0.000 000 000 000 000 000 008 535 79 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 79(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
0.000 000 000 000 000 000 008 535 79(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: 0.000 000 000 000 000 000 008 535 79.

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) 0.000 000 000 000 000 000 008 535 79 × 2 = 0 + 0.000 000 000 000 000 000 017 071 58;
  • 2) 0.000 000 000 000 000 000 017 071 58 × 2 = 0 + 0.000 000 000 000 000 000 034 143 16;
  • 3) 0.000 000 000 000 000 000 034 143 16 × 2 = 0 + 0.000 000 000 000 000 000 068 286 32;
  • 4) 0.000 000 000 000 000 000 068 286 32 × 2 = 0 + 0.000 000 000 000 000 000 136 572 64;
  • 5) 0.000 000 000 000 000 000 136 572 64 × 2 = 0 + 0.000 000 000 000 000 000 273 145 28;
  • 6) 0.000 000 000 000 000 000 273 145 28 × 2 = 0 + 0.000 000 000 000 000 000 546 290 56;
  • 7) 0.000 000 000 000 000 000 546 290 56 × 2 = 0 + 0.000 000 000 000 000 001 092 581 12;
  • 8) 0.000 000 000 000 000 001 092 581 12 × 2 = 0 + 0.000 000 000 000 000 002 185 162 24;
  • 9) 0.000 000 000 000 000 002 185 162 24 × 2 = 0 + 0.000 000 000 000 000 004 370 324 48;
  • 10) 0.000 000 000 000 000 004 370 324 48 × 2 = 0 + 0.000 000 000 000 000 008 740 648 96;
  • 11) 0.000 000 000 000 000 008 740 648 96 × 2 = 0 + 0.000 000 000 000 000 017 481 297 92;
  • 12) 0.000 000 000 000 000 017 481 297 92 × 2 = 0 + 0.000 000 000 000 000 034 962 595 84;
  • 13) 0.000 000 000 000 000 034 962 595 84 × 2 = 0 + 0.000 000 000 000 000 069 925 191 68;
  • 14) 0.000 000 000 000 000 069 925 191 68 × 2 = 0 + 0.000 000 000 000 000 139 850 383 36;
  • 15) 0.000 000 000 000 000 139 850 383 36 × 2 = 0 + 0.000 000 000 000 000 279 700 766 72;
  • 16) 0.000 000 000 000 000 279 700 766 72 × 2 = 0 + 0.000 000 000 000 000 559 401 533 44;
  • 17) 0.000 000 000 000 000 559 401 533 44 × 2 = 0 + 0.000 000 000 000 001 118 803 066 88;
  • 18) 0.000 000 000 000 001 118 803 066 88 × 2 = 0 + 0.000 000 000 000 002 237 606 133 76;
  • 19) 0.000 000 000 000 002 237 606 133 76 × 2 = 0 + 0.000 000 000 000 004 475 212 267 52;
  • 20) 0.000 000 000 000 004 475 212 267 52 × 2 = 0 + 0.000 000 000 000 008 950 424 535 04;
  • 21) 0.000 000 000 000 008 950 424 535 04 × 2 = 0 + 0.000 000 000 000 017 900 849 070 08;
  • 22) 0.000 000 000 000 017 900 849 070 08 × 2 = 0 + 0.000 000 000 000 035 801 698 140 16;
  • 23) 0.000 000 000 000 035 801 698 140 16 × 2 = 0 + 0.000 000 000 000 071 603 396 280 32;
  • 24) 0.000 000 000 000 071 603 396 280 32 × 2 = 0 + 0.000 000 000 000 143 206 792 560 64;
  • 25) 0.000 000 000 000 143 206 792 560 64 × 2 = 0 + 0.000 000 000 000 286 413 585 121 28;
  • 26) 0.000 000 000 000 286 413 585 121 28 × 2 = 0 + 0.000 000 000 000 572 827 170 242 56;
  • 27) 0.000 000 000 000 572 827 170 242 56 × 2 = 0 + 0.000 000 000 001 145 654 340 485 12;
  • 28) 0.000 000 000 001 145 654 340 485 12 × 2 = 0 + 0.000 000 000 002 291 308 680 970 24;
  • 29) 0.000 000 000 002 291 308 680 970 24 × 2 = 0 + 0.000 000 000 004 582 617 361 940 48;
  • 30) 0.000 000 000 004 582 617 361 940 48 × 2 = 0 + 0.000 000 000 009 165 234 723 880 96;
  • 31) 0.000 000 000 009 165 234 723 880 96 × 2 = 0 + 0.000 000 000 018 330 469 447 761 92;
  • 32) 0.000 000 000 018 330 469 447 761 92 × 2 = 0 + 0.000 000 000 036 660 938 895 523 84;
  • 33) 0.000 000 000 036 660 938 895 523 84 × 2 = 0 + 0.000 000 000 073 321 877 791 047 68;
  • 34) 0.000 000 000 073 321 877 791 047 68 × 2 = 0 + 0.000 000 000 146 643 755 582 095 36;
  • 35) 0.000 000 000 146 643 755 582 095 36 × 2 = 0 + 0.000 000 000 293 287 511 164 190 72;
  • 36) 0.000 000 000 293 287 511 164 190 72 × 2 = 0 + 0.000 000 000 586 575 022 328 381 44;
  • 37) 0.000 000 000 586 575 022 328 381 44 × 2 = 0 + 0.000 000 001 173 150 044 656 762 88;
  • 38) 0.000 000 001 173 150 044 656 762 88 × 2 = 0 + 0.000 000 002 346 300 089 313 525 76;
  • 39) 0.000 000 002 346 300 089 313 525 76 × 2 = 0 + 0.000 000 004 692 600 178 627 051 52;
  • 40) 0.000 000 004 692 600 178 627 051 52 × 2 = 0 + 0.000 000 009 385 200 357 254 103 04;
  • 41) 0.000 000 009 385 200 357 254 103 04 × 2 = 0 + 0.000 000 018 770 400 714 508 206 08;
  • 42) 0.000 000 018 770 400 714 508 206 08 × 2 = 0 + 0.000 000 037 540 801 429 016 412 16;
  • 43) 0.000 000 037 540 801 429 016 412 16 × 2 = 0 + 0.000 000 075 081 602 858 032 824 32;
  • 44) 0.000 000 075 081 602 858 032 824 32 × 2 = 0 + 0.000 000 150 163 205 716 065 648 64;
  • 45) 0.000 000 150 163 205 716 065 648 64 × 2 = 0 + 0.000 000 300 326 411 432 131 297 28;
  • 46) 0.000 000 300 326 411 432 131 297 28 × 2 = 0 + 0.000 000 600 652 822 864 262 594 56;
  • 47) 0.000 000 600 652 822 864 262 594 56 × 2 = 0 + 0.000 001 201 305 645 728 525 189 12;
  • 48) 0.000 001 201 305 645 728 525 189 12 × 2 = 0 + 0.000 002 402 611 291 457 050 378 24;
  • 49) 0.000 002 402 611 291 457 050 378 24 × 2 = 0 + 0.000 004 805 222 582 914 100 756 48;
  • 50) 0.000 004 805 222 582 914 100 756 48 × 2 = 0 + 0.000 009 610 445 165 828 201 512 96;
  • 51) 0.000 009 610 445 165 828 201 512 96 × 2 = 0 + 0.000 019 220 890 331 656 403 025 92;
  • 52) 0.000 019 220 890 331 656 403 025 92 × 2 = 0 + 0.000 038 441 780 663 312 806 051 84;
  • 53) 0.000 038 441 780 663 312 806 051 84 × 2 = 0 + 0.000 076 883 561 326 625 612 103 68;
  • 54) 0.000 076 883 561 326 625 612 103 68 × 2 = 0 + 0.000 153 767 122 653 251 224 207 36;
  • 55) 0.000 153 767 122 653 251 224 207 36 × 2 = 0 + 0.000 307 534 245 306 502 448 414 72;
  • 56) 0.000 307 534 245 306 502 448 414 72 × 2 = 0 + 0.000 615 068 490 613 004 896 829 44;
  • 57) 0.000 615 068 490 613 004 896 829 44 × 2 = 0 + 0.001 230 136 981 226 009 793 658 88;
  • 58) 0.001 230 136 981 226 009 793 658 88 × 2 = 0 + 0.002 460 273 962 452 019 587 317 76;
  • 59) 0.002 460 273 962 452 019 587 317 76 × 2 = 0 + 0.004 920 547 924 904 039 174 635 52;
  • 60) 0.004 920 547 924 904 039 174 635 52 × 2 = 0 + 0.009 841 095 849 808 078 349 271 04;
  • 61) 0.009 841 095 849 808 078 349 271 04 × 2 = 0 + 0.019 682 191 699 616 156 698 542 08;
  • 62) 0.019 682 191 699 616 156 698 542 08 × 2 = 0 + 0.039 364 383 399 232 313 397 084 16;
  • 63) 0.039 364 383 399 232 313 397 084 16 × 2 = 0 + 0.078 728 766 798 464 626 794 168 32;
  • 64) 0.078 728 766 798 464 626 794 168 32 × 2 = 0 + 0.157 457 533 596 929 253 588 336 64;
  • 65) 0.157 457 533 596 929 253 588 336 64 × 2 = 0 + 0.314 915 067 193 858 507 176 673 28;
  • 66) 0.314 915 067 193 858 507 176 673 28 × 2 = 0 + 0.629 830 134 387 717 014 353 346 56;
  • 67) 0.629 830 134 387 717 014 353 346 56 × 2 = 1 + 0.259 660 268 775 434 028 706 693 12;
  • 68) 0.259 660 268 775 434 028 706 693 12 × 2 = 0 + 0.519 320 537 550 868 057 413 386 24;
  • 69) 0.519 320 537 550 868 057 413 386 24 × 2 = 1 + 0.038 641 075 101 736 114 826 772 48;
  • 70) 0.038 641 075 101 736 114 826 772 48 × 2 = 0 + 0.077 282 150 203 472 229 653 544 96;
  • 71) 0.077 282 150 203 472 229 653 544 96 × 2 = 0 + 0.154 564 300 406 944 459 307 089 92;
  • 72) 0.154 564 300 406 944 459 307 089 92 × 2 = 0 + 0.309 128 600 813 888 918 614 179 84;
  • 73) 0.309 128 600 813 888 918 614 179 84 × 2 = 0 + 0.618 257 201 627 777 837 228 359 68;
  • 74) 0.618 257 201 627 777 837 228 359 68 × 2 = 1 + 0.236 514 403 255 555 674 456 719 36;
  • 75) 0.236 514 403 255 555 674 456 719 36 × 2 = 0 + 0.473 028 806 511 111 348 913 438 72;
  • 76) 0.473 028 806 511 111 348 913 438 72 × 2 = 0 + 0.946 057 613 022 222 697 826 877 44;
  • 77) 0.946 057 613 022 222 697 826 877 44 × 2 = 1 + 0.892 115 226 044 445 395 653 754 88;
  • 78) 0.892 115 226 044 445 395 653 754 88 × 2 = 1 + 0.784 230 452 088 890 791 307 509 76;
  • 79) 0.784 230 452 088 890 791 307 509 76 × 2 = 1 + 0.568 460 904 177 781 582 615 019 52;
  • 80) 0.568 460 904 177 781 582 615 019 52 × 2 = 1 + 0.136 921 808 355 563 165 230 039 04;
  • 81) 0.136 921 808 355 563 165 230 039 04 × 2 = 0 + 0.273 843 616 711 126 330 460 078 08;
  • 82) 0.273 843 616 711 126 330 460 078 08 × 2 = 0 + 0.547 687 233 422 252 660 920 156 16;
  • 83) 0.547 687 233 422 252 660 920 156 16 × 2 = 1 + 0.095 374 466 844 505 321 840 312 32;
  • 84) 0.095 374 466 844 505 321 840 312 32 × 2 = 0 + 0.190 748 933 689 010 643 680 624 64;
  • 85) 0.190 748 933 689 010 643 680 624 64 × 2 = 0 + 0.381 497 867 378 021 287 361 249 28;
  • 86) 0.381 497 867 378 021 287 361 249 28 × 2 = 0 + 0.762 995 734 756 042 574 722 498 56;
  • 87) 0.762 995 734 756 042 574 722 498 56 × 2 = 1 + 0.525 991 469 512 085 149 444 997 12;
  • 88) 0.525 991 469 512 085 149 444 997 12 × 2 = 1 + 0.051 982 939 024 170 298 889 994 24;
  • 89) 0.051 982 939 024 170 298 889 994 24 × 2 = 0 + 0.103 965 878 048 340 597 779 988 48;
  • 90) 0.103 965 878 048 340 597 779 988 48 × 2 = 0 + 0.207 931 756 096 681 195 559 976 96;
  • 91) 0.207 931 756 096 681 195 559 976 96 × 2 = 0 + 0.415 863 512 193 362 391 119 953 92;
  • 92) 0.415 863 512 193 362 391 119 953 92 × 2 = 0 + 0.831 727 024 386 724 782 239 907 84;
  • 93) 0.831 727 024 386 724 782 239 907 84 × 2 = 1 + 0.663 454 048 773 449 564 479 815 68;
  • 94) 0.663 454 048 773 449 564 479 815 68 × 2 = 1 + 0.326 908 097 546 899 128 959 631 36;
  • 95) 0.326 908 097 546 899 128 959 631 36 × 2 = 0 + 0.653 816 195 093 798 257 919 262 72;
  • 96) 0.653 816 195 093 798 257 919 262 72 × 2 = 1 + 0.307 632 390 187 596 515 838 525 44;
  • 97) 0.307 632 390 187 596 515 838 525 44 × 2 = 0 + 0.615 264 780 375 193 031 677 050 88;
  • 98) 0.615 264 780 375 193 031 677 050 88 × 2 = 1 + 0.230 529 560 750 386 063 354 101 76;
  • 99) 0.230 529 560 750 386 063 354 101 76 × 2 = 0 + 0.461 059 121 500 772 126 708 203 52;
  • 100) 0.461 059 121 500 772 126 708 203 52 × 2 = 0 + 0.922 118 243 001 544 253 416 407 04;
  • 101) 0.922 118 243 001 544 253 416 407 04 × 2 = 1 + 0.844 236 486 003 088 506 832 814 08;
  • 102) 0.844 236 486 003 088 506 832 814 08 × 2 = 1 + 0.688 472 972 006 177 013 665 628 16;
  • 103) 0.688 472 972 006 177 013 665 628 16 × 2 = 1 + 0.376 945 944 012 354 027 331 256 32;
  • 104) 0.376 945 944 012 354 027 331 256 32 × 2 = 0 + 0.753 891 888 024 708 054 662 512 64;
  • 105) 0.753 891 888 024 708 054 662 512 64 × 2 = 1 + 0.507 783 776 049 416 109 325 025 28;
  • 106) 0.507 783 776 049 416 109 325 025 28 × 2 = 1 + 0.015 567 552 098 832 218 650 050 56;
  • 107) 0.015 567 552 098 832 218 650 050 56 × 2 = 0 + 0.031 135 104 197 664 437 300 101 12;
  • 108) 0.031 135 104 197 664 437 300 101 12 × 2 = 0 + 0.062 270 208 395 328 874 600 202 24;
  • 109) 0.062 270 208 395 328 874 600 202 24 × 2 = 0 + 0.124 540 416 790 657 749 200 404 48;
  • 110) 0.124 540 416 790 657 749 200 404 48 × 2 = 0 + 0.249 080 833 581 315 498 400 808 96;
  • 111) 0.249 080 833 581 315 498 400 808 96 × 2 = 0 + 0.498 161 667 162 630 996 801 617 92;
  • 112) 0.498 161 667 162 630 996 801 617 92 × 2 = 0 + 0.996 323 334 325 261 993 603 235 84;
  • 113) 0.996 323 334 325 261 993 603 235 84 × 2 = 1 + 0.992 646 668 650 523 987 206 471 68;
  • 114) 0.992 646 668 650 523 987 206 471 68 × 2 = 1 + 0.985 293 337 301 047 974 412 943 36;
  • 115) 0.985 293 337 301 047 974 412 943 36 × 2 = 1 + 0.970 586 674 602 095 948 825 886 72;
  • 116) 0.970 586 674 602 095 948 825 886 72 × 2 = 1 + 0.941 173 349 204 191 897 651 773 44;
  • 117) 0.941 173 349 204 191 897 651 773 44 × 2 = 1 + 0.882 346 698 408 383 795 303 546 88;
  • 118) 0.882 346 698 408 383 795 303 546 88 × 2 = 1 + 0.764 693 396 816 767 590 607 093 76;
  • 119) 0.764 693 396 816 767 590 607 093 76 × 2 = 1 + 0.529 386 793 633 535 181 214 187 52;

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:


0.000 000 000 000 000 000 008 535 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0010 0011 0000 1101 0100 1110 1100 0000 1111 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0010 0011 0000 1101 0100 1110 1100 0000 1111 111(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:


0.000 000 000 000 000 000 008 535 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0010 0011 0000 1101 0100 1110 1100 0000 1111 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0010 0011 0000 1101 0100 1110 1100 0000 1111 111(2) × 20 =

1.0100 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111(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 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111


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 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111 =

0100 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111


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 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111


Decimal number 0.000 000 000 000 000 000 008 535 79 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1001 0001 1000 0110 1010 0111 0110 0000 0111 1111

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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