0.000 000 000 000 000 000 008 519 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 519(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
0.000 000 000 000 000 000 008 519(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: 0.000 000 000 000 000 000 008 519.

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) 0.000 000 000 000 000 000 008 519 × 2 = 0 + 0.000 000 000 000 000 000 017 038;
  • 2) 0.000 000 000 000 000 000 017 038 × 2 = 0 + 0.000 000 000 000 000 000 034 076;
  • 3) 0.000 000 000 000 000 000 034 076 × 2 = 0 + 0.000 000 000 000 000 000 068 152;
  • 4) 0.000 000 000 000 000 000 068 152 × 2 = 0 + 0.000 000 000 000 000 000 136 304;
  • 5) 0.000 000 000 000 000 000 136 304 × 2 = 0 + 0.000 000 000 000 000 000 272 608;
  • 6) 0.000 000 000 000 000 000 272 608 × 2 = 0 + 0.000 000 000 000 000 000 545 216;
  • 7) 0.000 000 000 000 000 000 545 216 × 2 = 0 + 0.000 000 000 000 000 001 090 432;
  • 8) 0.000 000 000 000 000 001 090 432 × 2 = 0 + 0.000 000 000 000 000 002 180 864;
  • 9) 0.000 000 000 000 000 002 180 864 × 2 = 0 + 0.000 000 000 000 000 004 361 728;
  • 10) 0.000 000 000 000 000 004 361 728 × 2 = 0 + 0.000 000 000 000 000 008 723 456;
  • 11) 0.000 000 000 000 000 008 723 456 × 2 = 0 + 0.000 000 000 000 000 017 446 912;
  • 12) 0.000 000 000 000 000 017 446 912 × 2 = 0 + 0.000 000 000 000 000 034 893 824;
  • 13) 0.000 000 000 000 000 034 893 824 × 2 = 0 + 0.000 000 000 000 000 069 787 648;
  • 14) 0.000 000 000 000 000 069 787 648 × 2 = 0 + 0.000 000 000 000 000 139 575 296;
  • 15) 0.000 000 000 000 000 139 575 296 × 2 = 0 + 0.000 000 000 000 000 279 150 592;
  • 16) 0.000 000 000 000 000 279 150 592 × 2 = 0 + 0.000 000 000 000 000 558 301 184;
  • 17) 0.000 000 000 000 000 558 301 184 × 2 = 0 + 0.000 000 000 000 001 116 602 368;
  • 18) 0.000 000 000 000 001 116 602 368 × 2 = 0 + 0.000 000 000 000 002 233 204 736;
  • 19) 0.000 000 000 000 002 233 204 736 × 2 = 0 + 0.000 000 000 000 004 466 409 472;
  • 20) 0.000 000 000 000 004 466 409 472 × 2 = 0 + 0.000 000 000 000 008 932 818 944;
  • 21) 0.000 000 000 000 008 932 818 944 × 2 = 0 + 0.000 000 000 000 017 865 637 888;
  • 22) 0.000 000 000 000 017 865 637 888 × 2 = 0 + 0.000 000 000 000 035 731 275 776;
  • 23) 0.000 000 000 000 035 731 275 776 × 2 = 0 + 0.000 000 000 000 071 462 551 552;
  • 24) 0.000 000 000 000 071 462 551 552 × 2 = 0 + 0.000 000 000 000 142 925 103 104;
  • 25) 0.000 000 000 000 142 925 103 104 × 2 = 0 + 0.000 000 000 000 285 850 206 208;
  • 26) 0.000 000 000 000 285 850 206 208 × 2 = 0 + 0.000 000 000 000 571 700 412 416;
  • 27) 0.000 000 000 000 571 700 412 416 × 2 = 0 + 0.000 000 000 001 143 400 824 832;
  • 28) 0.000 000 000 001 143 400 824 832 × 2 = 0 + 0.000 000 000 002 286 801 649 664;
  • 29) 0.000 000 000 002 286 801 649 664 × 2 = 0 + 0.000 000 000 004 573 603 299 328;
  • 30) 0.000 000 000 004 573 603 299 328 × 2 = 0 + 0.000 000 000 009 147 206 598 656;
  • 31) 0.000 000 000 009 147 206 598 656 × 2 = 0 + 0.000 000 000 018 294 413 197 312;
  • 32) 0.000 000 000 018 294 413 197 312 × 2 = 0 + 0.000 000 000 036 588 826 394 624;
  • 33) 0.000 000 000 036 588 826 394 624 × 2 = 0 + 0.000 000 000 073 177 652 789 248;
  • 34) 0.000 000 000 073 177 652 789 248 × 2 = 0 + 0.000 000 000 146 355 305 578 496;
  • 35) 0.000 000 000 146 355 305 578 496 × 2 = 0 + 0.000 000 000 292 710 611 156 992;
  • 36) 0.000 000 000 292 710 611 156 992 × 2 = 0 + 0.000 000 000 585 421 222 313 984;
  • 37) 0.000 000 000 585 421 222 313 984 × 2 = 0 + 0.000 000 001 170 842 444 627 968;
  • 38) 0.000 000 001 170 842 444 627 968 × 2 = 0 + 0.000 000 002 341 684 889 255 936;
  • 39) 0.000 000 002 341 684 889 255 936 × 2 = 0 + 0.000 000 004 683 369 778 511 872;
  • 40) 0.000 000 004 683 369 778 511 872 × 2 = 0 + 0.000 000 009 366 739 557 023 744;
  • 41) 0.000 000 009 366 739 557 023 744 × 2 = 0 + 0.000 000 018 733 479 114 047 488;
  • 42) 0.000 000 018 733 479 114 047 488 × 2 = 0 + 0.000 000 037 466 958 228 094 976;
  • 43) 0.000 000 037 466 958 228 094 976 × 2 = 0 + 0.000 000 074 933 916 456 189 952;
  • 44) 0.000 000 074 933 916 456 189 952 × 2 = 0 + 0.000 000 149 867 832 912 379 904;
  • 45) 0.000 000 149 867 832 912 379 904 × 2 = 0 + 0.000 000 299 735 665 824 759 808;
  • 46) 0.000 000 299 735 665 824 759 808 × 2 = 0 + 0.000 000 599 471 331 649 519 616;
  • 47) 0.000 000 599 471 331 649 519 616 × 2 = 0 + 0.000 001 198 942 663 299 039 232;
  • 48) 0.000 001 198 942 663 299 039 232 × 2 = 0 + 0.000 002 397 885 326 598 078 464;
  • 49) 0.000 002 397 885 326 598 078 464 × 2 = 0 + 0.000 004 795 770 653 196 156 928;
  • 50) 0.000 004 795 770 653 196 156 928 × 2 = 0 + 0.000 009 591 541 306 392 313 856;
  • 51) 0.000 009 591 541 306 392 313 856 × 2 = 0 + 0.000 019 183 082 612 784 627 712;
  • 52) 0.000 019 183 082 612 784 627 712 × 2 = 0 + 0.000 038 366 165 225 569 255 424;
  • 53) 0.000 038 366 165 225 569 255 424 × 2 = 0 + 0.000 076 732 330 451 138 510 848;
  • 54) 0.000 076 732 330 451 138 510 848 × 2 = 0 + 0.000 153 464 660 902 277 021 696;
  • 55) 0.000 153 464 660 902 277 021 696 × 2 = 0 + 0.000 306 929 321 804 554 043 392;
  • 56) 0.000 306 929 321 804 554 043 392 × 2 = 0 + 0.000 613 858 643 609 108 086 784;
  • 57) 0.000 613 858 643 609 108 086 784 × 2 = 0 + 0.001 227 717 287 218 216 173 568;
  • 58) 0.001 227 717 287 218 216 173 568 × 2 = 0 + 0.002 455 434 574 436 432 347 136;
  • 59) 0.002 455 434 574 436 432 347 136 × 2 = 0 + 0.004 910 869 148 872 864 694 272;
  • 60) 0.004 910 869 148 872 864 694 272 × 2 = 0 + 0.009 821 738 297 745 729 388 544;
  • 61) 0.009 821 738 297 745 729 388 544 × 2 = 0 + 0.019 643 476 595 491 458 777 088;
  • 62) 0.019 643 476 595 491 458 777 088 × 2 = 0 + 0.039 286 953 190 982 917 554 176;
  • 63) 0.039 286 953 190 982 917 554 176 × 2 = 0 + 0.078 573 906 381 965 835 108 352;
  • 64) 0.078 573 906 381 965 835 108 352 × 2 = 0 + 0.157 147 812 763 931 670 216 704;
  • 65) 0.157 147 812 763 931 670 216 704 × 2 = 0 + 0.314 295 625 527 863 340 433 408;
  • 66) 0.314 295 625 527 863 340 433 408 × 2 = 0 + 0.628 591 251 055 726 680 866 816;
  • 67) 0.628 591 251 055 726 680 866 816 × 2 = 1 + 0.257 182 502 111 453 361 733 632;
  • 68) 0.257 182 502 111 453 361 733 632 × 2 = 0 + 0.514 365 004 222 906 723 467 264;
  • 69) 0.514 365 004 222 906 723 467 264 × 2 = 1 + 0.028 730 008 445 813 446 934 528;
  • 70) 0.028 730 008 445 813 446 934 528 × 2 = 0 + 0.057 460 016 891 626 893 869 056;
  • 71) 0.057 460 016 891 626 893 869 056 × 2 = 0 + 0.114 920 033 783 253 787 738 112;
  • 72) 0.114 920 033 783 253 787 738 112 × 2 = 0 + 0.229 840 067 566 507 575 476 224;
  • 73) 0.229 840 067 566 507 575 476 224 × 2 = 0 + 0.459 680 135 133 015 150 952 448;
  • 74) 0.459 680 135 133 015 150 952 448 × 2 = 0 + 0.919 360 270 266 030 301 904 896;
  • 75) 0.919 360 270 266 030 301 904 896 × 2 = 1 + 0.838 720 540 532 060 603 809 792;
  • 76) 0.838 720 540 532 060 603 809 792 × 2 = 1 + 0.677 441 081 064 121 207 619 584;
  • 77) 0.677 441 081 064 121 207 619 584 × 2 = 1 + 0.354 882 162 128 242 415 239 168;
  • 78) 0.354 882 162 128 242 415 239 168 × 2 = 0 + 0.709 764 324 256 484 830 478 336;
  • 79) 0.709 764 324 256 484 830 478 336 × 2 = 1 + 0.419 528 648 512 969 660 956 672;
  • 80) 0.419 528 648 512 969 660 956 672 × 2 = 0 + 0.839 057 297 025 939 321 913 344;
  • 81) 0.839 057 297 025 939 321 913 344 × 2 = 1 + 0.678 114 594 051 878 643 826 688;
  • 82) 0.678 114 594 051 878 643 826 688 × 2 = 1 + 0.356 229 188 103 757 287 653 376;
  • 83) 0.356 229 188 103 757 287 653 376 × 2 = 0 + 0.712 458 376 207 514 575 306 752;
  • 84) 0.712 458 376 207 514 575 306 752 × 2 = 1 + 0.424 916 752 415 029 150 613 504;
  • 85) 0.424 916 752 415 029 150 613 504 × 2 = 0 + 0.849 833 504 830 058 301 227 008;
  • 86) 0.849 833 504 830 058 301 227 008 × 2 = 1 + 0.699 667 009 660 116 602 454 016;
  • 87) 0.699 667 009 660 116 602 454 016 × 2 = 1 + 0.399 334 019 320 233 204 908 032;
  • 88) 0.399 334 019 320 233 204 908 032 × 2 = 0 + 0.798 668 038 640 466 409 816 064;
  • 89) 0.798 668 038 640 466 409 816 064 × 2 = 1 + 0.597 336 077 280 932 819 632 128;
  • 90) 0.597 336 077 280 932 819 632 128 × 2 = 1 + 0.194 672 154 561 865 639 264 256;
  • 91) 0.194 672 154 561 865 639 264 256 × 2 = 0 + 0.389 344 309 123 731 278 528 512;
  • 92) 0.389 344 309 123 731 278 528 512 × 2 = 0 + 0.778 688 618 247 462 557 057 024;
  • 93) 0.778 688 618 247 462 557 057 024 × 2 = 1 + 0.557 377 236 494 925 114 114 048;
  • 94) 0.557 377 236 494 925 114 114 048 × 2 = 1 + 0.114 754 472 989 850 228 228 096;
  • 95) 0.114 754 472 989 850 228 228 096 × 2 = 0 + 0.229 508 945 979 700 456 456 192;
  • 96) 0.229 508 945 979 700 456 456 192 × 2 = 0 + 0.459 017 891 959 400 912 912 384;
  • 97) 0.459 017 891 959 400 912 912 384 × 2 = 0 + 0.918 035 783 918 801 825 824 768;
  • 98) 0.918 035 783 918 801 825 824 768 × 2 = 1 + 0.836 071 567 837 603 651 649 536;
  • 99) 0.836 071 567 837 603 651 649 536 × 2 = 1 + 0.672 143 135 675 207 303 299 072;
  • 100) 0.672 143 135 675 207 303 299 072 × 2 = 1 + 0.344 286 271 350 414 606 598 144;
  • 101) 0.344 286 271 350 414 606 598 144 × 2 = 0 + 0.688 572 542 700 829 213 196 288;
  • 102) 0.688 572 542 700 829 213 196 288 × 2 = 1 + 0.377 145 085 401 658 426 392 576;
  • 103) 0.377 145 085 401 658 426 392 576 × 2 = 0 + 0.754 290 170 803 316 852 785 152;
  • 104) 0.754 290 170 803 316 852 785 152 × 2 = 1 + 0.508 580 341 606 633 705 570 304;
  • 105) 0.508 580 341 606 633 705 570 304 × 2 = 1 + 0.017 160 683 213 267 411 140 608;
  • 106) 0.017 160 683 213 267 411 140 608 × 2 = 0 + 0.034 321 366 426 534 822 281 216;
  • 107) 0.034 321 366 426 534 822 281 216 × 2 = 0 + 0.068 642 732 853 069 644 562 432;
  • 108) 0.068 642 732 853 069 644 562 432 × 2 = 0 + 0.137 285 465 706 139 289 124 864;
  • 109) 0.137 285 465 706 139 289 124 864 × 2 = 0 + 0.274 570 931 412 278 578 249 728;
  • 110) 0.274 570 931 412 278 578 249 728 × 2 = 0 + 0.549 141 862 824 557 156 499 456;
  • 111) 0.549 141 862 824 557 156 499 456 × 2 = 1 + 0.098 283 725 649 114 312 998 912;
  • 112) 0.098 283 725 649 114 312 998 912 × 2 = 0 + 0.196 567 451 298 228 625 997 824;
  • 113) 0.196 567 451 298 228 625 997 824 × 2 = 0 + 0.393 134 902 596 457 251 995 648;
  • 114) 0.393 134 902 596 457 251 995 648 × 2 = 0 + 0.786 269 805 192 914 503 991 296;
  • 115) 0.786 269 805 192 914 503 991 296 × 2 = 1 + 0.572 539 610 385 829 007 982 592;
  • 116) 0.572 539 610 385 829 007 982 592 × 2 = 1 + 0.145 079 220 771 658 015 965 184;
  • 117) 0.145 079 220 771 658 015 965 184 × 2 = 0 + 0.290 158 441 543 316 031 930 368;
  • 118) 0.290 158 441 543 316 031 930 368 × 2 = 0 + 0.580 316 883 086 632 063 860 736;
  • 119) 0.580 316 883 086 632 063 860 736 × 2 = 1 + 0.160 633 766 173 264 127 721 472;

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:


0.000 000 000 000 000 000 008 519(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1010 1101 0110 1100 1100 0111 0101 1000 0010 0011 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 519(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1010 1101 0110 1100 1100 0111 0101 1000 0010 0011 001(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:


0.000 000 000 000 000 000 008 519(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1010 1101 0110 1100 1100 0111 0101 1000 0010 0011 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1010 1101 0110 1100 1100 0111 0101 1000 0010 0011 001(2) × 20 =

1.0100 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001(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 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001


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 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001 =

0100 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001


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 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001


Decimal number 0.000 000 000 000 000 000 008 519 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0001 1101 0110 1011 0110 0110 0011 1010 1100 0001 0001 1001

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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