64bit IEEE 754: Decimal -> Double Precision Floating Point Binary: -0.000 000 000 000 014 391 905 333 7 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -0.000 000 000 000 014 391 905 333 7(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 000 000 000 014 391 905 333 7| = 0.000 000 000 000 014 391 905 333 7

2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 014 391 905 333 7.

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) 0.000 000 000 000 014 391 905 333 7 × 2 = 0 + 0.000 000 000 000 028 783 810 667 4;
  • 2) 0.000 000 000 000 028 783 810 667 4 × 2 = 0 + 0.000 000 000 000 057 567 621 334 8;
  • 3) 0.000 000 000 000 057 567 621 334 8 × 2 = 0 + 0.000 000 000 000 115 135 242 669 6;
  • 4) 0.000 000 000 000 115 135 242 669 6 × 2 = 0 + 0.000 000 000 000 230 270 485 339 2;
  • 5) 0.000 000 000 000 230 270 485 339 2 × 2 = 0 + 0.000 000 000 000 460 540 970 678 4;
  • 6) 0.000 000 000 000 460 540 970 678 4 × 2 = 0 + 0.000 000 000 000 921 081 941 356 8;
  • 7) 0.000 000 000 000 921 081 941 356 8 × 2 = 0 + 0.000 000 000 001 842 163 882 713 6;
  • 8) 0.000 000 000 001 842 163 882 713 6 × 2 = 0 + 0.000 000 000 003 684 327 765 427 2;
  • 9) 0.000 000 000 003 684 327 765 427 2 × 2 = 0 + 0.000 000 000 007 368 655 530 854 4;
  • 10) 0.000 000 000 007 368 655 530 854 4 × 2 = 0 + 0.000 000 000 014 737 311 061 708 8;
  • 11) 0.000 000 000 014 737 311 061 708 8 × 2 = 0 + 0.000 000 000 029 474 622 123 417 6;
  • 12) 0.000 000 000 029 474 622 123 417 6 × 2 = 0 + 0.000 000 000 058 949 244 246 835 2;
  • 13) 0.000 000 000 058 949 244 246 835 2 × 2 = 0 + 0.000 000 000 117 898 488 493 670 4;
  • 14) 0.000 000 000 117 898 488 493 670 4 × 2 = 0 + 0.000 000 000 235 796 976 987 340 8;
  • 15) 0.000 000 000 235 796 976 987 340 8 × 2 = 0 + 0.000 000 000 471 593 953 974 681 6;
  • 16) 0.000 000 000 471 593 953 974 681 6 × 2 = 0 + 0.000 000 000 943 187 907 949 363 2;
  • 17) 0.000 000 000 943 187 907 949 363 2 × 2 = 0 + 0.000 000 001 886 375 815 898 726 4;
  • 18) 0.000 000 001 886 375 815 898 726 4 × 2 = 0 + 0.000 000 003 772 751 631 797 452 8;
  • 19) 0.000 000 003 772 751 631 797 452 8 × 2 = 0 + 0.000 000 007 545 503 263 594 905 6;
  • 20) 0.000 000 007 545 503 263 594 905 6 × 2 = 0 + 0.000 000 015 091 006 527 189 811 2;
  • 21) 0.000 000 015 091 006 527 189 811 2 × 2 = 0 + 0.000 000 030 182 013 054 379 622 4;
  • 22) 0.000 000 030 182 013 054 379 622 4 × 2 = 0 + 0.000 000 060 364 026 108 759 244 8;
  • 23) 0.000 000 060 364 026 108 759 244 8 × 2 = 0 + 0.000 000 120 728 052 217 518 489 6;
  • 24) 0.000 000 120 728 052 217 518 489 6 × 2 = 0 + 0.000 000 241 456 104 435 036 979 2;
  • 25) 0.000 000 241 456 104 435 036 979 2 × 2 = 0 + 0.000 000 482 912 208 870 073 958 4;
  • 26) 0.000 000 482 912 208 870 073 958 4 × 2 = 0 + 0.000 000 965 824 417 740 147 916 8;
  • 27) 0.000 000 965 824 417 740 147 916 8 × 2 = 0 + 0.000 001 931 648 835 480 295 833 6;
  • 28) 0.000 001 931 648 835 480 295 833 6 × 2 = 0 + 0.000 003 863 297 670 960 591 667 2;
  • 29) 0.000 003 863 297 670 960 591 667 2 × 2 = 0 + 0.000 007 726 595 341 921 183 334 4;
  • 30) 0.000 007 726 595 341 921 183 334 4 × 2 = 0 + 0.000 015 453 190 683 842 366 668 8;
  • 31) 0.000 015 453 190 683 842 366 668 8 × 2 = 0 + 0.000 030 906 381 367 684 733 337 6;
  • 32) 0.000 030 906 381 367 684 733 337 6 × 2 = 0 + 0.000 061 812 762 735 369 466 675 2;
  • 33) 0.000 061 812 762 735 369 466 675 2 × 2 = 0 + 0.000 123 625 525 470 738 933 350 4;
  • 34) 0.000 123 625 525 470 738 933 350 4 × 2 = 0 + 0.000 247 251 050 941 477 866 700 8;
  • 35) 0.000 247 251 050 941 477 866 700 8 × 2 = 0 + 0.000 494 502 101 882 955 733 401 6;
  • 36) 0.000 494 502 101 882 955 733 401 6 × 2 = 0 + 0.000 989 004 203 765 911 466 803 2;
  • 37) 0.000 989 004 203 765 911 466 803 2 × 2 = 0 + 0.001 978 008 407 531 822 933 606 4;
  • 38) 0.001 978 008 407 531 822 933 606 4 × 2 = 0 + 0.003 956 016 815 063 645 867 212 8;
  • 39) 0.003 956 016 815 063 645 867 212 8 × 2 = 0 + 0.007 912 033 630 127 291 734 425 6;
  • 40) 0.007 912 033 630 127 291 734 425 6 × 2 = 0 + 0.015 824 067 260 254 583 468 851 2;
  • 41) 0.015 824 067 260 254 583 468 851 2 × 2 = 0 + 0.031 648 134 520 509 166 937 702 4;
  • 42) 0.031 648 134 520 509 166 937 702 4 × 2 = 0 + 0.063 296 269 041 018 333 875 404 8;
  • 43) 0.063 296 269 041 018 333 875 404 8 × 2 = 0 + 0.126 592 538 082 036 667 750 809 6;
  • 44) 0.126 592 538 082 036 667 750 809 6 × 2 = 0 + 0.253 185 076 164 073 335 501 619 2;
  • 45) 0.253 185 076 164 073 335 501 619 2 × 2 = 0 + 0.506 370 152 328 146 671 003 238 4;
  • 46) 0.506 370 152 328 146 671 003 238 4 × 2 = 1 + 0.012 740 304 656 293 342 006 476 8;
  • 47) 0.012 740 304 656 293 342 006 476 8 × 2 = 0 + 0.025 480 609 312 586 684 012 953 6;
  • 48) 0.025 480 609 312 586 684 012 953 6 × 2 = 0 + 0.050 961 218 625 173 368 025 907 2;
  • 49) 0.050 961 218 625 173 368 025 907 2 × 2 = 0 + 0.101 922 437 250 346 736 051 814 4;
  • 50) 0.101 922 437 250 346 736 051 814 4 × 2 = 0 + 0.203 844 874 500 693 472 103 628 8;
  • 51) 0.203 844 874 500 693 472 103 628 8 × 2 = 0 + 0.407 689 749 001 386 944 207 257 6;
  • 52) 0.407 689 749 001 386 944 207 257 6 × 2 = 0 + 0.815 379 498 002 773 888 414 515 2;
  • 53) 0.815 379 498 002 773 888 414 515 2 × 2 = 1 + 0.630 758 996 005 547 776 829 030 4;
  • 54) 0.630 758 996 005 547 776 829 030 4 × 2 = 1 + 0.261 517 992 011 095 553 658 060 8;
  • 55) 0.261 517 992 011 095 553 658 060 8 × 2 = 0 + 0.523 035 984 022 191 107 316 121 6;
  • 56) 0.523 035 984 022 191 107 316 121 6 × 2 = 1 + 0.046 071 968 044 382 214 632 243 2;
  • 57) 0.046 071 968 044 382 214 632 243 2 × 2 = 0 + 0.092 143 936 088 764 429 264 486 4;
  • 58) 0.092 143 936 088 764 429 264 486 4 × 2 = 0 + 0.184 287 872 177 528 858 528 972 8;
  • 59) 0.184 287 872 177 528 858 528 972 8 × 2 = 0 + 0.368 575 744 355 057 717 057 945 6;
  • 60) 0.368 575 744 355 057 717 057 945 6 × 2 = 0 + 0.737 151 488 710 115 434 115 891 2;
  • 61) 0.737 151 488 710 115 434 115 891 2 × 2 = 1 + 0.474 302 977 420 230 868 231 782 4;
  • 62) 0.474 302 977 420 230 868 231 782 4 × 2 = 0 + 0.948 605 954 840 461 736 463 564 8;
  • 63) 0.948 605 954 840 461 736 463 564 8 × 2 = 1 + 0.897 211 909 680 923 472 927 129 6;
  • 64) 0.897 211 909 680 923 472 927 129 6 × 2 = 1 + 0.794 423 819 361 846 945 854 259 2;
  • 65) 0.794 423 819 361 846 945 854 259 2 × 2 = 1 + 0.588 847 638 723 693 891 708 518 4;
  • 66) 0.588 847 638 723 693 891 708 518 4 × 2 = 1 + 0.177 695 277 447 387 783 417 036 8;
  • 67) 0.177 695 277 447 387 783 417 036 8 × 2 = 0 + 0.355 390 554 894 775 566 834 073 6;
  • 68) 0.355 390 554 894 775 566 834 073 6 × 2 = 0 + 0.710 781 109 789 551 133 668 147 2;
  • 69) 0.710 781 109 789 551 133 668 147 2 × 2 = 1 + 0.421 562 219 579 102 267 336 294 4;
  • 70) 0.421 562 219 579 102 267 336 294 4 × 2 = 0 + 0.843 124 439 158 204 534 672 588 8;
  • 71) 0.843 124 439 158 204 534 672 588 8 × 2 = 1 + 0.686 248 878 316 409 069 345 177 6;
  • 72) 0.686 248 878 316 409 069 345 177 6 × 2 = 1 + 0.372 497 756 632 818 138 690 355 2;
  • 73) 0.372 497 756 632 818 138 690 355 2 × 2 = 0 + 0.744 995 513 265 636 277 380 710 4;
  • 74) 0.744 995 513 265 636 277 380 710 4 × 2 = 1 + 0.489 991 026 531 272 554 761 420 8;
  • 75) 0.489 991 026 531 272 554 761 420 8 × 2 = 0 + 0.979 982 053 062 545 109 522 841 6;
  • 76) 0.979 982 053 062 545 109 522 841 6 × 2 = 1 + 0.959 964 106 125 090 219 045 683 2;
  • 77) 0.959 964 106 125 090 219 045 683 2 × 2 = 1 + 0.919 928 212 250 180 438 091 366 4;
  • 78) 0.919 928 212 250 180 438 091 366 4 × 2 = 1 + 0.839 856 424 500 360 876 182 732 8;
  • 79) 0.839 856 424 500 360 876 182 732 8 × 2 = 1 + 0.679 712 849 000 721 752 365 465 6;
  • 80) 0.679 712 849 000 721 752 365 465 6 × 2 = 1 + 0.359 425 698 001 443 504 730 931 2;
  • 81) 0.359 425 698 001 443 504 730 931 2 × 2 = 0 + 0.718 851 396 002 887 009 461 862 4;
  • 82) 0.718 851 396 002 887 009 461 862 4 × 2 = 1 + 0.437 702 792 005 774 018 923 724 8;
  • 83) 0.437 702 792 005 774 018 923 724 8 × 2 = 0 + 0.875 405 584 011 548 037 847 449 6;
  • 84) 0.875 405 584 011 548 037 847 449 6 × 2 = 1 + 0.750 811 168 023 096 075 694 899 2;
  • 85) 0.750 811 168 023 096 075 694 899 2 × 2 = 1 + 0.501 622 336 046 192 151 389 798 4;
  • 86) 0.501 622 336 046 192 151 389 798 4 × 2 = 1 + 0.003 244 672 092 384 302 779 596 8;
  • 87) 0.003 244 672 092 384 302 779 596 8 × 2 = 0 + 0.006 489 344 184 768 605 559 193 6;
  • 88) 0.006 489 344 184 768 605 559 193 6 × 2 = 0 + 0.012 978 688 369 537 211 118 387 2;
  • 89) 0.012 978 688 369 537 211 118 387 2 × 2 = 0 + 0.025 957 376 739 074 422 236 774 4;
  • 90) 0.025 957 376 739 074 422 236 774 4 × 2 = 0 + 0.051 914 753 478 148 844 473 548 8;
  • 91) 0.051 914 753 478 148 844 473 548 8 × 2 = 0 + 0.103 829 506 956 297 688 947 097 6;
  • 92) 0.103 829 506 956 297 688 947 097 6 × 2 = 0 + 0.207 659 013 912 595 377 894 195 2;
  • 93) 0.207 659 013 912 595 377 894 195 2 × 2 = 0 + 0.415 318 027 825 190 755 788 390 4;
  • 94) 0.415 318 027 825 190 755 788 390 4 × 2 = 0 + 0.830 636 055 650 381 511 576 780 8;
  • 95) 0.830 636 055 650 381 511 576 780 8 × 2 = 1 + 0.661 272 111 300 763 023 153 561 6;
  • 96) 0.661 272 111 300 763 023 153 561 6 × 2 = 1 + 0.322 544 222 601 526 046 307 123 2;
  • 97) 0.322 544 222 601 526 046 307 123 2 × 2 = 0 + 0.645 088 445 203 052 092 614 246 4;
  • 98) 0.645 088 445 203 052 092 614 246 4 × 2 = 1 + 0.290 176 890 406 104 185 228 492 8;

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...)


5. 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:


0.000 000 000 000 014 391 905 333 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0000 1011 1100 1011 0101 1111 0101 1100 0000 0011 01(2)


6. Positive number before normalization:

0.000 000 000 000 014 391 905 333 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0000 1011 1100 1011 0101 1111 0101 1100 0000 0011 01(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 46 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 014 391 905 333 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0000 1011 1100 1011 0101 1111 0101 1100 0000 0011 01(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0000 1011 1100 1011 0101 1111 0101 1100 0000 0011 01(2) × 20 =

1.0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101(2) × 2-46


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): -46

Mantissa (not normalized):
1.0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-46 + 2(11-1) - 1 =

(-46 + 1 023)(10) =

977(10)


10. 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;
  • 977 ÷ 2 = 488 + 1;
  • 488 ÷ 2 = 244 + 0;
  • 244 ÷ 2 = 122 + 0;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

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


Exponent (adjusted) =

977(10) =

011 1101 0001(2)


12. 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. 0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101 =

0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1101 0001

Mantissa (52 bits) =
0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101


The base ten decimal number -0.000 000 000 000 014 391 905 333 7 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 011 1101 0001 - 0000 0011 0100 0010 1111 0010 1101 0111 1101 0111 0000 0000 1101

(64 bits IEEE 754)

  • Sign (1 bit):

    • 1

      63

  • Exponent (11 bits):

    • 0

      62

    • 1

      61

    • 1

      60

    • 1

      59

    • 1

      58

    • 0

      57

    • 1

      56

    • 0

      55

    • 0

      54

    • 0

      53

    • 1

      52

  • Mantissa (52 bits):

    • 0

      51

    • 0

      50

    • 0

      49

    • 0

      48

    • 0

      47

    • 0

      46

    • 1

      45

    • 1

      44

    • 0

      43

    • 1

      42

    • 0

      41

    • 0

      40

    • 0

      39

    • 0

      38

    • 1

      37

    • 0

      36

    • 1

      35

    • 1

      34

    • 1

      33

    • 1

      32

    • 0

      31

    • 0

      30

    • 1

      29

    • 0

      28

    • 1

      27

    • 1

      26

    • 0

      25

    • 1

      24

    • 0

      23

    • 1

      22

    • 1

      21

    • 1

      20

    • 1

      19

    • 1

      18

    • 0

      17

    • 1

      16

    • 0

      15

    • 1

      14

    • 1

      13

    • 1

      12

    • 0

      11

    • 0

      10

    • 0

      9

    • 0

      8

    • 0

      7

    • 0

      6

    • 0

      5

    • 0

      4

    • 1

      3

    • 1

      2

    • 0

      1

    • 1

      0

Number -0.000 000 000 000 014 391 905 333 8 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

Number -0.000 000 000 000 014 391 905 333 6 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

Convert to 64 bit double precision IEEE 754 binary floating point representation standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

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All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

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