0.000 000 000 000 000 000 008 536 77 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 77(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
0.000 000 000 000 000 000 008 536 77(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: 0.000 000 000 000 000 000 008 536 77.

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) 0.000 000 000 000 000 000 008 536 77 × 2 = 0 + 0.000 000 000 000 000 000 017 073 54;
  • 2) 0.000 000 000 000 000 000 017 073 54 × 2 = 0 + 0.000 000 000 000 000 000 034 147 08;
  • 3) 0.000 000 000 000 000 000 034 147 08 × 2 = 0 + 0.000 000 000 000 000 000 068 294 16;
  • 4) 0.000 000 000 000 000 000 068 294 16 × 2 = 0 + 0.000 000 000 000 000 000 136 588 32;
  • 5) 0.000 000 000 000 000 000 136 588 32 × 2 = 0 + 0.000 000 000 000 000 000 273 176 64;
  • 6) 0.000 000 000 000 000 000 273 176 64 × 2 = 0 + 0.000 000 000 000 000 000 546 353 28;
  • 7) 0.000 000 000 000 000 000 546 353 28 × 2 = 0 + 0.000 000 000 000 000 001 092 706 56;
  • 8) 0.000 000 000 000 000 001 092 706 56 × 2 = 0 + 0.000 000 000 000 000 002 185 413 12;
  • 9) 0.000 000 000 000 000 002 185 413 12 × 2 = 0 + 0.000 000 000 000 000 004 370 826 24;
  • 10) 0.000 000 000 000 000 004 370 826 24 × 2 = 0 + 0.000 000 000 000 000 008 741 652 48;
  • 11) 0.000 000 000 000 000 008 741 652 48 × 2 = 0 + 0.000 000 000 000 000 017 483 304 96;
  • 12) 0.000 000 000 000 000 017 483 304 96 × 2 = 0 + 0.000 000 000 000 000 034 966 609 92;
  • 13) 0.000 000 000 000 000 034 966 609 92 × 2 = 0 + 0.000 000 000 000 000 069 933 219 84;
  • 14) 0.000 000 000 000 000 069 933 219 84 × 2 = 0 + 0.000 000 000 000 000 139 866 439 68;
  • 15) 0.000 000 000 000 000 139 866 439 68 × 2 = 0 + 0.000 000 000 000 000 279 732 879 36;
  • 16) 0.000 000 000 000 000 279 732 879 36 × 2 = 0 + 0.000 000 000 000 000 559 465 758 72;
  • 17) 0.000 000 000 000 000 559 465 758 72 × 2 = 0 + 0.000 000 000 000 001 118 931 517 44;
  • 18) 0.000 000 000 000 001 118 931 517 44 × 2 = 0 + 0.000 000 000 000 002 237 863 034 88;
  • 19) 0.000 000 000 000 002 237 863 034 88 × 2 = 0 + 0.000 000 000 000 004 475 726 069 76;
  • 20) 0.000 000 000 000 004 475 726 069 76 × 2 = 0 + 0.000 000 000 000 008 951 452 139 52;
  • 21) 0.000 000 000 000 008 951 452 139 52 × 2 = 0 + 0.000 000 000 000 017 902 904 279 04;
  • 22) 0.000 000 000 000 017 902 904 279 04 × 2 = 0 + 0.000 000 000 000 035 805 808 558 08;
  • 23) 0.000 000 000 000 035 805 808 558 08 × 2 = 0 + 0.000 000 000 000 071 611 617 116 16;
  • 24) 0.000 000 000 000 071 611 617 116 16 × 2 = 0 + 0.000 000 000 000 143 223 234 232 32;
  • 25) 0.000 000 000 000 143 223 234 232 32 × 2 = 0 + 0.000 000 000 000 286 446 468 464 64;
  • 26) 0.000 000 000 000 286 446 468 464 64 × 2 = 0 + 0.000 000 000 000 572 892 936 929 28;
  • 27) 0.000 000 000 000 572 892 936 929 28 × 2 = 0 + 0.000 000 000 001 145 785 873 858 56;
  • 28) 0.000 000 000 001 145 785 873 858 56 × 2 = 0 + 0.000 000 000 002 291 571 747 717 12;
  • 29) 0.000 000 000 002 291 571 747 717 12 × 2 = 0 + 0.000 000 000 004 583 143 495 434 24;
  • 30) 0.000 000 000 004 583 143 495 434 24 × 2 = 0 + 0.000 000 000 009 166 286 990 868 48;
  • 31) 0.000 000 000 009 166 286 990 868 48 × 2 = 0 + 0.000 000 000 018 332 573 981 736 96;
  • 32) 0.000 000 000 018 332 573 981 736 96 × 2 = 0 + 0.000 000 000 036 665 147 963 473 92;
  • 33) 0.000 000 000 036 665 147 963 473 92 × 2 = 0 + 0.000 000 000 073 330 295 926 947 84;
  • 34) 0.000 000 000 073 330 295 926 947 84 × 2 = 0 + 0.000 000 000 146 660 591 853 895 68;
  • 35) 0.000 000 000 146 660 591 853 895 68 × 2 = 0 + 0.000 000 000 293 321 183 707 791 36;
  • 36) 0.000 000 000 293 321 183 707 791 36 × 2 = 0 + 0.000 000 000 586 642 367 415 582 72;
  • 37) 0.000 000 000 586 642 367 415 582 72 × 2 = 0 + 0.000 000 001 173 284 734 831 165 44;
  • 38) 0.000 000 001 173 284 734 831 165 44 × 2 = 0 + 0.000 000 002 346 569 469 662 330 88;
  • 39) 0.000 000 002 346 569 469 662 330 88 × 2 = 0 + 0.000 000 004 693 138 939 324 661 76;
  • 40) 0.000 000 004 693 138 939 324 661 76 × 2 = 0 + 0.000 000 009 386 277 878 649 323 52;
  • 41) 0.000 000 009 386 277 878 649 323 52 × 2 = 0 + 0.000 000 018 772 555 757 298 647 04;
  • 42) 0.000 000 018 772 555 757 298 647 04 × 2 = 0 + 0.000 000 037 545 111 514 597 294 08;
  • 43) 0.000 000 037 545 111 514 597 294 08 × 2 = 0 + 0.000 000 075 090 223 029 194 588 16;
  • 44) 0.000 000 075 090 223 029 194 588 16 × 2 = 0 + 0.000 000 150 180 446 058 389 176 32;
  • 45) 0.000 000 150 180 446 058 389 176 32 × 2 = 0 + 0.000 000 300 360 892 116 778 352 64;
  • 46) 0.000 000 300 360 892 116 778 352 64 × 2 = 0 + 0.000 000 600 721 784 233 556 705 28;
  • 47) 0.000 000 600 721 784 233 556 705 28 × 2 = 0 + 0.000 001 201 443 568 467 113 410 56;
  • 48) 0.000 001 201 443 568 467 113 410 56 × 2 = 0 + 0.000 002 402 887 136 934 226 821 12;
  • 49) 0.000 002 402 887 136 934 226 821 12 × 2 = 0 + 0.000 004 805 774 273 868 453 642 24;
  • 50) 0.000 004 805 774 273 868 453 642 24 × 2 = 0 + 0.000 009 611 548 547 736 907 284 48;
  • 51) 0.000 009 611 548 547 736 907 284 48 × 2 = 0 + 0.000 019 223 097 095 473 814 568 96;
  • 52) 0.000 019 223 097 095 473 814 568 96 × 2 = 0 + 0.000 038 446 194 190 947 629 137 92;
  • 53) 0.000 038 446 194 190 947 629 137 92 × 2 = 0 + 0.000 076 892 388 381 895 258 275 84;
  • 54) 0.000 076 892 388 381 895 258 275 84 × 2 = 0 + 0.000 153 784 776 763 790 516 551 68;
  • 55) 0.000 153 784 776 763 790 516 551 68 × 2 = 0 + 0.000 307 569 553 527 581 033 103 36;
  • 56) 0.000 307 569 553 527 581 033 103 36 × 2 = 0 + 0.000 615 139 107 055 162 066 206 72;
  • 57) 0.000 615 139 107 055 162 066 206 72 × 2 = 0 + 0.001 230 278 214 110 324 132 413 44;
  • 58) 0.001 230 278 214 110 324 132 413 44 × 2 = 0 + 0.002 460 556 428 220 648 264 826 88;
  • 59) 0.002 460 556 428 220 648 264 826 88 × 2 = 0 + 0.004 921 112 856 441 296 529 653 76;
  • 60) 0.004 921 112 856 441 296 529 653 76 × 2 = 0 + 0.009 842 225 712 882 593 059 307 52;
  • 61) 0.009 842 225 712 882 593 059 307 52 × 2 = 0 + 0.019 684 451 425 765 186 118 615 04;
  • 62) 0.019 684 451 425 765 186 118 615 04 × 2 = 0 + 0.039 368 902 851 530 372 237 230 08;
  • 63) 0.039 368 902 851 530 372 237 230 08 × 2 = 0 + 0.078 737 805 703 060 744 474 460 16;
  • 64) 0.078 737 805 703 060 744 474 460 16 × 2 = 0 + 0.157 475 611 406 121 488 948 920 32;
  • 65) 0.157 475 611 406 121 488 948 920 32 × 2 = 0 + 0.314 951 222 812 242 977 897 840 64;
  • 66) 0.314 951 222 812 242 977 897 840 64 × 2 = 0 + 0.629 902 445 624 485 955 795 681 28;
  • 67) 0.629 902 445 624 485 955 795 681 28 × 2 = 1 + 0.259 804 891 248 971 911 591 362 56;
  • 68) 0.259 804 891 248 971 911 591 362 56 × 2 = 0 + 0.519 609 782 497 943 823 182 725 12;
  • 69) 0.519 609 782 497 943 823 182 725 12 × 2 = 1 + 0.039 219 564 995 887 646 365 450 24;
  • 70) 0.039 219 564 995 887 646 365 450 24 × 2 = 0 + 0.078 439 129 991 775 292 730 900 48;
  • 71) 0.078 439 129 991 775 292 730 900 48 × 2 = 0 + 0.156 878 259 983 550 585 461 800 96;
  • 72) 0.156 878 259 983 550 585 461 800 96 × 2 = 0 + 0.313 756 519 967 101 170 923 601 92;
  • 73) 0.313 756 519 967 101 170 923 601 92 × 2 = 0 + 0.627 513 039 934 202 341 847 203 84;
  • 74) 0.627 513 039 934 202 341 847 203 84 × 2 = 1 + 0.255 026 079 868 404 683 694 407 68;
  • 75) 0.255 026 079 868 404 683 694 407 68 × 2 = 0 + 0.510 052 159 736 809 367 388 815 36;
  • 76) 0.510 052 159 736 809 367 388 815 36 × 2 = 1 + 0.020 104 319 473 618 734 777 630 72;
  • 77) 0.020 104 319 473 618 734 777 630 72 × 2 = 0 + 0.040 208 638 947 237 469 555 261 44;
  • 78) 0.040 208 638 947 237 469 555 261 44 × 2 = 0 + 0.080 417 277 894 474 939 110 522 88;
  • 79) 0.080 417 277 894 474 939 110 522 88 × 2 = 0 + 0.160 834 555 788 949 878 221 045 76;
  • 80) 0.160 834 555 788 949 878 221 045 76 × 2 = 0 + 0.321 669 111 577 899 756 442 091 52;
  • 81) 0.321 669 111 577 899 756 442 091 52 × 2 = 0 + 0.643 338 223 155 799 512 884 183 04;
  • 82) 0.643 338 223 155 799 512 884 183 04 × 2 = 1 + 0.286 676 446 311 599 025 768 366 08;
  • 83) 0.286 676 446 311 599 025 768 366 08 × 2 = 0 + 0.573 352 892 623 198 051 536 732 16;
  • 84) 0.573 352 892 623 198 051 536 732 16 × 2 = 1 + 0.146 705 785 246 396 103 073 464 32;
  • 85) 0.146 705 785 246 396 103 073 464 32 × 2 = 0 + 0.293 411 570 492 792 206 146 928 64;
  • 86) 0.293 411 570 492 792 206 146 928 64 × 2 = 0 + 0.586 823 140 985 584 412 293 857 28;
  • 87) 0.586 823 140 985 584 412 293 857 28 × 2 = 1 + 0.173 646 281 971 168 824 587 714 56;
  • 88) 0.173 646 281 971 168 824 587 714 56 × 2 = 0 + 0.347 292 563 942 337 649 175 429 12;
  • 89) 0.347 292 563 942 337 649 175 429 12 × 2 = 0 + 0.694 585 127 884 675 298 350 858 24;
  • 90) 0.694 585 127 884 675 298 350 858 24 × 2 = 1 + 0.389 170 255 769 350 596 701 716 48;
  • 91) 0.389 170 255 769 350 596 701 716 48 × 2 = 0 + 0.778 340 511 538 701 193 403 432 96;
  • 92) 0.778 340 511 538 701 193 403 432 96 × 2 = 1 + 0.556 681 023 077 402 386 806 865 92;
  • 93) 0.556 681 023 077 402 386 806 865 92 × 2 = 1 + 0.113 362 046 154 804 773 613 731 84;
  • 94) 0.113 362 046 154 804 773 613 731 84 × 2 = 0 + 0.226 724 092 309 609 547 227 463 68;
  • 95) 0.226 724 092 309 609 547 227 463 68 × 2 = 0 + 0.453 448 184 619 219 094 454 927 36;
  • 96) 0.453 448 184 619 219 094 454 927 36 × 2 = 0 + 0.906 896 369 238 438 188 909 854 72;
  • 97) 0.906 896 369 238 438 188 909 854 72 × 2 = 1 + 0.813 792 738 476 876 377 819 709 44;
  • 98) 0.813 792 738 476 876 377 819 709 44 × 2 = 1 + 0.627 585 476 953 752 755 639 418 88;
  • 99) 0.627 585 476 953 752 755 639 418 88 × 2 = 1 + 0.255 170 953 907 505 511 278 837 76;
  • 100) 0.255 170 953 907 505 511 278 837 76 × 2 = 0 + 0.510 341 907 815 011 022 557 675 52;
  • 101) 0.510 341 907 815 011 022 557 675 52 × 2 = 1 + 0.020 683 815 630 022 045 115 351 04;
  • 102) 0.020 683 815 630 022 045 115 351 04 × 2 = 0 + 0.041 367 631 260 044 090 230 702 08;
  • 103) 0.041 367 631 260 044 090 230 702 08 × 2 = 0 + 0.082 735 262 520 088 180 461 404 16;
  • 104) 0.082 735 262 520 088 180 461 404 16 × 2 = 0 + 0.165 470 525 040 176 360 922 808 32;
  • 105) 0.165 470 525 040 176 360 922 808 32 × 2 = 0 + 0.330 941 050 080 352 721 845 616 64;
  • 106) 0.330 941 050 080 352 721 845 616 64 × 2 = 0 + 0.661 882 100 160 705 443 691 233 28;
  • 107) 0.661 882 100 160 705 443 691 233 28 × 2 = 1 + 0.323 764 200 321 410 887 382 466 56;
  • 108) 0.323 764 200 321 410 887 382 466 56 × 2 = 0 + 0.647 528 400 642 821 774 764 933 12;
  • 109) 0.647 528 400 642 821 774 764 933 12 × 2 = 1 + 0.295 056 801 285 643 549 529 866 24;
  • 110) 0.295 056 801 285 643 549 529 866 24 × 2 = 0 + 0.590 113 602 571 287 099 059 732 48;
  • 111) 0.590 113 602 571 287 099 059 732 48 × 2 = 1 + 0.180 227 205 142 574 198 119 464 96;
  • 112) 0.180 227 205 142 574 198 119 464 96 × 2 = 0 + 0.360 454 410 285 148 396 238 929 92;
  • 113) 0.360 454 410 285 148 396 238 929 92 × 2 = 0 + 0.720 908 820 570 296 792 477 859 84;
  • 114) 0.720 908 820 570 296 792 477 859 84 × 2 = 1 + 0.441 817 641 140 593 584 955 719 68;
  • 115) 0.441 817 641 140 593 584 955 719 68 × 2 = 0 + 0.883 635 282 281 187 169 911 439 36;
  • 116) 0.883 635 282 281 187 169 911 439 36 × 2 = 1 + 0.767 270 564 562 374 339 822 878 72;
  • 117) 0.767 270 564 562 374 339 822 878 72 × 2 = 1 + 0.534 541 129 124 748 679 645 757 44;
  • 118) 0.534 541 129 124 748 679 645 757 44 × 2 = 1 + 0.069 082 258 249 497 359 291 514 88;
  • 119) 0.069 082 258 249 497 359 291 514 88 × 2 = 0 + 0.138 164 516 498 994 718 583 029 76;

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:


0.000 000 000 000 000 000 008 536 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 0010 0101 1000 1110 1000 0010 1010 0101 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 0010 0101 1000 1110 1000 0010 1010 0101 110(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:


0.000 000 000 000 000 000 008 536 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 0010 0101 1000 1110 1000 0010 1010 0101 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0101 0010 0101 1000 1110 1000 0010 1010 0101 110(2) × 20 =

1.0100 0010 1000 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110(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 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110


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 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110 =

0100 0010 1000 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110


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 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110


Decimal number 0.000 000 000 000 000 000 008 536 77 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0010 1001 0010 1100 0111 0100 0001 0101 0010 1110

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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