0.000 000 000 000 000 000 008 529 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 529 4(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
0.000 000 000 000 000 000 008 529 4(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: 0.000 000 000 000 000 000 008 529 4.

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) 0.000 000 000 000 000 000 008 529 4 × 2 = 0 + 0.000 000 000 000 000 000 017 058 8;
  • 2) 0.000 000 000 000 000 000 017 058 8 × 2 = 0 + 0.000 000 000 000 000 000 034 117 6;
  • 3) 0.000 000 000 000 000 000 034 117 6 × 2 = 0 + 0.000 000 000 000 000 000 068 235 2;
  • 4) 0.000 000 000 000 000 000 068 235 2 × 2 = 0 + 0.000 000 000 000 000 000 136 470 4;
  • 5) 0.000 000 000 000 000 000 136 470 4 × 2 = 0 + 0.000 000 000 000 000 000 272 940 8;
  • 6) 0.000 000 000 000 000 000 272 940 8 × 2 = 0 + 0.000 000 000 000 000 000 545 881 6;
  • 7) 0.000 000 000 000 000 000 545 881 6 × 2 = 0 + 0.000 000 000 000 000 001 091 763 2;
  • 8) 0.000 000 000 000 000 001 091 763 2 × 2 = 0 + 0.000 000 000 000 000 002 183 526 4;
  • 9) 0.000 000 000 000 000 002 183 526 4 × 2 = 0 + 0.000 000 000 000 000 004 367 052 8;
  • 10) 0.000 000 000 000 000 004 367 052 8 × 2 = 0 + 0.000 000 000 000 000 008 734 105 6;
  • 11) 0.000 000 000 000 000 008 734 105 6 × 2 = 0 + 0.000 000 000 000 000 017 468 211 2;
  • 12) 0.000 000 000 000 000 017 468 211 2 × 2 = 0 + 0.000 000 000 000 000 034 936 422 4;
  • 13) 0.000 000 000 000 000 034 936 422 4 × 2 = 0 + 0.000 000 000 000 000 069 872 844 8;
  • 14) 0.000 000 000 000 000 069 872 844 8 × 2 = 0 + 0.000 000 000 000 000 139 745 689 6;
  • 15) 0.000 000 000 000 000 139 745 689 6 × 2 = 0 + 0.000 000 000 000 000 279 491 379 2;
  • 16) 0.000 000 000 000 000 279 491 379 2 × 2 = 0 + 0.000 000 000 000 000 558 982 758 4;
  • 17) 0.000 000 000 000 000 558 982 758 4 × 2 = 0 + 0.000 000 000 000 001 117 965 516 8;
  • 18) 0.000 000 000 000 001 117 965 516 8 × 2 = 0 + 0.000 000 000 000 002 235 931 033 6;
  • 19) 0.000 000 000 000 002 235 931 033 6 × 2 = 0 + 0.000 000 000 000 004 471 862 067 2;
  • 20) 0.000 000 000 000 004 471 862 067 2 × 2 = 0 + 0.000 000 000 000 008 943 724 134 4;
  • 21) 0.000 000 000 000 008 943 724 134 4 × 2 = 0 + 0.000 000 000 000 017 887 448 268 8;
  • 22) 0.000 000 000 000 017 887 448 268 8 × 2 = 0 + 0.000 000 000 000 035 774 896 537 6;
  • 23) 0.000 000 000 000 035 774 896 537 6 × 2 = 0 + 0.000 000 000 000 071 549 793 075 2;
  • 24) 0.000 000 000 000 071 549 793 075 2 × 2 = 0 + 0.000 000 000 000 143 099 586 150 4;
  • 25) 0.000 000 000 000 143 099 586 150 4 × 2 = 0 + 0.000 000 000 000 286 199 172 300 8;
  • 26) 0.000 000 000 000 286 199 172 300 8 × 2 = 0 + 0.000 000 000 000 572 398 344 601 6;
  • 27) 0.000 000 000 000 572 398 344 601 6 × 2 = 0 + 0.000 000 000 001 144 796 689 203 2;
  • 28) 0.000 000 000 001 144 796 689 203 2 × 2 = 0 + 0.000 000 000 002 289 593 378 406 4;
  • 29) 0.000 000 000 002 289 593 378 406 4 × 2 = 0 + 0.000 000 000 004 579 186 756 812 8;
  • 30) 0.000 000 000 004 579 186 756 812 8 × 2 = 0 + 0.000 000 000 009 158 373 513 625 6;
  • 31) 0.000 000 000 009 158 373 513 625 6 × 2 = 0 + 0.000 000 000 018 316 747 027 251 2;
  • 32) 0.000 000 000 018 316 747 027 251 2 × 2 = 0 + 0.000 000 000 036 633 494 054 502 4;
  • 33) 0.000 000 000 036 633 494 054 502 4 × 2 = 0 + 0.000 000 000 073 266 988 109 004 8;
  • 34) 0.000 000 000 073 266 988 109 004 8 × 2 = 0 + 0.000 000 000 146 533 976 218 009 6;
  • 35) 0.000 000 000 146 533 976 218 009 6 × 2 = 0 + 0.000 000 000 293 067 952 436 019 2;
  • 36) 0.000 000 000 293 067 952 436 019 2 × 2 = 0 + 0.000 000 000 586 135 904 872 038 4;
  • 37) 0.000 000 000 586 135 904 872 038 4 × 2 = 0 + 0.000 000 001 172 271 809 744 076 8;
  • 38) 0.000 000 001 172 271 809 744 076 8 × 2 = 0 + 0.000 000 002 344 543 619 488 153 6;
  • 39) 0.000 000 002 344 543 619 488 153 6 × 2 = 0 + 0.000 000 004 689 087 238 976 307 2;
  • 40) 0.000 000 004 689 087 238 976 307 2 × 2 = 0 + 0.000 000 009 378 174 477 952 614 4;
  • 41) 0.000 000 009 378 174 477 952 614 4 × 2 = 0 + 0.000 000 018 756 348 955 905 228 8;
  • 42) 0.000 000 018 756 348 955 905 228 8 × 2 = 0 + 0.000 000 037 512 697 911 810 457 6;
  • 43) 0.000 000 037 512 697 911 810 457 6 × 2 = 0 + 0.000 000 075 025 395 823 620 915 2;
  • 44) 0.000 000 075 025 395 823 620 915 2 × 2 = 0 + 0.000 000 150 050 791 647 241 830 4;
  • 45) 0.000 000 150 050 791 647 241 830 4 × 2 = 0 + 0.000 000 300 101 583 294 483 660 8;
  • 46) 0.000 000 300 101 583 294 483 660 8 × 2 = 0 + 0.000 000 600 203 166 588 967 321 6;
  • 47) 0.000 000 600 203 166 588 967 321 6 × 2 = 0 + 0.000 001 200 406 333 177 934 643 2;
  • 48) 0.000 001 200 406 333 177 934 643 2 × 2 = 0 + 0.000 002 400 812 666 355 869 286 4;
  • 49) 0.000 002 400 812 666 355 869 286 4 × 2 = 0 + 0.000 004 801 625 332 711 738 572 8;
  • 50) 0.000 004 801 625 332 711 738 572 8 × 2 = 0 + 0.000 009 603 250 665 423 477 145 6;
  • 51) 0.000 009 603 250 665 423 477 145 6 × 2 = 0 + 0.000 019 206 501 330 846 954 291 2;
  • 52) 0.000 019 206 501 330 846 954 291 2 × 2 = 0 + 0.000 038 413 002 661 693 908 582 4;
  • 53) 0.000 038 413 002 661 693 908 582 4 × 2 = 0 + 0.000 076 826 005 323 387 817 164 8;
  • 54) 0.000 076 826 005 323 387 817 164 8 × 2 = 0 + 0.000 153 652 010 646 775 634 329 6;
  • 55) 0.000 153 652 010 646 775 634 329 6 × 2 = 0 + 0.000 307 304 021 293 551 268 659 2;
  • 56) 0.000 307 304 021 293 551 268 659 2 × 2 = 0 + 0.000 614 608 042 587 102 537 318 4;
  • 57) 0.000 614 608 042 587 102 537 318 4 × 2 = 0 + 0.001 229 216 085 174 205 074 636 8;
  • 58) 0.001 229 216 085 174 205 074 636 8 × 2 = 0 + 0.002 458 432 170 348 410 149 273 6;
  • 59) 0.002 458 432 170 348 410 149 273 6 × 2 = 0 + 0.004 916 864 340 696 820 298 547 2;
  • 60) 0.004 916 864 340 696 820 298 547 2 × 2 = 0 + 0.009 833 728 681 393 640 597 094 4;
  • 61) 0.009 833 728 681 393 640 597 094 4 × 2 = 0 + 0.019 667 457 362 787 281 194 188 8;
  • 62) 0.019 667 457 362 787 281 194 188 8 × 2 = 0 + 0.039 334 914 725 574 562 388 377 6;
  • 63) 0.039 334 914 725 574 562 388 377 6 × 2 = 0 + 0.078 669 829 451 149 124 776 755 2;
  • 64) 0.078 669 829 451 149 124 776 755 2 × 2 = 0 + 0.157 339 658 902 298 249 553 510 4;
  • 65) 0.157 339 658 902 298 249 553 510 4 × 2 = 0 + 0.314 679 317 804 596 499 107 020 8;
  • 66) 0.314 679 317 804 596 499 107 020 8 × 2 = 0 + 0.629 358 635 609 192 998 214 041 6;
  • 67) 0.629 358 635 609 192 998 214 041 6 × 2 = 1 + 0.258 717 271 218 385 996 428 083 2;
  • 68) 0.258 717 271 218 385 996 428 083 2 × 2 = 0 + 0.517 434 542 436 771 992 856 166 4;
  • 69) 0.517 434 542 436 771 992 856 166 4 × 2 = 1 + 0.034 869 084 873 543 985 712 332 8;
  • 70) 0.034 869 084 873 543 985 712 332 8 × 2 = 0 + 0.069 738 169 747 087 971 424 665 6;
  • 71) 0.069 738 169 747 087 971 424 665 6 × 2 = 0 + 0.139 476 339 494 175 942 849 331 2;
  • 72) 0.139 476 339 494 175 942 849 331 2 × 2 = 0 + 0.278 952 678 988 351 885 698 662 4;
  • 73) 0.278 952 678 988 351 885 698 662 4 × 2 = 0 + 0.557 905 357 976 703 771 397 324 8;
  • 74) 0.557 905 357 976 703 771 397 324 8 × 2 = 1 + 0.115 810 715 953 407 542 794 649 6;
  • 75) 0.115 810 715 953 407 542 794 649 6 × 2 = 0 + 0.231 621 431 906 815 085 589 299 2;
  • 76) 0.231 621 431 906 815 085 589 299 2 × 2 = 0 + 0.463 242 863 813 630 171 178 598 4;
  • 77) 0.463 242 863 813 630 171 178 598 4 × 2 = 0 + 0.926 485 727 627 260 342 357 196 8;
  • 78) 0.926 485 727 627 260 342 357 196 8 × 2 = 1 + 0.852 971 455 254 520 684 714 393 6;
  • 79) 0.852 971 455 254 520 684 714 393 6 × 2 = 1 + 0.705 942 910 509 041 369 428 787 2;
  • 80) 0.705 942 910 509 041 369 428 787 2 × 2 = 1 + 0.411 885 821 018 082 738 857 574 4;
  • 81) 0.411 885 821 018 082 738 857 574 4 × 2 = 0 + 0.823 771 642 036 165 477 715 148 8;
  • 82) 0.823 771 642 036 165 477 715 148 8 × 2 = 1 + 0.647 543 284 072 330 955 430 297 6;
  • 83) 0.647 543 284 072 330 955 430 297 6 × 2 = 1 + 0.295 086 568 144 661 910 860 595 2;
  • 84) 0.295 086 568 144 661 910 860 595 2 × 2 = 0 + 0.590 173 136 289 323 821 721 190 4;
  • 85) 0.590 173 136 289 323 821 721 190 4 × 2 = 1 + 0.180 346 272 578 647 643 442 380 8;
  • 86) 0.180 346 272 578 647 643 442 380 8 × 2 = 0 + 0.360 692 545 157 295 286 884 761 6;
  • 87) 0.360 692 545 157 295 286 884 761 6 × 2 = 0 + 0.721 385 090 314 590 573 769 523 2;
  • 88) 0.721 385 090 314 590 573 769 523 2 × 2 = 1 + 0.442 770 180 629 181 147 539 046 4;
  • 89) 0.442 770 180 629 181 147 539 046 4 × 2 = 0 + 0.885 540 361 258 362 295 078 092 8;
  • 90) 0.885 540 361 258 362 295 078 092 8 × 2 = 1 + 0.771 080 722 516 724 590 156 185 6;
  • 91) 0.771 080 722 516 724 590 156 185 6 × 2 = 1 + 0.542 161 445 033 449 180 312 371 2;
  • 92) 0.542 161 445 033 449 180 312 371 2 × 2 = 1 + 0.084 322 890 066 898 360 624 742 4;
  • 93) 0.084 322 890 066 898 360 624 742 4 × 2 = 0 + 0.168 645 780 133 796 721 249 484 8;
  • 94) 0.168 645 780 133 796 721 249 484 8 × 2 = 0 + 0.337 291 560 267 593 442 498 969 6;
  • 95) 0.337 291 560 267 593 442 498 969 6 × 2 = 0 + 0.674 583 120 535 186 884 997 939 2;
  • 96) 0.674 583 120 535 186 884 997 939 2 × 2 = 1 + 0.349 166 241 070 373 769 995 878 4;
  • 97) 0.349 166 241 070 373 769 995 878 4 × 2 = 0 + 0.698 332 482 140 747 539 991 756 8;
  • 98) 0.698 332 482 140 747 539 991 756 8 × 2 = 1 + 0.396 664 964 281 495 079 983 513 6;
  • 99) 0.396 664 964 281 495 079 983 513 6 × 2 = 0 + 0.793 329 928 562 990 159 967 027 2;
  • 100) 0.793 329 928 562 990 159 967 027 2 × 2 = 1 + 0.586 659 857 125 980 319 934 054 4;
  • 101) 0.586 659 857 125 980 319 934 054 4 × 2 = 1 + 0.173 319 714 251 960 639 868 108 8;
  • 102) 0.173 319 714 251 960 639 868 108 8 × 2 = 0 + 0.346 639 428 503 921 279 736 217 6;
  • 103) 0.346 639 428 503 921 279 736 217 6 × 2 = 0 + 0.693 278 857 007 842 559 472 435 2;
  • 104) 0.693 278 857 007 842 559 472 435 2 × 2 = 1 + 0.386 557 714 015 685 118 944 870 4;
  • 105) 0.386 557 714 015 685 118 944 870 4 × 2 = 0 + 0.773 115 428 031 370 237 889 740 8;
  • 106) 0.773 115 428 031 370 237 889 740 8 × 2 = 1 + 0.546 230 856 062 740 475 779 481 6;
  • 107) 0.546 230 856 062 740 475 779 481 6 × 2 = 1 + 0.092 461 712 125 480 951 558 963 2;
  • 108) 0.092 461 712 125 480 951 558 963 2 × 2 = 0 + 0.184 923 424 250 961 903 117 926 4;
  • 109) 0.184 923 424 250 961 903 117 926 4 × 2 = 0 + 0.369 846 848 501 923 806 235 852 8;
  • 110) 0.369 846 848 501 923 806 235 852 8 × 2 = 0 + 0.739 693 697 003 847 612 471 705 6;
  • 111) 0.739 693 697 003 847 612 471 705 6 × 2 = 1 + 0.479 387 394 007 695 224 943 411 2;
  • 112) 0.479 387 394 007 695 224 943 411 2 × 2 = 0 + 0.958 774 788 015 390 449 886 822 4;
  • 113) 0.958 774 788 015 390 449 886 822 4 × 2 = 1 + 0.917 549 576 030 780 899 773 644 8;
  • 114) 0.917 549 576 030 780 899 773 644 8 × 2 = 1 + 0.835 099 152 061 561 799 547 289 6;
  • 115) 0.835 099 152 061 561 799 547 289 6 × 2 = 1 + 0.670 198 304 123 123 599 094 579 2;
  • 116) 0.670 198 304 123 123 599 094 579 2 × 2 = 1 + 0.340 396 608 246 247 198 189 158 4;
  • 117) 0.340 396 608 246 247 198 189 158 4 × 2 = 0 + 0.680 793 216 492 494 396 378 316 8;
  • 118) 0.680 793 216 492 494 396 378 316 8 × 2 = 1 + 0.361 586 432 984 988 792 756 633 6;
  • 119) 0.361 586 432 984 988 792 756 633 6 × 2 = 0 + 0.723 172 865 969 977 585 513 267 2;

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:


0.000 000 000 000 000 000 008 529 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0110 1001 0111 0001 0101 1001 0110 0010 1111 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 529 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0110 1001 0111 0001 0101 1001 0110 0010 1111 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:


0.000 000 000 000 000 000 008 529 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0110 1001 0111 0001 0101 1001 0110 0010 1111 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0111 0110 1001 0111 0001 0101 1001 0110 0010 1111 010(2) × 20 =

1.0100 0010 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 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 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 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 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 1010 =

0100 0010 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 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 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 1010


Decimal number 0.000 000 000 000 000 000 008 529 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0011 1011 0100 1011 1000 1010 1100 1011 0001 0111 1010

Share

» Convert decimal 0.000 000 000 000 000 000 008 523 5 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: Reputation: Bridge Between Company's Actions & Market Value. NotebookLM YouTube Video of 6.55 minutes

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