 830 592 028 270 331 722 465 28 × 2 = 1 + 0.037 661 184 056 540 663 444 930 56;
  • 70) 0.037 661 184 056 540 663 444 930 56 × 2 = 0 + 0.075 322 368 113 081 326 889 861 12;
  • 71) 0.075 322 368 113 081 326 889 861 12 × 2 = 0 + 0.150 644 736 226 162 653 779 722 24;
  • 72) 0.150 644 736 226 162 653 779 722 24 × 2 = 0 + 0.301 289 472 452 325 307 559 444 48;
  • 73) 0.301 289 472 452 325 307 559 444 48 × 2 = 0 + 0.602 578 944 904 650 615 118 888 96;
  • 74) 0.602 578 944 904 650 615 118 888 96 × 2 = 1 + 0.205 157 889 809 301 230 237 777 92;
  • 75) 0.205 157 889 809 301 230 237 777 92 × 2 = 0 + 0.410 315 779 618 602 460 475 555 84;
  • 76) 0.410 315 779 618 602 460 475 555 84 × 2 = 0 + 0.820 631 559 237 204 920 951 111 68;
  • 77) 0.820 631 559 237 204 920 951 111 68 × 2 = 1 + 0.641 263 118 474 409 841 902 223 36;
  • 78) 0.641 263 118 474 409 841 902 223 36 × 2 = 1 + 0.282 526 236 948 819 683 804 446 72;
  • 79) 0.282 526 236 948 819 683 804 446 72 × 2 = 0 + 0.565 052 473 897 639 367 608 893 44;
  • 80) 0.565 052 473 897 639 367 608 893 44 × 2 = 1 + 0.130 104 947 795 278 735 217 786 88;
  • 81) 0.130 104 947 795 278 735 217 786 88 × 2 = 0 + 0.260 209 895 590 557 470 435 573 76;
  • 82) 0.260 209 895 590 557 470 435 573 76 × 2 = 0 + 0.520 419 791 181 114 940 871 147 52;
  • 83) 0.520 419 791 181 114 940 871 147 52 × 2 = 1 + 0.040 839 582 362 229 881 742 295 04;
  • 84) 0.040 839 582 362 229 881 742 295 04 × 2 = 0 + 0.081 679 164 724 459 763 484 590 08;
  • 85) 0.081 679 164 724 459 763 484 590 08 × 2 = 0 + 0.163 358 329 448 919 526 969 180 16;
  • 86) 0.163 358 329 448 919 526 969 180 16 × 2 = 0 + 0.326 716 658 897 839 053 938 360 32;
  • 87) 0.326 716 658 897 839 053 938 360 32 × 2 = 0 + 0.653 433 317 795 678 107 876 720 64;
  • 88) 0.653 433 317 795 678 107 876 720 64 × 2 = 1 + 0.306 866 635 591 356 215 753 441 28;
  • 89) 0.306 866 635 591 356 215 753 441 28 × 2 = 0 + 0.613 733 271 182 712 431 506 882 56;
  • 90) 0.613 733 271 182 712 431 506 882 56 × 2 = 1 + 0.227 466 542 365 424 863 013 765 12;
  • 91) 0.227 466 542 365 424 863 013 765 12 × 2 = 0 + 0.454 933 084 730 849 726 027 530 24;
  • 92) 0.454 933 084 730 849 726 027 530 24 × 2 = 0 + 0.909 866 169 461 699 452 055 060 48;
  • 93) 0.909 866 169 461 699 452 055 060 48 × 2 = 1 + 0.819 732 338 923 398 904 110 120 96;
  • 94) 0.819 732 338 923 398 904 110 120 96 × 2 = 1 + 0.639 464 677 846 797 808 220 241 92;
  • 95) 0.639 464 677 846 797 808 220 241 92 × 2 = 1 + 0.278 929 355 693 595 616 440 483 84;
  • 96) 0.278 929 355 693 595 616 440 483 84 × 2 = 0 + 0.557 858 711 387 191 232 880 967 68;
  • 97) 0.557 858 711 387 191 232 880 967 68 × 2 = 1 + 0.115 717 422 774 382 465 761 935 36;
  • 98) 0.115 717 422 774 382 465 761 935 36 × 2 = 0 + 0.231 434 845 548 764 931 523 870 72;
  • 99) 0.231 434 845 548 764 931 523 870 72 × 2 = 0 + 0.462 869 691 097 529 863 047 741 44;
  • 100) 0.462 869 691 097 529 863 047 741 44 × 2 = 0 + 0.925 739 382 195 059 726 095 482 88;
  • 101) 0.925 739 382 195 059 726 095 482 88 × 2 = 1 + 0.851 478 764 390 119 452 190 965 76;
  • 102) 0.851 478 764 390 119 452 190 965 76 × 2 = 1 + 0.702 957 528 780 238 904 381 931 52;
  • 103) 0.702 957 528 780 238 904 381 931 52 × 2 = 1 + 0.405 915 057 560 477 808 763 863 04;
  • 104) 0.405 915 057 560 477 808 763 863 04 × 2 = 0 + 0.811 830 115 120 955 617 527 726 08;
  • 105) 0.811 830 115 120 955 617 527 726 08 × 2 = 1 + 0.623 660 230 241 911 235 055 452 16;
  • 106) 0.623 660 230 241 911 235 055 452 16 × 2 = 1 + 0.247 320 460 483 822 470 110 904 32;
  • 107) 0.247 320 460 483 822 470 110 904 32 × 2 = 0 + 0.494 640 920 967 644 940 221 808 64;
  • 108) 0.494 640 920 967 644 940 221 808 64 × 2 = 0 + 0.989 281 841 935 289 880 443 617 28;
  • 109) 0.989 281 841 935 289 880 443 617 28 × 2 = 1 + 0.978 563 683 870 579 760 887 234 56;
  • 110) 0.978 563 683 870 579 760 887 234 56 × 2 = 1 + 0.957 127 367 741 159 521 774 469 12;
  • 111) 0.957 127 367 741 159 521 774 469 12 × 2 = 1 + 0.914 254 735 482 319 043 548 938 24;
  • 112) 0.914 254 735 482 319 043 548 938 24 × 2 = 1 + 0.828 509 470 964 638 087 097 876 48;
  • 113) 0.828 509 470 964 638 087 097 876 48 × 2 = 1 + 0.657 018 941 929 276 174 195 752 96;
  • 114) 0.657 018 941 929 276 174 195 752 96 × 2 = 1 + 0.314 037 883 858 552 348 391 505 92;
  • 115) 0.314 037 883 858 552 348 391 505 92 × 2 = 0 + 0.628 075 767 717 104 696 783 011 84;
  • 116) 0.628 075 767 717 104 696 783 011 84 × 2 = 1 + 0.256 151 535 434 209 393 566 023 68;
  • 117) 0.256 151 535 434 209 393 566 023 68 × 2 = 0 + 0.512 303 070 868 418 787 132 047 36;
  • 118) 0.512 303 070 868 418 787 132 047 36 × 2 = 1 + 0.024 606 141 736 837 574 264 094 72;
  • 119) 0.024 606 141 736 837 574 264 094 72 × 2 = 0 + 0.049 212 283 473 675 148 528 189 44;

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:


0.000 000 000 000 000 000 008 534 13(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0010 0001 0100 1110 1000 1110 1100 1111 1101 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 13(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0010 0001 0100 1110 1000 1110 1100 1111 1101 010(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:


0.000 000 000 000 000 000 008 534 13(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0010 0001 0100 1110 1000 1110 1100 1111 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0010 0001 0100 1110 1000 1110 1100 1111 1101 010(2) × 20 =

1.0100 0010 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010(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 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010


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 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010 =

0100 0010 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010


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 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010


Decimal number 0.000 000 000 000 000 000 008 534 13 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1001 0000 1010 0111 0100 0111 0110 0111 1110 1010

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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