0.000 000 000 000 000 032 487 359 817 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 032 487 359 817(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
0.000 000 000 000 000 032 487 359 817(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: 0.000 000 000 000 000 032 487 359 817.

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) 0.000 000 000 000 000 032 487 359 817 × 2 = 0 + 0.000 000 000 000 000 064 974 719 634;
  • 2) 0.000 000 000 000 000 064 974 719 634 × 2 = 0 + 0.000 000 000 000 000 129 949 439 268;
  • 3) 0.000 000 000 000 000 129 949 439 268 × 2 = 0 + 0.000 000 000 000 000 259 898 878 536;
  • 4) 0.000 000 000 000 000 259 898 878 536 × 2 = 0 + 0.000 000 000 000 000 519 797 757 072;
  • 5) 0.000 000 000 000 000 519 797 757 072 × 2 = 0 + 0.000 000 000 000 001 039 595 514 144;
  • 6) 0.000 000 000 000 001 039 595 514 144 × 2 = 0 + 0.000 000 000 000 002 079 191 028 288;
  • 7) 0.000 000 000 000 002 079 191 028 288 × 2 = 0 + 0.000 000 000 000 004 158 382 056 576;
  • 8) 0.000 000 000 000 004 158 382 056 576 × 2 = 0 + 0.000 000 000 000 008 316 764 113 152;
  • 9) 0.000 000 000 000 008 316 764 113 152 × 2 = 0 + 0.000 000 000 000 016 633 528 226 304;
  • 10) 0.000 000 000 000 016 633 528 226 304 × 2 = 0 + 0.000 000 000 000 033 267 056 452 608;
  • 11) 0.000 000 000 000 033 267 056 452 608 × 2 = 0 + 0.000 000 000 000 066 534 112 905 216;
  • 12) 0.000 000 000 000 066 534 112 905 216 × 2 = 0 + 0.000 000 000 000 133 068 225 810 432;
  • 13) 0.000 000 000 000 133 068 225 810 432 × 2 = 0 + 0.000 000 000 000 266 136 451 620 864;
  • 14) 0.000 000 000 000 266 136 451 620 864 × 2 = 0 + 0.000 000 000 000 532 272 903 241 728;
  • 15) 0.000 000 000 000 532 272 903 241 728 × 2 = 0 + 0.000 000 000 001 064 545 806 483 456;
  • 16) 0.000 000 000 001 064 545 806 483 456 × 2 = 0 + 0.000 000 000 002 129 091 612 966 912;
  • 17) 0.000 000 000 002 129 091 612 966 912 × 2 = 0 + 0.000 000 000 004 258 183 225 933 824;
  • 18) 0.000 000 000 004 258 183 225 933 824 × 2 = 0 + 0.000 000 000 008 516 366 451 867 648;
  • 19) 0.000 000 000 008 516 366 451 867 648 × 2 = 0 + 0.000 000 000 017 032 732 903 735 296;
  • 20) 0.000 000 000 017 032 732 903 735 296 × 2 = 0 + 0.000 000 000 034 065 465 807 470 592;
  • 21) 0.000 000 000 034 065 465 807 470 592 × 2 = 0 + 0.000 000 000 068 130 931 614 941 184;
  • 22) 0.000 000 000 068 130 931 614 941 184 × 2 = 0 + 0.000 000 000 136 261 863 229 882 368;
  • 23) 0.000 000 000 136 261 863 229 882 368 × 2 = 0 + 0.000 000 000 272 523 726 459 764 736;
  • 24) 0.000 000 000 272 523 726 459 764 736 × 2 = 0 + 0.000 000 000 545 047 452 919 529 472;
  • 25) 0.000 000 000 545 047 452 919 529 472 × 2 = 0 + 0.000 000 001 090 094 905 839 058 944;
  • 26) 0.000 000 001 090 094 905 839 058 944 × 2 = 0 + 0.000 000 002 180 189 811 678 117 888;
  • 27) 0.000 000 002 180 189 811 678 117 888 × 2 = 0 + 0.000 000 004 360 379 623 356 235 776;
  • 28) 0.000 000 004 360 379 623 356 235 776 × 2 = 0 + 0.000 000 008 720 759 246 712 471 552;
  • 29) 0.000 000 008 720 759 246 712 471 552 × 2 = 0 + 0.000 000 017 441 518 493 424 943 104;
  • 30) 0.000 000 017 441 518 493 424 943 104 × 2 = 0 + 0.000 000 034 883 036 986 849 886 208;
  • 31) 0.000 000 034 883 036 986 849 886 208 × 2 = 0 + 0.000 000 069 766 073 973 699 772 416;
  • 32) 0.000 000 069 766 073 973 699 772 416 × 2 = 0 + 0.000 000 139 532 147 947 399 544 832;
  • 33) 0.000 000 139 532 147 947 399 544 832 × 2 = 0 + 0.000 000 279 064 295 894 799 089 664;
  • 34) 0.000 000 279 064 295 894 799 089 664 × 2 = 0 + 0.000 000 558 128 591 789 598 179 328;
  • 35) 0.000 000 558 128 591 789 598 179 328 × 2 = 0 + 0.000 001 116 257 183 579 196 358 656;
  • 36) 0.000 001 116 257 183 579 196 358 656 × 2 = 0 + 0.000 002 232 514 367 158 392 717 312;
  • 37) 0.000 002 232 514 367 158 392 717 312 × 2 = 0 + 0.000 004 465 028 734 316 785 434 624;
  • 38) 0.000 004 465 028 734 316 785 434 624 × 2 = 0 + 0.000 008 930 057 468 633 570 869 248;
  • 39) 0.000 008 930 057 468 633 570 869 248 × 2 = 0 + 0.000 017 860 114 937 267 141 738 496;
  • 40) 0.000 017 860 114 937 267 141 738 496 × 2 = 0 + 0.000 035 720 229 874 534 283 476 992;
  • 41) 0.000 035 720 229 874 534 283 476 992 × 2 = 0 + 0.000 071 440 459 749 068 566 953 984;
  • 42) 0.000 071 440 459 749 068 566 953 984 × 2 = 0 + 0.000 142 880 919 498 137 133 907 968;
  • 43) 0.000 142 880 919 498 137 133 907 968 × 2 = 0 + 0.000 285 761 838 996 274 267 815 936;
  • 44) 0.000 285 761 838 996 274 267 815 936 × 2 = 0 + 0.000 571 523 677 992 548 535 631 872;
  • 45) 0.000 571 523 677 992 548 535 631 872 × 2 = 0 + 0.001 143 047 355 985 097 071 263 744;
  • 46) 0.001 143 047 355 985 097 071 263 744 × 2 = 0 + 0.002 286 094 711 970 194 142 527 488;
  • 47) 0.002 286 094 711 970 194 142 527 488 × 2 = 0 + 0.004 572 189 423 940 388 285 054 976;
  • 48) 0.004 572 189 423 940 388 285 054 976 × 2 = 0 + 0.009 144 378 847 880 776 570 109 952;
  • 49) 0.009 144 378 847 880 776 570 109 952 × 2 = 0 + 0.018 288 757 695 761 553 140 219 904;
  • 50) 0.018 288 757 695 761 553 140 219 904 × 2 = 0 + 0.036 577 515 391 523 106 280 439 808;
  • 51) 0.036 577 515 391 523 106 280 439 808 × 2 = 0 + 0.073 155 030 783 046 212 560 879 616;
  • 52) 0.073 155 030 783 046 212 560 879 616 × 2 = 0 + 0.146 310 061 566 092 425 121 759 232;
  • 53) 0.146 310 061 566 092 425 121 759 232 × 2 = 0 + 0.292 620 123 132 184 850 243 518 464;
  • 54) 0.292 620 123 132 184 850 243 518 464 × 2 = 0 + 0.585 240 246 264 369 700 487 036 928;
  • 55) 0.585 240 246 264 369 700 487 036 928 × 2 = 1 + 0.170 480 492 528 739 400 974 073 856;
  • 56) 0.170 480 492 528 739 400 974 073 856 × 2 = 0 + 0.340 960 985 057 478 801 948 147 712;
  • 57) 0.340 960 985 057 478 801 948 147 712 × 2 = 0 + 0.681 921 970 114 957 603 896 295 424;
  • 58) 0.681 921 970 114 957 603 896 295 424 × 2 = 1 + 0.363 843 940 229 915 207 792 590 848;
  • 59) 0.363 843 940 229 915 207 792 590 848 × 2 = 0 + 0.727 687 880 459 830 415 585 181 696;
  • 60) 0.727 687 880 459 830 415 585 181 696 × 2 = 1 + 0.455 375 760 919 660 831 170 363 392;
  • 61) 0.455 375 760 919 660 831 170 363 392 × 2 = 0 + 0.910 751 521 839 321 662 340 726 784;
  • 62) 0.910 751 521 839 321 662 340 726 784 × 2 = 1 + 0.821 503 043 678 643 324 681 453 568;
  • 63) 0.821 503 043 678 643 324 681 453 568 × 2 = 1 + 0.643 006 087 357 286 649 362 907 136;
  • 64) 0.643 006 087 357 286 649 362 907 136 × 2 = 1 + 0.286 012 174 714 573 298 725 814 272;
  • 65) 0.286 012 174 714 573 298 725 814 272 × 2 = 0 + 0.572 024 349 429 146 597 451 628 544;
  • 66) 0.572 024 349 429 146 597 451 628 544 × 2 = 1 + 0.144 048 698 858 293 194 903 257 088;
  • 67) 0.144 048 698 858 293 194 903 257 088 × 2 = 0 + 0.288 097 397 716 586 389 806 514 176;
  • 68) 0.288 097 397 716 586 389 806 514 176 × 2 = 0 + 0.576 194 795 433 172 779 613 028 352;
  • 69) 0.576 194 795 433 172 779 613 028 352 × 2 = 1 + 0.152 389 590 866 345 559 226 056 704;
  • 70) 0.152 389 590 866 345 559 226 056 704 × 2 = 0 + 0.304 779 181 732 691 118 452 113 408;
  • 71) 0.304 779 181 732 691 118 452 113 408 × 2 = 0 + 0.609 558 363 465 382 236 904 226 816;
  • 72) 0.609 558 363 465 382 236 904 226 816 × 2 = 1 + 0.219 116 726 930 764 473 808 453 632;
  • 73) 0.219 116 726 930 764 473 808 453 632 × 2 = 0 + 0.438 233 453 861 528 947 616 907 264;
  • 74) 0.438 233 453 861 528 947 616 907 264 × 2 = 0 + 0.876 466 907 723 057 895 233 814 528;
  • 75) 0.876 466 907 723 057 895 233 814 528 × 2 = 1 + 0.752 933 815 446 115 790 467 629 056;
  • 76) 0.752 933 815 446 115 790 467 629 056 × 2 = 1 + 0.505 867 630 892 231 580 935 258 112;
  • 77) 0.505 867 630 892 231 580 935 258 112 × 2 = 1 + 0.011 735 261 784 463 161 870 516 224;
  • 78) 0.011 735 261 784 463 161 870 516 224 × 2 = 0 + 0.023 470 523 568 926 323 741 032 448;
  • 79) 0.023 470 523 568 926 323 741 032 448 × 2 = 0 + 0.046 941 047 137 852 647 482 064 896;
  • 80) 0.046 941 047 137 852 647 482 064 896 × 2 = 0 + 0.093 882 094 275 705 294 964 129 792;
  • 81) 0.093 882 094 275 705 294 964 129 792 × 2 = 0 + 0.187 764 188 551 410 589 928 259 584;
  • 82) 0.187 764 188 551 410 589 928 259 584 × 2 = 0 + 0.375 528 377 102 821 179 856 519 168;
  • 83) 0.375 528 377 102 821 179 856 519 168 × 2 = 0 + 0.751 056 754 205 642 359 713 038 336;
  • 84) 0.751 056 754 205 642 359 713 038 336 × 2 = 1 + 0.502 113 508 411 284 719 426 076 672;
  • 85) 0.502 113 508 411 284 719 426 076 672 × 2 = 1 + 0.004 227 016 822 569 438 852 153 344;
  • 86) 0.004 227 016 822 569 438 852 153 344 × 2 = 0 + 0.008 454 033 645 138 877 704 306 688;
  • 87) 0.008 454 033 645 138 877 704 306 688 × 2 = 0 + 0.016 908 067 290 277 755 408 613 376;
  • 88) 0.016 908 067 290 277 755 408 613 376 × 2 = 0 + 0.033 816 134 580 555 510 817 226 752;
  • 89) 0.033 816 134 580 555 510 817 226 752 × 2 = 0 + 0.067 632 269 161 111 021 634 453 504;
  • 90) 0.067 632 269 161 111 021 634 453 504 × 2 = 0 + 0.135 264 538 322 222 043 268 907 008;
  • 91) 0.135 264 538 322 222 043 268 907 008 × 2 = 0 + 0.270 529 076 644 444 086 537 814 016;
  • 92) 0.270 529 076 644 444 086 537 814 016 × 2 = 0 + 0.541 058 153 288 888 173 075 628 032;
  • 93) 0.541 058 153 288 888 173 075 628 032 × 2 = 1 + 0.082 116 306 577 776 346 151 256 064;
  • 94) 0.082 116 306 577 776 346 151 256 064 × 2 = 0 + 0.164 232 613 155 552 692 302 512 128;
  • 95) 0.164 232 613 155 552 692 302 512 128 × 2 = 0 + 0.328 465 226 311 105 384 605 024 256;
  • 96) 0.328 465 226 311 105 384 605 024 256 × 2 = 0 + 0.656 930 452 622 210 769 210 048 512;
  • 97) 0.656 930 452 622 210 769 210 048 512 × 2 = 1 + 0.313 860 905 244 421 538 420 097 024;
  • 98) 0.313 860 905 244 421 538 420 097 024 × 2 = 0 + 0.627 721 810 488 843 076 840 194 048;
  • 99) 0.627 721 810 488 843 076 840 194 048 × 2 = 1 + 0.255 443 620 977 686 153 680 388 096;
  • 100) 0.255 443 620 977 686 153 680 388 096 × 2 = 0 + 0.510 887 241 955 372 307 360 776 192;
  • 101) 0.510 887 241 955 372 307 360 776 192 × 2 = 1 + 0.021 774 483 910 744 614 721 552 384;
  • 102) 0.021 774 483 910 744 614 721 552 384 × 2 = 0 + 0.043 548 967 821 489 229 443 104 768;
  • 103) 0.043 548 967 821 489 229 443 104 768 × 2 = 0 + 0.087 097 935 642 978 458 886 209 536;
  • 104) 0.087 097 935 642 978 458 886 209 536 × 2 = 0 + 0.174 195 871 285 956 917 772 419 072;
  • 105) 0.174 195 871 285 956 917 772 419 072 × 2 = 0 + 0.348 391 742 571 913 835 544 838 144;
  • 106) 0.348 391 742 571 913 835 544 838 144 × 2 = 0 + 0.696 783 485 143 827 671 089 676 288;
  • 107) 0.696 783 485 143 827 671 089 676 288 × 2 = 1 + 0.393 566 970 287 655 342 179 352 576;

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:


0.000 000 000 000 000 032 487 359 817(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0001 1000 0000 1000 1010 1000 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 032 487 359 817(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0001 1000 0000 1000 1010 1000 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 55 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 032 487 359 817(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0001 1000 0000 1000 1010 1000 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0001 1000 0000 1000 1010 1000 001(2) × 20 =

1.0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001(2) × 2-55


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): -55

Mantissa (not normalized):
1.0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-55 + 2(11-1) - 1 =

(-55 + 1 023)(10) =

968(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;
  • 968 ÷ 2 = 484 + 0;
  • 484 ÷ 2 = 242 + 0;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

968(10) =

011 1100 1000(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. 0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001 =

0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001


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 1100 1000

Mantissa (52 bits) =
0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001


Decimal number 0.000 000 000 000 000 032 487 359 817 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 1000 - 0010 1011 1010 0100 1001 1100 0000 1100 0000 0100 0101 0100 0001

» Convert decimal 0.000 000 000 000 000 032 487 359 776 to 64 bit double precision IEEE 754 binary floating point representation standard

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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