0.000 000 000 000 000 000 008 559 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 559 3(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
0.000 000 000 000 000 000 008 559 3(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: 0.000 000 000 000 000 000 008 559 3.

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) 0.000 000 000 000 000 000 008 559 3 × 2 = 0 + 0.000 000 000 000 000 000 017 118 6;
  • 2) 0.000 000 000 000 000 000 017 118 6 × 2 = 0 + 0.000 000 000 000 000 000 034 237 2;
  • 3) 0.000 000 000 000 000 000 034 237 2 × 2 = 0 + 0.000 000 000 000 000 000 068 474 4;
  • 4) 0.000 000 000 000 000 000 068 474 4 × 2 = 0 + 0.000 000 000 000 000 000 136 948 8;
  • 5) 0.000 000 000 000 000 000 136 948 8 × 2 = 0 + 0.000 000 000 000 000 000 273 897 6;
  • 6) 0.000 000 000 000 000 000 273 897 6 × 2 = 0 + 0.000 000 000 000 000 000 547 795 2;
  • 7) 0.000 000 000 000 000 000 547 795 2 × 2 = 0 + 0.000 000 000 000 000 001 095 590 4;
  • 8) 0.000 000 000 000 000 001 095 590 4 × 2 = 0 + 0.000 000 000 000 000 002 191 180 8;
  • 9) 0.000 000 000 000 000 002 191 180 8 × 2 = 0 + 0.000 000 000 000 000 004 382 361 6;
  • 10) 0.000 000 000 000 000 004 382 361 6 × 2 = 0 + 0.000 000 000 000 000 008 764 723 2;
  • 11) 0.000 000 000 000 000 008 764 723 2 × 2 = 0 + 0.000 000 000 000 000 017 529 446 4;
  • 12) 0.000 000 000 000 000 017 529 446 4 × 2 = 0 + 0.000 000 000 000 000 035 058 892 8;
  • 13) 0.000 000 000 000 000 035 058 892 8 × 2 = 0 + 0.000 000 000 000 000 070 117 785 6;
  • 14) 0.000 000 000 000 000 070 117 785 6 × 2 = 0 + 0.000 000 000 000 000 140 235 571 2;
  • 15) 0.000 000 000 000 000 140 235 571 2 × 2 = 0 + 0.000 000 000 000 000 280 471 142 4;
  • 16) 0.000 000 000 000 000 280 471 142 4 × 2 = 0 + 0.000 000 000 000 000 560 942 284 8;
  • 17) 0.000 000 000 000 000 560 942 284 8 × 2 = 0 + 0.000 000 000 000 001 121 884 569 6;
  • 18) 0.000 000 000 000 001 121 884 569 6 × 2 = 0 + 0.000 000 000 000 002 243 769 139 2;
  • 19) 0.000 000 000 000 002 243 769 139 2 × 2 = 0 + 0.000 000 000 000 004 487 538 278 4;
  • 20) 0.000 000 000 000 004 487 538 278 4 × 2 = 0 + 0.000 000 000 000 008 975 076 556 8;
  • 21) 0.000 000 000 000 008 975 076 556 8 × 2 = 0 + 0.000 000 000 000 017 950 153 113 6;
  • 22) 0.000 000 000 000 017 950 153 113 6 × 2 = 0 + 0.000 000 000 000 035 900 306 227 2;
  • 23) 0.000 000 000 000 035 900 306 227 2 × 2 = 0 + 0.000 000 000 000 071 800 612 454 4;
  • 24) 0.000 000 000 000 071 800 612 454 4 × 2 = 0 + 0.000 000 000 000 143 601 224 908 8;
  • 25) 0.000 000 000 000 143 601 224 908 8 × 2 = 0 + 0.000 000 000 000 287 202 449 817 6;
  • 26) 0.000 000 000 000 287 202 449 817 6 × 2 = 0 + 0.000 000 000 000 574 404 899 635 2;
  • 27) 0.000 000 000 000 574 404 899 635 2 × 2 = 0 + 0.000 000 000 001 148 809 799 270 4;
  • 28) 0.000 000 000 001 148 809 799 270 4 × 2 = 0 + 0.000 000 000 002 297 619 598 540 8;
  • 29) 0.000 000 000 002 297 619 598 540 8 × 2 = 0 + 0.000 000 000 004 595 239 197 081 6;
  • 30) 0.000 000 000 004 595 239 197 081 6 × 2 = 0 + 0.000 000 000 009 190 478 394 163 2;
  • 31) 0.000 000 000 009 190 478 394 163 2 × 2 = 0 + 0.000 000 000 018 380 956 788 326 4;
  • 32) 0.000 000 000 018 380 956 788 326 4 × 2 = 0 + 0.000 000 000 036 761 913 576 652 8;
  • 33) 0.000 000 000 036 761 913 576 652 8 × 2 = 0 + 0.000 000 000 073 523 827 153 305 6;
  • 34) 0.000 000 000 073 523 827 153 305 6 × 2 = 0 + 0.000 000 000 147 047 654 306 611 2;
  • 35) 0.000 000 000 147 047 654 306 611 2 × 2 = 0 + 0.000 000 000 294 095 308 613 222 4;
  • 36) 0.000 000 000 294 095 308 613 222 4 × 2 = 0 + 0.000 000 000 588 190 617 226 444 8;
  • 37) 0.000 000 000 588 190 617 226 444 8 × 2 = 0 + 0.000 000 001 176 381 234 452 889 6;
  • 38) 0.000 000 001 176 381 234 452 889 6 × 2 = 0 + 0.000 000 002 352 762 468 905 779 2;
  • 39) 0.000 000 002 352 762 468 905 779 2 × 2 = 0 + 0.000 000 004 705 524 937 811 558 4;
  • 40) 0.000 000 004 705 524 937 811 558 4 × 2 = 0 + 0.000 000 009 411 049 875 623 116 8;
  • 41) 0.000 000 009 411 049 875 623 116 8 × 2 = 0 + 0.000 000 018 822 099 751 246 233 6;
  • 42) 0.000 000 018 822 099 751 246 233 6 × 2 = 0 + 0.000 000 037 644 199 502 492 467 2;
  • 43) 0.000 000 037 644 199 502 492 467 2 × 2 = 0 + 0.000 000 075 288 399 004 984 934 4;
  • 44) 0.000 000 075 288 399 004 984 934 4 × 2 = 0 + 0.000 000 150 576 798 009 969 868 8;
  • 45) 0.000 000 150 576 798 009 969 868 8 × 2 = 0 + 0.000 000 301 153 596 019 939 737 6;
  • 46) 0.000 000 301 153 596 019 939 737 6 × 2 = 0 + 0.000 000 602 307 192 039 879 475 2;
  • 47) 0.000 000 602 307 192 039 879 475 2 × 2 = 0 + 0.000 001 204 614 384 079 758 950 4;
  • 48) 0.000 001 204 614 384 079 758 950 4 × 2 = 0 + 0.000 002 409 228 768 159 517 900 8;
  • 49) 0.000 002 409 228 768 159 517 900 8 × 2 = 0 + 0.000 004 818 457 536 319 035 801 6;
  • 50) 0.000 004 818 457 536 319 035 801 6 × 2 = 0 + 0.000 009 636 915 072 638 071 603 2;
  • 51) 0.000 009 636 915 072 638 071 603 2 × 2 = 0 + 0.000 019 273 830 145 276 143 206 4;
  • 52) 0.000 019 273 830 145 276 143 206 4 × 2 = 0 + 0.000 038 547 660 290 552 286 412 8;
  • 53) 0.000 038 547 660 290 552 286 412 8 × 2 = 0 + 0.000 077 095 320 581 104 572 825 6;
  • 54) 0.000 077 095 320 581 104 572 825 6 × 2 = 0 + 0.000 154 190 641 162 209 145 651 2;
  • 55) 0.000 154 190 641 162 209 145 651 2 × 2 = 0 + 0.000 308 381 282 324 418 291 302 4;
  • 56) 0.000 308 381 282 324 418 291 302 4 × 2 = 0 + 0.000 616 762 564 648 836 582 604 8;
  • 57) 0.000 616 762 564 648 836 582 604 8 × 2 = 0 + 0.001 233 525 129 297 673 165 209 6;
  • 58) 0.001 233 525 129 297 673 165 209 6 × 2 = 0 + 0.002 467 050 258 595 346 330 419 2;
  • 59) 0.002 467 050 258 595 346 330 419 2 × 2 = 0 + 0.004 934 100 517 190 692 660 838 4;
  • 60) 0.004 934 100 517 190 692 660 838 4 × 2 = 0 + 0.009 868 201 034 381 385 321 676 8;
  • 61) 0.009 868 201 034 381 385 321 676 8 × 2 = 0 + 0.019 736 402 068 762 770 643 353 6;
  • 62) 0.019 736 402 068 762 770 643 353 6 × 2 = 0 + 0.039 472 804 137 525 541 286 707 2;
  • 63) 0.039 472 804 137 525 541 286 707 2 × 2 = 0 + 0.078 945 608 275 051 082 573 414 4;
  • 64) 0.078 945 608 275 051 082 573 414 4 × 2 = 0 + 0.157 891 216 550 102 165 146 828 8;
  • 65) 0.157 891 216 550 102 165 146 828 8 × 2 = 0 + 0.315 782 433 100 204 330 293 657 6;
  • 66) 0.315 782 433 100 204 330 293 657 6 × 2 = 0 + 0.631 564 866 200 408 660 587 315 2;
  • 67) 0.631 564 866 200 408 660 587 315 2 × 2 = 1 + 0.263 129 732 400 817 321 174 630 4;
  • 68) 0.263 129 732 400 817 321 174 630 4 × 2 = 0 + 0.526 259 464 801 634 642 349 260 8;
  • 69) 0.526 259 464 801 634 642 349 260 8 × 2 = 1 + 0.052 518 929 603 269 284 698 521 6;
  • 70) 0.052 518 929 603 269 284 698 521 6 × 2 = 0 + 0.105 037 859 206 538 569 397 043 2;
  • 71) 0.105 037 859 206 538 569 397 043 2 × 2 = 0 + 0.210 075 718 413 077 138 794 086 4;
  • 72) 0.210 075 718 413 077 138 794 086 4 × 2 = 0 + 0.420 151 436 826 154 277 588 172 8;
  • 73) 0.420 151 436 826 154 277 588 172 8 × 2 = 0 + 0.840 302 873 652 308 555 176 345 6;
  • 74) 0.840 302 873 652 308 555 176 345 6 × 2 = 1 + 0.680 605 747 304 617 110 352 691 2;
  • 75) 0.680 605 747 304 617 110 352 691 2 × 2 = 1 + 0.361 211 494 609 234 220 705 382 4;
  • 76) 0.361 211 494 609 234 220 705 382 4 × 2 = 0 + 0.722 422 989 218 468 441 410 764 8;
  • 77) 0.722 422 989 218 468 441 410 764 8 × 2 = 1 + 0.444 845 978 436 936 882 821 529 6;
  • 78) 0.444 845 978 436 936 882 821 529 6 × 2 = 0 + 0.889 691 956 873 873 765 643 059 2;
  • 79) 0.889 691 956 873 873 765 643 059 2 × 2 = 1 + 0.779 383 913 747 747 531 286 118 4;
  • 80) 0.779 383 913 747 747 531 286 118 4 × 2 = 1 + 0.558 767 827 495 495 062 572 236 8;
  • 81) 0.558 767 827 495 495 062 572 236 8 × 2 = 1 + 0.117 535 654 990 990 125 144 473 6;
  • 82) 0.117 535 654 990 990 125 144 473 6 × 2 = 0 + 0.235 071 309 981 980 250 288 947 2;
  • 83) 0.235 071 309 981 980 250 288 947 2 × 2 = 0 + 0.470 142 619 963 960 500 577 894 4;
  • 84) 0.470 142 619 963 960 500 577 894 4 × 2 = 0 + 0.940 285 239 927 921 001 155 788 8;
  • 85) 0.940 285 239 927 921 001 155 788 8 × 2 = 1 + 0.880 570 479 855 842 002 311 577 6;
  • 86) 0.880 570 479 855 842 002 311 577 6 × 2 = 1 + 0.761 140 959 711 684 004 623 155 2;
  • 87) 0.761 140 959 711 684 004 623 155 2 × 2 = 1 + 0.522 281 919 423 368 009 246 310 4;
  • 88) 0.522 281 919 423 368 009 246 310 4 × 2 = 1 + 0.044 563 838 846 736 018 492 620 8;
  • 89) 0.044 563 838 846 736 018 492 620 8 × 2 = 0 + 0.089 127 677 693 472 036 985 241 6;
  • 90) 0.089 127 677 693 472 036 985 241 6 × 2 = 0 + 0.178 255 355 386 944 073 970 483 2;
  • 91) 0.178 255 355 386 944 073 970 483 2 × 2 = 0 + 0.356 510 710 773 888 147 940 966 4;
  • 92) 0.356 510 710 773 888 147 940 966 4 × 2 = 0 + 0.713 021 421 547 776 295 881 932 8;
  • 93) 0.713 021 421 547 776 295 881 932 8 × 2 = 1 + 0.426 042 843 095 552 591 763 865 6;
  • 94) 0.426 042 843 095 552 591 763 865 6 × 2 = 0 + 0.852 085 686 191 105 183 527 731 2;
  • 95) 0.852 085 686 191 105 183 527 731 2 × 2 = 1 + 0.704 171 372 382 210 367 055 462 4;
  • 96) 0.704 171 372 382 210 367 055 462 4 × 2 = 1 + 0.408 342 744 764 420 734 110 924 8;
  • 97) 0.408 342 744 764 420 734 110 924 8 × 2 = 0 + 0.816 685 489 528 841 468 221 849 6;
  • 98) 0.816 685 489 528 841 468 221 849 6 × 2 = 1 + 0.633 370 979 057 682 936 443 699 2;
  • 99) 0.633 370 979 057 682 936 443 699 2 × 2 = 1 + 0.266 741 958 115 365 872 887 398 4;
  • 100) 0.266 741 958 115 365 872 887 398 4 × 2 = 0 + 0.533 483 916 230 731 745 774 796 8;
  • 101) 0.533 483 916 230 731 745 774 796 8 × 2 = 1 + 0.066 967 832 461 463 491 549 593 6;
  • 102) 0.066 967 832 461 463 491 549 593 6 × 2 = 0 + 0.133 935 664 922 926 983 099 187 2;
  • 103) 0.133 935 664 922 926 983 099 187 2 × 2 = 0 + 0.267 871 329 845 853 966 198 374 4;
  • 104) 0.267 871 329 845 853 966 198 374 4 × 2 = 0 + 0.535 742 659 691 707 932 396 748 8;
  • 105) 0.535 742 659 691 707 932 396 748 8 × 2 = 1 + 0.071 485 319 383 415 864 793 497 6;
  • 106) 0.071 485 319 383 415 864 793 497 6 × 2 = 0 + 0.142 970 638 766 831 729 586 995 2;
  • 107) 0.142 970 638 766 831 729 586 995 2 × 2 = 0 + 0.285 941 277 533 663 459 173 990 4;
  • 108) 0.285 941 277 533 663 459 173 990 4 × 2 = 0 + 0.571 882 555 067 326 918 347 980 8;
  • 109) 0.571 882 555 067 326 918 347 980 8 × 2 = 1 + 0.143 765 110 134 653 836 695 961 6;
  • 110) 0.143 765 110 134 653 836 695 961 6 × 2 = 0 + 0.287 530 220 269 307 673 391 923 2;
  • 111) 0.287 530 220 269 307 673 391 923 2 × 2 = 0 + 0.575 060 440 538 615 346 783 846 4;
  • 112) 0.575 060 440 538 615 346 783 846 4 × 2 = 1 + 0.150 120 881 077 230 693 567 692 8;
  • 113) 0.150 120 881 077 230 693 567 692 8 × 2 = 0 + 0.300 241 762 154 461 387 135 385 6;
  • 114) 0.300 241 762 154 461 387 135 385 6 × 2 = 0 + 0.600 483 524 308 922 774 270 771 2;
  • 115) 0.600 483 524 308 922 774 270 771 2 × 2 = 1 + 0.200 967 048 617 845 548 541 542 4;
  • 116) 0.200 967 048 617 845 548 541 542 4 × 2 = 0 + 0.401 934 097 235 691 097 083 084 8;
  • 117) 0.401 934 097 235 691 097 083 084 8 × 2 = 0 + 0.803 868 194 471 382 194 166 169 6;
  • 118) 0.803 868 194 471 382 194 166 169 6 × 2 = 1 + 0.607 736 388 942 764 388 332 339 2;
  • 119) 0.607 736 388 942 764 388 332 339 2 × 2 = 1 + 0.215 472 777 885 528 776 664 678 4;

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:


0.000 000 000 000 000 000 008 559 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1011 1000 1111 0000 1011 0110 1000 1000 1001 0010 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 559 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1011 1000 1111 0000 1011 0110 1000 1000 1001 0010 011(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:


0.000 000 000 000 000 000 008 559 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1011 1000 1111 0000 1011 0110 1000 1000 1001 0010 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1011 1000 1111 0000 1011 0110 1000 1000 1001 0010 011(2) × 20 =

1.0100 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011(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 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011


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 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011 =

0100 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011


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 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011


Decimal number 0.000 000 000 000 000 000 008 559 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0101 1100 0111 1000 0101 1011 0100 0100 0100 1001 0011

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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