0.000 000 000 000 000 000 008 537 67 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 67(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 008 537 67(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 008 537 67.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 000 000 000 008 537 67 × 2 = 0 + 0.000 000 000 000 000 000 017 075 34;
  • 2) 0.000 000 000 000 000 000 017 075 34 × 2 = 0 + 0.000 000 000 000 000 000 034 150 68;
  • 3) 0.000 000 000 000 000 000 034 150 68 × 2 = 0 + 0.000 000 000 000 000 000 068 301 36;
  • 4) 0.000 000 000 000 000 000 068 301 36 × 2 = 0 + 0.000 000 000 000 000 000 136 602 72;
  • 5) 0.000 000 000 000 000 000 136 602 72 × 2 = 0 + 0.000 000 000 000 000 000 273 205 44;
  • 6) 0.000 000 000 000 000 000 273 205 44 × 2 = 0 + 0.000 000 000 000 000 000 546 410 88;
  • 7) 0.000 000 000 000 000 000 546 410 88 × 2 = 0 + 0.000 000 000 000 000 001 092 821 76;
  • 8) 0.000 000 000 000 000 001 092 821 76 × 2 = 0 + 0.000 000 000 000 000 002 185 643 52;
  • 9) 0.000 000 000 000 000 002 185 643 52 × 2 = 0 + 0.000 000 000 000 000 004 371 287 04;
  • 10) 0.000 000 000 000 000 004 371 287 04 × 2 = 0 + 0.000 000 000 000 000 008 742 574 08;
  • 11) 0.000 000 000 000 000 008 742 574 08 × 2 = 0 + 0.000 000 000 000 000 017 485 148 16;
  • 12) 0.000 000 000 000 000 017 485 148 16 × 2 = 0 + 0.000 000 000 000 000 034 970 296 32;
  • 13) 0.000 000 000 000 000 034 970 296 32 × 2 = 0 + 0.000 000 000 000 000 069 940 592 64;
  • 14) 0.000 000 000 000 000 069 940 592 64 × 2 = 0 + 0.000 000 000 000 000 139 881 185 28;
  • 15) 0.000 000 000 000 000 139 881 185 28 × 2 = 0 + 0.000 000 000 000 000 279 762 370 56;
  • 16) 0.000 000 000 000 000 279 762 370 56 × 2 = 0 + 0.000 000 000 000 000 559 524 741 12;
  • 17) 0.000 000 000 000 000 559 524 741 12 × 2 = 0 + 0.000 000 000 000 001 119 049 482 24;
  • 18) 0.000 000 000 000 001 119 049 482 24 × 2 = 0 + 0.000 000 000 000 002 238 098 964 48;
  • 19) 0.000 000 000 000 002 238 098 964 48 × 2 = 0 + 0.000 000 000 000 004 476 197 928 96;
  • 20) 0.000 000 000 000 004 476 197 928 96 × 2 = 0 + 0.000 000 000 000 008 952 395 857 92;
  • 21) 0.000 000 000 000 008 952 395 857 92 × 2 = 0 + 0.000 000 000 000 017 904 791 715 84;
  • 22) 0.000 000 000 000 017 904 791 715 84 × 2 = 0 + 0.000 000 000 000 035 809 583 431 68;
  • 23) 0.000 000 000 000 035 809 583 431 68 × 2 = 0 + 0.000 000 000 000 071 619 166 863 36;
  • 24) 0.000 000 000 000 071 619 166 863 36 × 2 = 0 + 0.000 000 000 000 143 238 333 726 72;
  • 25) 0.000 000 000 000 143 238 333 726 72 × 2 = 0 + 0.000 000 000 000 286 476 667 453 44;
  • 26) 0.000 000 000 000 286 476 667 453 44 × 2 = 0 + 0.000 000 000 000 572 953 334 906 88;
  • 27) 0.000 000 000 000 572 953 334 906 88 × 2 = 0 + 0.000 000 000 001 145 906 669 813 76;
  • 28) 0.000 000 000 001 145 906 669 813 76 × 2 = 0 + 0.000 000 000 002 291 813 339 627 52;
  • 29) 0.000 000 000 002 291 813 339 627 52 × 2 = 0 + 0.000 000 000 004 583 626 679 255 04;
  • 30) 0.000 000 000 004 583 626 679 255 04 × 2 = 0 + 0.000 000 000 009 167 253 358 510 08;
  • 31) 0.000 000 000 009 167 253 358 510 08 × 2 = 0 + 0.000 000 000 018 334 506 717 020 16;
  • 32) 0.000 000 000 018 334 506 717 020 16 × 2 = 0 + 0.000 000 000 036 669 013 434 040 32;
  • 33) 0.000 000 000 036 669 013 434 040 32 × 2 = 0 + 0.000 000 000 073 338 026 868 080 64;
  • 34) 0.000 000 000 073 338 026 868 080 64 × 2 = 0 + 0.000 000 000 146 676 053 736 161 28;
  • 35) 0.000 000 000 146 676 053 736 161 28 × 2 = 0 + 0.000 000 000 293 352 107 472 322 56;
  • 36) 0.000 000 000 293 352 107 472 322 56 × 2 = 0 + 0.000 000 000 586 704 214 944 645 12;
  • 37) 0.000 000 000 586 704 214 944 645 12 × 2 = 0 + 0.000 000 001 173 408 429 889 290 24;
  • 38) 0.000 000 001 173 408 429 889 290 24 × 2 = 0 + 0.000 000 002 346 816 859 778 580 48;
  • 39) 0.000 000 002 346 816 859 778 580 48 × 2 = 0 + 0.000 000 004 693 633 719 557 160 96;
  • 40) 0.000 000 004 693 633 719 557 160 96 × 2 = 0 + 0.000 000 009 387 267 439 114 321 92;
  • 41) 0.000 000 009 387 267 439 114 321 92 × 2 = 0 + 0.000 000 018 774 534 878 228 643 84;
  • 42) 0.000 000 018 774 534 878 228 643 84 × 2 = 0 + 0.000 000 037 549 069 756 457 287 68;
  • 43) 0.000 000 037 549 069 756 457 287 68 × 2 = 0 + 0.000 000 075 098 139 512 914 575 36;
  • 44) 0.000 000 075 098 139 512 914 575 36 × 2 = 0 + 0.000 000 150 196 279 025 829 150 72;
  • 45) 0.000 000 150 196 279 025 829 150 72 × 2 = 0 + 0.000 000 300 392 558 051 658 301 44;
  • 46) 0.000 000 300 392 558 051 658 301 44 × 2 = 0 + 0.000 000 600 785 116 103 316 602 88;
  • 47) 0.000 000 600 785 116 103 316 602 88 × 2 = 0 + 0.000 001 201 570 232 206 633 205 76;
  • 48) 0.000 001 201 570 232 206 633 205 76 × 2 = 0 + 0.000 002 403 140 464 413 266 411 52;
  • 49) 0.000 002 403 140 464 413 266 411 52 × 2 = 0 + 0.000 004 806 280 928 826 532 823 04;
  • 50) 0.000 004 806 280 928 826 532 823 04 × 2 = 0 + 0.000 009 612 561 857 653 065 646 08;
  • 51) 0.000 009 612 561 857 653 065 646 08 × 2 = 0 + 0.000 019 225 123 715 306 131 292 16;
  • 52) 0.000 019 225 123 715 306 131 292 16 × 2 = 0 + 0.000 038 450 247 430 612 262 584 32;
  • 53) 0.000 038 450 247 430 612 262 584 32 × 2 = 0 + 0.000 076 900 494 861 224 525 168 64;
  • 54) 0.000 076 900 494 861 224 525 168 64 × 2 = 0 + 0.000 153 800 989 722 449 050 337 28;
  • 55) 0.000 153 800 989 722 449 050 337 28 × 2 = 0 + 0.000 307 601 979 444 898 100 674 56;
  • 56) 0.000 307 601 979 444 898 100 674 56 × 2 = 0 + 0.000 615 203 958 889 796 201 349 12;
  • 57) 0.000 615 203 958 889 796 201 349 12 × 2 = 0 + 0.001 230 407 917 779 592 402 698 24;
  • 58) 0.001 230 407 917 779 592 402 698 24 × 2 = 0 + 0.002 460 815 835 559 184 805 396 48;
  • 59) 0.002 460 815 835 559 184 805 396 48 × 2 = 0 + 0.004 921 631 671 118 369 610 792 96;
  • 60) 0.004 921 631 671 118 369 610 792 96 × 2 = 0 + 0.009 843 263 342 236 739 221 585 92;
  • 61) 0.009 843 263 342 236 739 221 585 92 × 2 = 0 + 0.019 686 526 684 473 478 443 171 84;
  • 62) 0.019 686 526 684 473 478 443 171 84 × 2 = 0 + 0.039 373 053 368 946 956 886 343 68;
  • 63) 0.039 373 053 368 946 956 886 343 68 × 2 = 0 + 0.078 746 106 737 893 913 772 687 36;
  • 64) 0.078 746 106 737 893 913 772 687 36 × 2 = 0 + 0.157 492 213 475 787 827 545 374 72;
  • 65) 0.157 492 213 475 787 827 545 374 72 × 2 = 0 + 0.314 984 426 951 575 655 090 749 44;
  • 66) 0.314 984 426 951 575 655 090 749 44 × 2 = 0 + 0.629 968 853 903 151 310 181 498 88;
  • 67) 0.629 968 853 903 151 310 181 498 88 × 2 = 1 + 0.259 937 707 806 302 620 362 997 76;
  • 68) 0.259 937 707 806 302 620 362 997 76 × 2 = 0 + 0.519 875 415 612 605 240 725 995 52;
  • 69) 0.519 875 415 612 605 240 725 995 52 × 2 = 1 + 0.039 750 831 225 210 481 451 991 04;
  • 70) 0.039 750 831 225 210 481 451 991 04 × 2 = 0 + 0.079 501 662 450 420 962 903 982 08;
  • 71) 0.079 501 662 450 420 962 903 982 08 × 2 = 0 + 0.159 003 324 900 841 925 807 964 16;
  • 72) 0.159 003 324 900 841 925 807 964 16 × 2 = 0 + 0.318 006 649 801 683 851 615 928 32;
  • 73) 0.318 006 649 801 683 851 615 928 32 × 2 = 0 + 0.636 013 299 603 367 703 231 856 64;
  • 74) 0.636 013 299 603 367 703 231 856 64 × 2 = 1 + 0.272 026 599 206 735 406 463 713 28;
  • 75) 0.272 026 599 206 735 406 463 713 28 × 2 = 0 + 0.544 053 198 413 470 812 927 426 56;
  • 76) 0.544 053 198 413 470 812 927 426 56 × 2 = 1 + 0.088 106 396 826 941 625 854 853 12;
  • 77) 0.088 106 396 826 941 625 854 853 12 × 2 = 0 + 0.176 212 793 653 883 251 709 706 24;
  • 78) 0.176 212 793 653 883 251 709 706 24 × 2 = 0 + 0.352 425 587 307 766 503 419 412 48;
  • 79) 0.352 425 587 307 766 503 419 412 48 × 2 = 0 + 0.704 851 174 615 533 006 838 824 96;
  • 80) 0.704 851 174 615 533 006 838 824 96 × 2 = 1 + 0.409 702 349 231 066 013 677 649 92;
  • 81) 0.409 702 349 231 066 013 677 649 92 × 2 = 0 + 0.819 404 698 462 132 027 355 299 84;
  • 82) 0.819 404 698 462 132 027 355 299 84 × 2 = 1 + 0.638 809 396 924 264 054 710 599 68;
  • 83) 0.638 809 396 924 264 054 710 599 68 × 2 = 1 + 0.277 618 793 848 528 109 421 199 36;
  • 84) 0.277 618 793 848 528 109 421 199 36 × 2 = 0 + 0.555 237 587 697 056 218 842 398 72;
  • 85) 0.555 237 587 697 056 218 842 398 72 × 2 = 1 + 0.110 475 175 394 112 437 684 797 44;
  • 86) 0.110 475 175 394 112 437 684 797 44 × 2 = 0 + 0.220 950 350 788 224 875 369 594 88;
  • 87) 0.220 950 350 788 224 875 369 594 88 × 2 = 0 + 0.441 900 701 576 449 750 739 189 76;
  • 88) 0.441 900 701 576 449 750 739 189 76 × 2 = 0 + 0.883 801 403 152 899 501 478 379 52;
  • 89) 0.883 801 403 152 899 501 478 379 52 × 2 = 1 + 0.767 602 806 305 799 002 956 759 04;
  • 90) 0.767 602 806 305 799 002 956 759 04 × 2 = 1 + 0.535 205 612 611 598 005 913 518 08;
  • 91) 0.535 205 612 611 598 005 913 518 08 × 2 = 1 + 0.070 411 225 223 196 011 827 036 16;
  • 92) 0.070 411 225 223 196 011 827 036 16 × 2 = 0 + 0.140 822 450 446 392 023 654 072 32;
  • 93) 0.140 822 450 446 392 023 654 072 32 × 2 = 0 + 0.281 644 900 892 784 047 308 144 64;
  • 94) 0.281 644 900 892 784 047 308 144 64 × 2 = 0 + 0.563 289 801 785 568 094 616 289 28;
  • 95) 0.563 289 801 785 568 094 616 289 28 × 2 = 1 + 0.126 579 603 571 136 189 232 578 56;
  • 96) 0.126 579 603 571 136 189 232 578 56 × 2 = 0 + 0.253 159 207 142 272 378 465 157 12;
  • 97) 0.253 159 207 142 272 378 465 157 12 × 2 = 0 + 0.506 318 414 284 544 756 930 314 24;
  • 98) 0.506 318 414 284 544 756 930 314 24 × 2 = 1 + 0.012 636 828 569 089 513 860 628 48;
  • 99) 0.012 636 828 569 089 513 860 628 48 × 2 = 0 + 0.025 273 657 138 179 027 721 256 96;
  • 100) 0.025 273 657 138 179 027 721 256 96 × 2 = 0 + 0.050 547 314 276 358 055 442 513 92;
  • 101) 0.050 547 314 276 358 055 442 513 92 × 2 = 0 + 0.101 094 628 552 716 110 885 027 84;
  • 102) 0.101 094 628 552 716 110 885 027 84 × 2 = 0 + 0.202 189 257 105 432 221 770 055 68;
  • 103) 0.202 189 257 105 432 221 770 055 68 × 2 = 0 + 0.404 378 514 210 864 443 540 111 36;
  • 104) 0.404 378 514 210 864 443 540 111 36 × 2 = 0 + 0.808 757 028 421 728 887 080 222 72;
  • 105) 0.808 757 028 421 728 887 080 222 72 × 2 = 1 + 0.617 514 056 843 457 774 160 445 44;
  • 106) 0.617 514 056 843 457 774 160 445 44 × 2 = 1 + 0.235 028 113 686 915 548 320 890 88;
  • 107) 0.235 028 113 686 915 548 320 890 88 × 2 = 0 + 0.470 056 227 373 831 096 641 781 76;
  • 108) 0.470 056 227 373 831 096 641 781 76 × 2 = 0 + 0.940 112 454 747 662 193 283 563 52;
  • 109) 0.940 112 454 747 662 193 283 563 52 × 2 = 1 + 0.880 224 909 495 324 386 567 127 04;
  • 110) 0.880 224 909 495 324 386 567 127 04 × 2 = 1 + 0.760 449 818 990 648 773 134 254 08;
  • 111) 0.760 449 818 990 648 773 134 254 08 × 2 = 1 + 0.520 899 637 981 297 546 268 508 16;
  • 112) 0.520 899 637 981 297 546 268 508 16 × 2 = 1 + 0.041 799 275 962 595 092 537 016 32;
  • 113) 0.041 799 275 962 595 092 537 016 32 × 2 = 0 + 0.083 598 551 925 190 185 074 032 64;
  • 114) 0.083 598 551 925 190 185 074 032 64 × 2 = 0 + 0.167 197 103 850 380 370 148 065 28;
  • 115) 0.167 197 103 850 380 370 148 065 28 × 2 = 0 + 0.334 394 207 700 760 740 296 130 56;
  • 116) 0.334 394 207 700 760 740 296 130 56 × 2 = 0 + 0.668 788 415 401 521 480 592 261 12;
  • 117) 0.668 788 415 401 521 480 592 261 12 × 2 = 1 + 0.337 576 830 803 042 961 184 522 24;
  • 118) 0.337 576 830 803 042 961 184 522 24 × 2 = 0 + 0.675 153 661 606 085 922 369 044 48;
  • 119) 0.675 153 661 606 085 922 369 044 48 × 2 = 1 + 0.350 307 323 212 171 844 738 088 96;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 000 008 537 67(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0110 1000 1110 0010 0100 0000 1100 1111 0000 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 67(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0110 1000 1110 0010 0100 0000 1100 1111 0000 101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 537 67(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0110 1000 1110 0010 0100 0000 1100 1111 0000 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0110 1000 1110 0010 0100 0000 1100 1111 0000 101(2) × 20 =

1.0100 0010 1000 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101(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 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101


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 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101 =

0100 0010 1000 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101


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 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101


Decimal number 0.000 000 000 000 000 000 008 537 67 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1011 0100 0111 0001 0010 0000 0110 0111 1000 0101

Share

» Convert decimal 0.000 000 000 000 000 000 008 538 35 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: Pull vs. Push Leadership: The Invisible Power. NotebookLM YouTube Video of 11.31 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