0.000 000 000 000 000 000 008 534 56 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 56(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 56(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 56.

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 56 × 2 = 0 + 0.000 000 000 000 000 000 017 069 12;
  • 2) 0.000 000 000 000 000 000 017 069 12 × 2 = 0 + 0.000 000 000 000 000 000 034 138 24;
  • 3) 0.000 000 000 000 000 000 034 138 24 × 2 = 0 + 0.000 000 000 000 000 000 068 276 48;
  • 4) 0.000 000 000 000 000 000 068 276 48 × 2 = 0 + 0.000 000 000 000 000 000 136 552 96;
  • 5) 0.000 000 000 000 000 000 136 552 96 × 2 = 0 + 0.000 000 000 000 000 000 273 105 92;
  • 6) 0.000 000 000 000 000 000 273 105 92 × 2 = 0 + 0.000 000 000 000 000 000 546 211 84;
  • 7) 0.000 000 000 000 000 000 546 211 84 × 2 = 0 + 0.000 000 000 000 000 001 092 423 68;
  • 8) 0.000 000 000 000 000 001 092 423 68 × 2 = 0 + 0.000 000 000 000 000 002 184 847 36;
  • 9) 0.000 000 000 000 000 002 184 847 36 × 2 = 0 + 0.000 000 000 000 000 004 369 694 72;
  • 10) 0.000 000 000 000 000 004 369 694 72 × 2 = 0 + 0.000 000 000 000 000 008 739 389 44;
  • 11) 0.000 000 000 000 000 008 739 389 44 × 2 = 0 + 0.000 000 000 000 000 017 478 778 88;
  • 12) 0.000 000 000 000 000 017 478 778 88 × 2 = 0 + 0.000 000 000 000 000 034 957 557 76;
  • 13) 0.000 000 000 000 000 034 957 557 76 × 2 = 0 + 0.000 000 000 000 000 069 915 115 52;
  • 14) 0.000 000 000 000 000 069 915 115 52 × 2 = 0 + 0.000 000 000 000 000 139 830 231 04;
  • 15) 0.000 000 000 000 000 139 830 231 04 × 2 = 0 + 0.000 000 000 000 000 279 660 462 08;
  • 16) 0.000 000 000 000 000 279 660 462 08 × 2 = 0 + 0.000 000 000 000 000 559 320 924 16;
  • 17) 0.000 000 000 000 000 559 320 924 16 × 2 = 0 + 0.000 000 000 000 001 118 641 848 32;
  • 18) 0.000 000 000 000 001 118 641 848 32 × 2 = 0 + 0.000 000 000 000 002 237 283 696 64;
  • 19) 0.000 000 000 000 002 237 283 696 64 × 2 = 0 + 0.000 000 000 000 004 474 567 393 28;
  • 20) 0.000 000 000 000 004 474 567 393 28 × 2 = 0 + 0.000 000 000 000 008 949 134 786 56;
  • 21) 0.000 000 000 000 008 949 134 786 56 × 2 = 0 + 0.000 000 000 000 017 898 269 573 12;
  • 22) 0.000 000 000 000 017 898 269 573 12 × 2 = 0 + 0.000 000 000 000 035 796 539 146 24;
  • 23) 0.000 000 000 000 035 796 539 146 24 × 2 = 0 + 0.000 000 000 000 071 593 078 292 48;
  • 24) 0.000 000 000 000 071 593 078 292 48 × 2 = 0 + 0.000 000 000 000 143 186 156 584 96;
  • 25) 0.000 000 000 000 143 186 156 584 96 × 2 = 0 + 0.000 000 000 000 286 372 313 169 92;
  • 26) 0.000 000 000 000 286 372 313 169 92 × 2 = 0 + 0.000 000 000 000 572 744 626 339 84;
  • 27) 0.000 000 000 000 572 744 626 339 84 × 2 = 0 + 0.000 000 000 001 145 489 252 679 68;
  • 28) 0.000 000 000 001 145 489 252 679 68 × 2 = 0 + 0.000 000 000 002 290 978 505 359 36;
  • 29) 0.000 000 000 002 290 978 505 359 36 × 2 = 0 + 0.000 000 000 004 581 957 010 718 72;
  • 30) 0.000 000 000 004 581 957 010 718 72 × 2 = 0 + 0.000 000 000 009 163 914 021 437 44;
  • 31) 0.000 000 000 009 163 914 021 437 44 × 2 = 0 + 0.000 000 000 018 327 828 042 874 88;
  • 32) 0.000 000 000 018 327 828 042 874 88 × 2 = 0 + 0.000 000 000 036 655 656 085 749 76;
  • 33) 0.000 000 000 036 655 656 085 749 76 × 2 = 0 + 0.000 000 000 073 311 312 171 499 52;
  • 34) 0.000 000 000 073 311 312 171 499 52 × 2 = 0 + 0.000 000 000 146 622 624 342 999 04;
  • 35) 0.000 000 000 146 622 624 342 999 04 × 2 = 0 + 0.000 000 000 293 245 248 685 998 08;
  • 36) 0.000 000 000 293 245 248 685 998 08 × 2 = 0 + 0.000 000 000 586 490 497 371 996 16;
  • 37) 0.000 000 000 586 490 497 371 996 16 × 2 = 0 + 0.000 000 001 172 980 994 743 992 32;
  • 38) 0.000 000 001 172 980 994 743 992 32 × 2 = 0 + 0.000 000 002 345 961 989 487 984 64;
  • 39) 0.000 000 002 345 961 989 487 984 64 × 2 = 0 + 0.000 000 004 691 923 978 975 969 28;
  • 40) 0.000 000 004 691 923 978 975 969 28 × 2 = 0 + 0.000 000 009 383 847 957 951 938 56;
  • 41) 0.000 000 009 383 847 957 951 938 56 × 2 = 0 + 0.000 000 018 767 695 915 903 877 12;
  • 42) 0.000 000 018 767 695 915 903 877 12 × 2 = 0 + 0.000 000 037 535 391 831 807 754 24;
  • 43) 0.000 000 037 535 391 831 807 754 24 × 2 = 0 + 0.000 000 075 070 783 663 615 508 48;
  • 44) 0.000 000 075 070 783 663 615 508 48 × 2 = 0 + 0.000 000 150 141 567 327 231 016 96;
  • 45) 0.000 000 150 141 567 327 231 016 96 × 2 = 0 + 0.000 000 300 283 134 654 462 033 92;
  • 46) 0.000 000 300 283 134 654 462 033 92 × 2 = 0 + 0.000 000 600 566 269 308 924 067 84;
  • 47) 0.000 000 600 566 269 308 924 067 84 × 2 = 0 + 0.000 001 201 132 538 617 848 135 68;
  • 48) 0.000 001 201 132 538 617 848 135 68 × 2 = 0 + 0.000 002 402 265 077 235 696 271 36;
  • 49) 0.000 002 402 265 077 235 696 271 36 × 2 = 0 + 0.000 004 804 530 154 471 392 542 72;
  • 50) 0.000 004 804 530 154 471 392 542 72 × 2 = 0 + 0.000 009 609 060 308 942 785 085 44;
  • 51) 0.000 009 609 060 308 942 785 085 44 × 2 = 0 + 0.000 019 218 120 617 885 570 170 88;
  • 52) 0.000 019 218 120 617 885 570 170 88 × 2 = 0 + 0.000 038 436 241 235 771 140 341 76;
  • 53) 0.000 038 436 241 235 771 140 341 76 × 2 = 0 + 0.000 076 872 482 471 542 280 683 52;
  • 54) 0.000 076 872 482 471 542 280 683 52 × 2 = 0 + 0.000 153 744 964 943 084 561 367 04;
  • 55) 0.000 153 744 964 943 084 561 367 04 × 2 = 0 + 0.000 307 489 929 886 169 122 734 08;
  • 56) 0.000 307 489 929 886 169 122 734 08 × 2 = 0 + 0.000 614 979 859 772 338 245 468 16;
  • 57) 0.000 614 979 859 772 338 245 468 16 × 2 = 0 + 0.001 229 959 719 544 676 490 936 32;
  • 58) 0.001 229 959 719 544 676 490 936 32 × 2 = 0 + 0.002 459 919 439 089 352 981 872 64;
  • 59) 0.002 459 919 439 089 352 981 872 64 × 2 = 0 + 0.004 919 838 878 178 705 963 745 28;
  • 60) 0.004 919 838 878 178 705 963 745 28 × 2 = 0 + 0.009 839 677 756 357 411 927 490 56;
  • 61) 0.009 839 677 756 357 411 927 490 56 × 2 = 0 + 0.019 679 355 512 714 823 854 981 12;
  • 62) 0.019 679 355 512 714 823 854 981 12 × 2 = 0 + 0.039 358 711 025 429 647 709 962 24;
  • 63) 0.039 358 711 025 429 647 709 962 24 × 2 = 0 + 0.078 717 422 050 859 295 419 924 48;
  • 64) 0.078 717 422 050 859 295 419 924 48 × 2 = 0 + 0.157 434 844 101 718 590 839 848 96;
  • 65) 0.157 434 844 101 718 590 839 848 96 × 2 = 0 + 0.314 869 688 203 437 181 679 697 92;
  • 66) 0.314 869 688 203 437 181 679 697 92 × 2 = 0 + 0.629 739 376 406 874 363 359 395 84;
  • 67) 0.629 739 376 406 874 363 359 395 84 × 2 = 1 + 0.259 478 752 813 748 726 718 791 68;
  • 68) 0.259 478 752 813 748 726 718 791 68 × 2 = 0 + 0.518 957 505 627 497 453 437 583 36;
  • 69) 0.518 957 505 627 497 453 437 583 36 × 2 = 1 + 0.037 915 011 254 994 906 875 166 72;
  • 70) 0.037 915 011 254 994 906 875 166 72 × 2 = 0 + 0.075 830 022 509 989 813 750 333 44;
  • 71) 0.075 830 022 509 989 813 750 333 44 × 2 = 0 + 0.151 660 045 019 979 627 500 666 88;
  • 72) 0.151 660 045 019 979 627 500 666 88 × 2 = 0 + 0.303 320 090 039 959 255 001 333 76;
  • 73) 0.303 320 090 039 959 255 001 333 76 × 2 = 0 + 0.606 640 180 079 918 510 002 667 52;
  • 74) 0.606 640 180 079 918 510 002 667 52 × 2 = 1 + 0.213 280 360 159 837 020 005 335 04;
  • 75) 0.213 280 360 159 837 020 005 335 04 × 2 = 0 + 0.426 560 720 319 674 040 010 670 08;
  • 76) 0.426 560 720 319 674 040 010 670 08 × 2 = 0 + 0.853 121 440 639 348 080 021 340 16;
  • 77) 0.853 121 440 639 348 080 021 340 16 × 2 = 1 + 0.706 242 881 278 696 160 042 680 32;
  • 78) 0.706 242 881 278 696 160 042 680 32 × 2 = 1 + 0.412 485 762 557 392 320 085 360 64;
  • 79) 0.412 485 762 557 392 320 085 360 64 × 2 = 0 + 0.824 971 525 114 784 640 170 721 28;
  • 80) 0.824 971 525 114 784 640 170 721 28 × 2 = 1 + 0.649 943 050 229 569 280 341 442 56;
  • 81) 0.649 943 050 229 569 280 341 442 56 × 2 = 1 + 0.299 886 100 459 138 560 682 885 12;
  • 82) 0.299 886 100 459 138 560 682 885 12 × 2 = 0 + 0.599 772 200 918 277 121 365 770 24;
  • 83) 0.599 772 200 918 277 121 365 770 24 × 2 = 1 + 0.199 544 401 836 554 242 731 540 48;
  • 84) 0.199 544 401 836 554 242 731 540 48 × 2 = 0 + 0.399 088 803 673 108 485 463 080 96;
  • 85) 0.399 088 803 673 108 485 463 080 96 × 2 = 0 + 0.798 177 607 346 216 970 926 161 92;
  • 86) 0.798 177 607 346 216 970 926 161 92 × 2 = 1 + 0.596 355 214 692 433 941 852 323 84;
  • 87) 0.596 355 214 692 433 941 852 323 84 × 2 = 1 + 0.192 710 429 384 867 883 704 647 68;
  • 88) 0.192 710 429 384 867 883 704 647 68 × 2 = 0 + 0.385 420 858 769 735 767 409 295 36;
  • 89) 0.385 420 858 769 735 767 409 295 36 × 2 = 0 + 0.770 841 717 539 471 534 818 590 72;
  • 90) 0.770 841 717 539 471 534 818 590 72 × 2 = 1 + 0.541 683 435 078 943 069 637 181 44;
  • 91) 0.541 683 435 078 943 069 637 181 44 × 2 = 1 + 0.083 366 870 157 886 139 274 362 88;
  • 92) 0.083 366 870 157 886 139 274 362 88 × 2 = 0 + 0.166 733 740 315 772 278 548 725 76;
  • 93) 0.166 733 740 315 772 278 548 725 76 × 2 = 0 + 0.333 467 480 631 544 557 097 451 52;
  • 94) 0.333 467 480 631 544 557 097 451 52 × 2 = 0 + 0.666 934 961 263 089 114 194 903 04;
  • 95) 0.666 934 961 263 089 114 194 903 04 × 2 = 1 + 0.333 869 922 526 178 228 389 806 08;
  • 96) 0.333 869 922 526 178 228 389 806 08 × 2 = 0 + 0.667 739 845 052 356 456 779 612 16;
  • 97) 0.667 739 845 052 356 456 779 612 16 × 2 = 1 + 0.335 479 690 104 712 913 559 224 32;
  • 98) 0.335 479 690 104 712 913 559 224 32 × 2 = 0 + 0.670 959 380 209 425 827 118 448 64;
  • 99) 0.670 959 380 209 425 827 118 448 64 × 2 = 1 + 0.341 918 760 418 851 654 236 897 28;
  • 100) 0.341 918 760 418 851 654 236 897 28 × 2 = 0 + 0.683 837 520 837 703 308 473 794 56;
  • 101) 0.683 837 520 837 703 308 473 794 56 × 2 = 1 + 0.367 675 041 675 406 616 947 589 12;
  • 102) 0.367 675 041 675 406 616 947 589 12 × 2 = 0 + 0.735 350 083 350 813 233 895 178 24;
  • 103) 0.735 350 083 350 813 233 895 178 24 × 2 = 1 + 0.470 700 166 701 626 467 790 356 48;
  • 104) 0.470 700 166 701 626 467 790 356 48 × 2 = 0 + 0.941 400 333 403 252 935 580 712 96;
  • 105) 0.941 400 333 403 252 935 580 712 96 × 2 = 1 + 0.882 800 666 806 505 871 161 425 92;
  • 106) 0.882 800 666 806 505 871 161 425 92 × 2 = 1 + 0.765 601 333 613 011 742 322 851 84;
  • 107) 0.765 601 333 613 011 742 322 851 84 × 2 = 1 + 0.531 202 667 226 023 484 645 703 68;
  • 108) 0.531 202 667 226 023 484 645 703 68 × 2 = 1 + 0.062 405 334 452 046 969 291 407 36;
  • 109) 0.062 405 334 452 046 969 291 407 36 × 2 = 0 + 0.124 810 668 904 093 938 582 814 72;
  • 110) 0.124 810 668 904 093 938 582 814 72 × 2 = 0 + 0.249 621 337 808 187 877 165 629 44;
  • 111) 0.249 621 337 808 187 877 165 629 44 × 2 = 0 + 0.499 242 675 616 375 754 331 258 88;
  • 112) 0.499 242 675 616 375 754 331 258 88 × 2 = 0 + 0.998 485 351 232 751 508 662 517 76;
  • 113) 0.998 485 351 232 751 508 662 517 76 × 2 = 1 + 0.996 970 702 465 503 017 325 035 52;
  • 114) 0.996 970 702 465 503 017 325 035 52 × 2 = 1 + 0.993 941 404 931 006 034 650 071 04;
  • 115) 0.993 941 404 931 006 034 650 071 04 × 2 = 1 + 0.987 882 809 862 012 069 300 142 08;
  • 116) 0.987 882 809 862 012 069 300 142 08 × 2 = 1 + 0.975 765 619 724 024 138 600 284 16;
  • 117) 0.975 765 619 724 024 138 600 284 16 × 2 = 1 + 0.951 531 239 448 048 277 200 568 32;
  • 118) 0.951 531 239 448 048 277 200 568 32 × 2 = 1 + 0.903 062 478 896 096 554 401 136 64;
  • 119) 0.903 062 478 896 096 554 401 136 64 × 2 = 1 + 0.806 124 957 792 193 108 802 273 28;

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 56(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1010 0110 0110 0010 1010 1010 1111 0000 1111 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 56(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1010 0110 0110 0010 1010 1010 1111 0000 1111 111(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 56(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1010 0110 0110 0010 1010 1010 1111 0000 1111 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1010 0110 0110 0010 1010 1010 1111 0000 1111 111(2) × 20 =

1.0100 0010 0110 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111(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 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111


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 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111 =

0100 0010 0110 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111


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 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111


Decimal number 0.000 000 000 000 000 000 008 534 56 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1101 0011 0011 0001 0101 0101 0111 1000 0111 1111

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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