0.000 000 000 000 000 000 008 57 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 57(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 57(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 57.

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 57 × 2 = 0 + 0.000 000 000 000 000 000 017 14;
  • 2) 0.000 000 000 000 000 000 017 14 × 2 = 0 + 0.000 000 000 000 000 000 034 28;
  • 3) 0.000 000 000 000 000 000 034 28 × 2 = 0 + 0.000 000 000 000 000 000 068 56;
  • 4) 0.000 000 000 000 000 000 068 56 × 2 = 0 + 0.000 000 000 000 000 000 137 12;
  • 5) 0.000 000 000 000 000 000 137 12 × 2 = 0 + 0.000 000 000 000 000 000 274 24;
  • 6) 0.000 000 000 000 000 000 274 24 × 2 = 0 + 0.000 000 000 000 000 000 548 48;
  • 7) 0.000 000 000 000 000 000 548 48 × 2 = 0 + 0.000 000 000 000 000 001 096 96;
  • 8) 0.000 000 000 000 000 001 096 96 × 2 = 0 + 0.000 000 000 000 000 002 193 92;
  • 9) 0.000 000 000 000 000 002 193 92 × 2 = 0 + 0.000 000 000 000 000 004 387 84;
  • 10) 0.000 000 000 000 000 004 387 84 × 2 = 0 + 0.000 000 000 000 000 008 775 68;
  • 11) 0.000 000 000 000 000 008 775 68 × 2 = 0 + 0.000 000 000 000 000 017 551 36;
  • 12) 0.000 000 000 000 000 017 551 36 × 2 = 0 + 0.000 000 000 000 000 035 102 72;
  • 13) 0.000 000 000 000 000 035 102 72 × 2 = 0 + 0.000 000 000 000 000 070 205 44;
  • 14) 0.000 000 000 000 000 070 205 44 × 2 = 0 + 0.000 000 000 000 000 140 410 88;
  • 15) 0.000 000 000 000 000 140 410 88 × 2 = 0 + 0.000 000 000 000 000 280 821 76;
  • 16) 0.000 000 000 000 000 280 821 76 × 2 = 0 + 0.000 000 000 000 000 561 643 52;
  • 17) 0.000 000 000 000 000 561 643 52 × 2 = 0 + 0.000 000 000 000 001 123 287 04;
  • 18) 0.000 000 000 000 001 123 287 04 × 2 = 0 + 0.000 000 000 000 002 246 574 08;
  • 19) 0.000 000 000 000 002 246 574 08 × 2 = 0 + 0.000 000 000 000 004 493 148 16;
  • 20) 0.000 000 000 000 004 493 148 16 × 2 = 0 + 0.000 000 000 000 008 986 296 32;
  • 21) 0.000 000 000 000 008 986 296 32 × 2 = 0 + 0.000 000 000 000 017 972 592 64;
  • 22) 0.000 000 000 000 017 972 592 64 × 2 = 0 + 0.000 000 000 000 035 945 185 28;
  • 23) 0.000 000 000 000 035 945 185 28 × 2 = 0 + 0.000 000 000 000 071 890 370 56;
  • 24) 0.000 000 000 000 071 890 370 56 × 2 = 0 + 0.000 000 000 000 143 780 741 12;
  • 25) 0.000 000 000 000 143 780 741 12 × 2 = 0 + 0.000 000 000 000 287 561 482 24;
  • 26) 0.000 000 000 000 287 561 482 24 × 2 = 0 + 0.000 000 000 000 575 122 964 48;
  • 27) 0.000 000 000 000 575 122 964 48 × 2 = 0 + 0.000 000 000 001 150 245 928 96;
  • 28) 0.000 000 000 001 150 245 928 96 × 2 = 0 + 0.000 000 000 002 300 491 857 92;
  • 29) 0.000 000 000 002 300 491 857 92 × 2 = 0 + 0.000 000 000 004 600 983 715 84;
  • 30) 0.000 000 000 004 600 983 715 84 × 2 = 0 + 0.000 000 000 009 201 967 431 68;
  • 31) 0.000 000 000 009 201 967 431 68 × 2 = 0 + 0.000 000 000 018 403 934 863 36;
  • 32) 0.000 000 000 018 403 934 863 36 × 2 = 0 + 0.000 000 000 036 807 869 726 72;
  • 33) 0.000 000 000 036 807 869 726 72 × 2 = 0 + 0.000 000 000 073 615 739 453 44;
  • 34) 0.000 000 000 073 615 739 453 44 × 2 = 0 + 0.000 000 000 147 231 478 906 88;
  • 35) 0.000 000 000 147 231 478 906 88 × 2 = 0 + 0.000 000 000 294 462 957 813 76;
  • 36) 0.000 000 000 294 462 957 813 76 × 2 = 0 + 0.000 000 000 588 925 915 627 52;
  • 37) 0.000 000 000 588 925 915 627 52 × 2 = 0 + 0.000 000 001 177 851 831 255 04;
  • 38) 0.000 000 001 177 851 831 255 04 × 2 = 0 + 0.000 000 002 355 703 662 510 08;
  • 39) 0.000 000 002 355 703 662 510 08 × 2 = 0 + 0.000 000 004 711 407 325 020 16;
  • 40) 0.000 000 004 711 407 325 020 16 × 2 = 0 + 0.000 000 009 422 814 650 040 32;
  • 41) 0.000 000 009 422 814 650 040 32 × 2 = 0 + 0.000 000 018 845 629 300 080 64;
  • 42) 0.000 000 018 845 629 300 080 64 × 2 = 0 + 0.000 000 037 691 258 600 161 28;
  • 43) 0.000 000 037 691 258 600 161 28 × 2 = 0 + 0.000 000 075 382 517 200 322 56;
  • 44) 0.000 000 075 382 517 200 322 56 × 2 = 0 + 0.000 000 150 765 034 400 645 12;
  • 45) 0.000 000 150 765 034 400 645 12 × 2 = 0 + 0.000 000 301 530 068 801 290 24;
  • 46) 0.000 000 301 530 068 801 290 24 × 2 = 0 + 0.000 000 603 060 137 602 580 48;
  • 47) 0.000 000 603 060 137 602 580 48 × 2 = 0 + 0.000 001 206 120 275 205 160 96;
  • 48) 0.000 001 206 120 275 205 160 96 × 2 = 0 + 0.000 002 412 240 550 410 321 92;
  • 49) 0.000 002 412 240 550 410 321 92 × 2 = 0 + 0.000 004 824 481 100 820 643 84;
  • 50) 0.000 004 824 481 100 820 643 84 × 2 = 0 + 0.000 009 648 962 201 641 287 68;
  • 51) 0.000 009 648 962 201 641 287 68 × 2 = 0 + 0.000 019 297 924 403 282 575 36;
  • 52) 0.000 019 297 924 403 282 575 36 × 2 = 0 + 0.000 038 595 848 806 565 150 72;
  • 53) 0.000 038 595 848 806 565 150 72 × 2 = 0 + 0.000 077 191 697 613 130 301 44;
  • 54) 0.000 077 191 697 613 130 301 44 × 2 = 0 + 0.000 154 383 395 226 260 602 88;
  • 55) 0.000 154 383 395 226 260 602 88 × 2 = 0 + 0.000 308 766 790 452 521 205 76;
  • 56) 0.000 308 766 790 452 521 205 76 × 2 = 0 + 0.000 617 533 580 905 042 411 52;
  • 57) 0.000 617 533 580 905 042 411 52 × 2 = 0 + 0.001 235 067 161 810 084 823 04;
  • 58) 0.001 235 067 161 810 084 823 04 × 2 = 0 + 0.002 470 134 323 620 169 646 08;
  • 59) 0.002 470 134 323 620 169 646 08 × 2 = 0 + 0.004 940 268 647 240 339 292 16;
  • 60) 0.004 940 268 647 240 339 292 16 × 2 = 0 + 0.009 880 537 294 480 678 584 32;
  • 61) 0.009 880 537 294 480 678 584 32 × 2 = 0 + 0.019 761 074 588 961 357 168 64;
  • 62) 0.019 761 074 588 961 357 168 64 × 2 = 0 + 0.039 522 149 177 922 714 337 28;
  • 63) 0.039 522 149 177 922 714 337 28 × 2 = 0 + 0.079 044 298 355 845 428 674 56;
  • 64) 0.079 044 298 355 845 428 674 56 × 2 = 0 + 0.158 088 596 711 690 857 349 12;
  • 65) 0.158 088 596 711 690 857 349 12 × 2 = 0 + 0.316 177 193 423 381 714 698 24;
  • 66) 0.316 177 193 423 381 714 698 24 × 2 = 0 + 0.632 354 386 846 763 429 396 48;
  • 67) 0.632 354 386 846 763 429 396 48 × 2 = 1 + 0.264 708 773 693 526 858 792 96;
  • 68) 0.264 708 773 693 526 858 792 96 × 2 = 0 + 0.529 417 547 387 053 717 585 92;
  • 69) 0.529 417 547 387 053 717 585 92 × 2 = 1 + 0.058 835 094 774 107 435 171 84;
  • 70) 0.058 835 094 774 107 435 171 84 × 2 = 0 + 0.117 670 189 548 214 870 343 68;
  • 71) 0.117 670 189 548 214 870 343 68 × 2 = 0 + 0.235 340 379 096 429 740 687 36;
  • 72) 0.235 340 379 096 429 740 687 36 × 2 = 0 + 0.470 680 758 192 859 481 374 72;
  • 73) 0.470 680 758 192 859 481 374 72 × 2 = 0 + 0.941 361 516 385 718 962 749 44;
  • 74) 0.941 361 516 385 718 962 749 44 × 2 = 1 + 0.882 723 032 771 437 925 498 88;
  • 75) 0.882 723 032 771 437 925 498 88 × 2 = 1 + 0.765 446 065 542 875 850 997 76;
  • 76) 0.765 446 065 542 875 850 997 76 × 2 = 1 + 0.530 892 131 085 751 701 995 52;
  • 77) 0.530 892 131 085 751 701 995 52 × 2 = 1 + 0.061 784 262 171 503 403 991 04;
  • 78) 0.061 784 262 171 503 403 991 04 × 2 = 0 + 0.123 568 524 343 006 807 982 08;
  • 79) 0.123 568 524 343 006 807 982 08 × 2 = 0 + 0.247 137 048 686 013 615 964 16;
  • 80) 0.247 137 048 686 013 615 964 16 × 2 = 0 + 0.494 274 097 372 027 231 928 32;
  • 81) 0.494 274 097 372 027 231 928 32 × 2 = 0 + 0.988 548 194 744 054 463 856 64;
  • 82) 0.988 548 194 744 054 463 856 64 × 2 = 1 + 0.977 096 389 488 108 927 713 28;
  • 83) 0.977 096 389 488 108 927 713 28 × 2 = 1 + 0.954 192 778 976 217 855 426 56;
  • 84) 0.954 192 778 976 217 855 426 56 × 2 = 1 + 0.908 385 557 952 435 710 853 12;
  • 85) 0.908 385 557 952 435 710 853 12 × 2 = 1 + 0.816 771 115 904 871 421 706 24;
  • 86) 0.816 771 115 904 871 421 706 24 × 2 = 1 + 0.633 542 231 809 742 843 412 48;
  • 87) 0.633 542 231 809 742 843 412 48 × 2 = 1 + 0.267 084 463 619 485 686 824 96;
  • 88) 0.267 084 463 619 485 686 824 96 × 2 = 0 + 0.534 168 927 238 971 373 649 92;
  • 89) 0.534 168 927 238 971 373 649 92 × 2 = 1 + 0.068 337 854 477 942 747 299 84;
  • 90) 0.068 337 854 477 942 747 299 84 × 2 = 0 + 0.136 675 708 955 885 494 599 68;
  • 91) 0.136 675 708 955 885 494 599 68 × 2 = 0 + 0.273 351 417 911 770 989 199 36;
  • 92) 0.273 351 417 911 770 989 199 36 × 2 = 0 + 0.546 702 835 823 541 978 398 72;
  • 93) 0.546 702 835 823 541 978 398 72 × 2 = 1 + 0.093 405 671 647 083 956 797 44;
  • 94) 0.093 405 671 647 083 956 797 44 × 2 = 0 + 0.186 811 343 294 167 913 594 88;
  • 95) 0.186 811 343 294 167 913 594 88 × 2 = 0 + 0.373 622 686 588 335 827 189 76;
  • 96) 0.373 622 686 588 335 827 189 76 × 2 = 0 + 0.747 245 373 176 671 654 379 52;
  • 97) 0.747 245 373 176 671 654 379 52 × 2 = 1 + 0.494 490 746 353 343 308 759 04;
  • 98) 0.494 490 746 353 343 308 759 04 × 2 = 0 + 0.988 981 492 706 686 617 518 08;
  • 99) 0.988 981 492 706 686 617 518 08 × 2 = 1 + 0.977 962 985 413 373 235 036 16;
  • 100) 0.977 962 985 413 373 235 036 16 × 2 = 1 + 0.955 925 970 826 746 470 072 32;
  • 101) 0.955 925 970 826 746 470 072 32 × 2 = 1 + 0.911 851 941 653 492 940 144 64;
  • 102) 0.911 851 941 653 492 940 144 64 × 2 = 1 + 0.823 703 883 306 985 880 289 28;
  • 103) 0.823 703 883 306 985 880 289 28 × 2 = 1 + 0.647 407 766 613 971 760 578 56;
  • 104) 0.647 407 766 613 971 760 578 56 × 2 = 1 + 0.294 815 533 227 943 521 157 12;
  • 105) 0.294 815 533 227 943 521 157 12 × 2 = 0 + 0.589 631 066 455 887 042 314 24;
  • 106) 0.589 631 066 455 887 042 314 24 × 2 = 1 + 0.179 262 132 911 774 084 628 48;
  • 107) 0.179 262 132 911 774 084 628 48 × 2 = 0 + 0.358 524 265 823 548 169 256 96;
  • 108) 0.358 524 265 823 548 169 256 96 × 2 = 0 + 0.717 048 531 647 096 338 513 92;
  • 109) 0.717 048 531 647 096 338 513 92 × 2 = 1 + 0.434 097 063 294 192 677 027 84;
  • 110) 0.434 097 063 294 192 677 027 84 × 2 = 0 + 0.868 194 126 588 385 354 055 68;
  • 111) 0.868 194 126 588 385 354 055 68 × 2 = 1 + 0.736 388 253 176 770 708 111 36;
  • 112) 0.736 388 253 176 770 708 111 36 × 2 = 1 + 0.472 776 506 353 541 416 222 72;
  • 113) 0.472 776 506 353 541 416 222 72 × 2 = 0 + 0.945 553 012 707 082 832 445 44;
  • 114) 0.945 553 012 707 082 832 445 44 × 2 = 1 + 0.891 106 025 414 165 664 890 88;
  • 115) 0.891 106 025 414 165 664 890 88 × 2 = 1 + 0.782 212 050 828 331 329 781 76;
  • 116) 0.782 212 050 828 331 329 781 76 × 2 = 1 + 0.564 424 101 656 662 659 563 52;
  • 117) 0.564 424 101 656 662 659 563 52 × 2 = 1 + 0.128 848 203 313 325 319 127 04;
  • 118) 0.128 848 203 313 325 319 127 04 × 2 = 0 + 0.257 696 406 626 650 638 254 08;
  • 119) 0.257 696 406 626 650 638 254 08 × 2 = 0 + 0.515 392 813 253 301 276 508 16;

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 57(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 1000 0111 1110 1000 1000 1011 1111 0100 1011 0111 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 57(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 1000 0111 1110 1000 1000 1011 1111 0100 1011 0111 100(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 57(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 1000 0111 1110 1000 1000 1011 1111 0100 1011 0111 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 1000 0111 1110 1000 1000 1011 1111 0100 1011 0111 100(2) × 20 =

1.0100 0011 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100(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 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100


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 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100 =

0100 0011 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100


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 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100


Decimal number 0.000 000 000 000 000 000 008 57 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 1100 0011 1111 0100 0100 0101 1111 1010 0101 1011 1100

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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