0.000 000 000 000 000 000 008 537 73 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 73(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
0.000 000 000 000 000 000 008 537 73(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: 0.000 000 000 000 000 000 008 537 73.

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) 0.000 000 000 000 000 000 008 537 73 × 2 = 0 + 0.000 000 000 000 000 000 017 075 46;
  • 2) 0.000 000 000 000 000 000 017 075 46 × 2 = 0 + 0.000 000 000 000 000 000 034 150 92;
  • 3) 0.000 000 000 000 000 000 034 150 92 × 2 = 0 + 0.000 000 000 000 000 000 068 301 84;
  • 4) 0.000 000 000 000 000 000 068 301 84 × 2 = 0 + 0.000 000 000 000 000 000 136 603 68;
  • 5) 0.000 000 000 000 000 000 136 603 68 × 2 = 0 + 0.000 000 000 000 000 000 273 207 36;
  • 6) 0.000 000 000 000 000 000 273 207 36 × 2 = 0 + 0.000 000 000 000 000 000 546 414 72;
  • 7) 0.000 000 000 000 000 000 546 414 72 × 2 = 0 + 0.000 000 000 000 000 001 092 829 44;
  • 8) 0.000 000 000 000 000 001 092 829 44 × 2 = 0 + 0.000 000 000 000 000 002 185 658 88;
  • 9) 0.000 000 000 000 000 002 185 658 88 × 2 = 0 + 0.000 000 000 000 000 004 371 317 76;
  • 10) 0.000 000 000 000 000 004 371 317 76 × 2 = 0 + 0.000 000 000 000 000 008 742 635 52;
  • 11) 0.000 000 000 000 000 008 742 635 52 × 2 = 0 + 0.000 000 000 000 000 017 485 271 04;
  • 12) 0.000 000 000 000 000 017 485 271 04 × 2 = 0 + 0.000 000 000 000 000 034 970 542 08;
  • 13) 0.000 000 000 000 000 034 970 542 08 × 2 = 0 + 0.000 000 000 000 000 069 941 084 16;
  • 14) 0.000 000 000 000 000 069 941 084 16 × 2 = 0 + 0.000 000 000 000 000 139 882 168 32;
  • 15) 0.000 000 000 000 000 139 882 168 32 × 2 = 0 + 0.000 000 000 000 000 279 764 336 64;
  • 16) 0.000 000 000 000 000 279 764 336 64 × 2 = 0 + 0.000 000 000 000 000 559 528 673 28;
  • 17) 0.000 000 000 000 000 559 528 673 28 × 2 = 0 + 0.000 000 000 000 001 119 057 346 56;
  • 18) 0.000 000 000 000 001 119 057 346 56 × 2 = 0 + 0.000 000 000 000 002 238 114 693 12;
  • 19) 0.000 000 000 000 002 238 114 693 12 × 2 = 0 + 0.000 000 000 000 004 476 229 386 24;
  • 20) 0.000 000 000 000 004 476 229 386 24 × 2 = 0 + 0.000 000 000 000 008 952 458 772 48;
  • 21) 0.000 000 000 000 008 952 458 772 48 × 2 = 0 + 0.000 000 000 000 017 904 917 544 96;
  • 22) 0.000 000 000 000 017 904 917 544 96 × 2 = 0 + 0.000 000 000 000 035 809 835 089 92;
  • 23) 0.000 000 000 000 035 809 835 089 92 × 2 = 0 + 0.000 000 000 000 071 619 670 179 84;
  • 24) 0.000 000 000 000 071 619 670 179 84 × 2 = 0 + 0.000 000 000 000 143 239 340 359 68;
  • 25) 0.000 000 000 000 143 239 340 359 68 × 2 = 0 + 0.000 000 000 000 286 478 680 719 36;
  • 26) 0.000 000 000 000 286 478 680 719 36 × 2 = 0 + 0.000 000 000 000 572 957 361 438 72;
  • 27) 0.000 000 000 000 572 957 361 438 72 × 2 = 0 + 0.000 000 000 001 145 914 722 877 44;
  • 28) 0.000 000 000 001 145 914 722 877 44 × 2 = 0 + 0.000 000 000 002 291 829 445 754 88;
  • 29) 0.000 000 000 002 291 829 445 754 88 × 2 = 0 + 0.000 000 000 004 583 658 891 509 76;
  • 30) 0.000 000 000 004 583 658 891 509 76 × 2 = 0 + 0.000 000 000 009 167 317 783 019 52;
  • 31) 0.000 000 000 009 167 317 783 019 52 × 2 = 0 + 0.000 000 000 018 334 635 566 039 04;
  • 32) 0.000 000 000 018 334 635 566 039 04 × 2 = 0 + 0.000 000 000 036 669 271 132 078 08;
  • 33) 0.000 000 000 036 669 271 132 078 08 × 2 = 0 + 0.000 000 000 073 338 542 264 156 16;
  • 34) 0.000 000 000 073 338 542 264 156 16 × 2 = 0 + 0.000 000 000 146 677 084 528 312 32;
  • 35) 0.000 000 000 146 677 084 528 312 32 × 2 = 0 + 0.000 000 000 293 354 169 056 624 64;
  • 36) 0.000 000 000 293 354 169 056 624 64 × 2 = 0 + 0.000 000 000 586 708 338 113 249 28;
  • 37) 0.000 000 000 586 708 338 113 249 28 × 2 = 0 + 0.000 000 001 173 416 676 226 498 56;
  • 38) 0.000 000 001 173 416 676 226 498 56 × 2 = 0 + 0.000 000 002 346 833 352 452 997 12;
  • 39) 0.000 000 002 346 833 352 452 997 12 × 2 = 0 + 0.000 000 004 693 666 704 905 994 24;
  • 40) 0.000 000 004 693 666 704 905 994 24 × 2 = 0 + 0.000 000 009 387 333 409 811 988 48;
  • 41) 0.000 000 009 387 333 409 811 988 48 × 2 = 0 + 0.000 000 018 774 666 819 623 976 96;
  • 42) 0.000 000 018 774 666 819 623 976 96 × 2 = 0 + 0.000 000 037 549 333 639 247 953 92;
  • 43) 0.000 000 037 549 333 639 247 953 92 × 2 = 0 + 0.000 000 075 098 667 278 495 907 84;
  • 44) 0.000 000 075 098 667 278 495 907 84 × 2 = 0 + 0.000 000 150 197 334 556 991 815 68;
  • 45) 0.000 000 150 197 334 556 991 815 68 × 2 = 0 + 0.000 000 300 394 669 113 983 631 36;
  • 46) 0.000 000 300 394 669 113 983 631 36 × 2 = 0 + 0.000 000 600 789 338 227 967 262 72;
  • 47) 0.000 000 600 789 338 227 967 262 72 × 2 = 0 + 0.000 001 201 578 676 455 934 525 44;
  • 48) 0.000 001 201 578 676 455 934 525 44 × 2 = 0 + 0.000 002 403 157 352 911 869 050 88;
  • 49) 0.000 002 403 157 352 911 869 050 88 × 2 = 0 + 0.000 004 806 314 705 823 738 101 76;
  • 50) 0.000 004 806 314 705 823 738 101 76 × 2 = 0 + 0.000 009 612 629 411 647 476 203 52;
  • 51) 0.000 009 612 629 411 647 476 203 52 × 2 = 0 + 0.000 019 225 258 823 294 952 407 04;
  • 52) 0.000 019 225 258 823 294 952 407 04 × 2 = 0 + 0.000 038 450 517 646 589 904 814 08;
  • 53) 0.000 038 450 517 646 589 904 814 08 × 2 = 0 + 0.000 076 901 035 293 179 809 628 16;
  • 54) 0.000 076 901 035 293 179 809 628 16 × 2 = 0 + 0.000 153 802 070 586 359 619 256 32;
  • 55) 0.000 153 802 070 586 359 619 256 32 × 2 = 0 + 0.000 307 604 141 172 719 238 512 64;
  • 56) 0.000 307 604 141 172 719 238 512 64 × 2 = 0 + 0.000 615 208 282 345 438 477 025 28;
  • 57) 0.000 615 208 282 345 438 477 025 28 × 2 = 0 + 0.001 230 416 564 690 876 954 050 56;
  • 58) 0.001 230 416 564 690 876 954 050 56 × 2 = 0 + 0.002 460 833 129 381 753 908 101 12;
  • 59) 0.002 460 833 129 381 753 908 101 12 × 2 = 0 + 0.004 921 666 258 763 507 816 202 24;
  • 60) 0.004 921 666 258 763 507 816 202 24 × 2 = 0 + 0.009 843 332 517 527 015 632 404 48;
  • 61) 0.009 843 332 517 527 015 632 404 48 × 2 = 0 + 0.019 686 665 035 054 031 264 808 96;
  • 62) 0.019 686 665 035 054 031 264 808 96 × 2 = 0 + 0.039 373 330 070 108 062 529 617 92;
  • 63) 0.039 373 330 070 108 062 529 617 92 × 2 = 0 + 0.078 746 660 140 216 125 059 235 84;
  • 64) 0.078 746 660 140 216 125 059 235 84 × 2 = 0 + 0.157 493 320 280 432 250 118 471 68;
  • 65) 0.157 493 320 280 432 250 118 471 68 × 2 = 0 + 0.314 986 640 560 864 500 236 943 36;
  • 66) 0.314 986 640 560 864 500 236 943 36 × 2 = 0 + 0.629 973 281 121 729 000 473 886 72;
  • 67) 0.629 973 281 121 729 000 473 886 72 × 2 = 1 + 0.259 946 562 243 458 000 947 773 44;
  • 68) 0.259 946 562 243 458 000 947 773 44 × 2 = 0 + 0.519 893 124 486 916 001 895 546 88;
  • 69) 0.519 893 124 486 916 001 895 546 88 × 2 = 1 + 0.039 786 248 973 832 003 791 093 76;
  • 70) 0.039 786 248 973 832 003 791 093 76 × 2 = 0 + 0.079 572 497 947 664 007 582 187 52;
  • 71) 0.079 572 497 947 664 007 582 187 52 × 2 = 0 + 0.159 144 995 895 328 015 164 375 04;
  • 72) 0.159 144 995 895 328 015 164 375 04 × 2 = 0 + 0.318 289 991 790 656 030 328 750 08;
  • 73) 0.318 289 991 790 656 030 328 750 08 × 2 = 0 + 0.636 579 983 581 312 060 657 500 16;
  • 74) 0.636 579 983 581 312 060 657 500 16 × 2 = 1 + 0.273 159 967 162 624 121 315 000 32;
  • 75) 0.273 159 967 162 624 121 315 000 32 × 2 = 0 + 0.546 319 934 325 248 242 630 000 64;
  • 76) 0.546 319 934 325 248 242 630 000 64 × 2 = 1 + 0.092 639 868 650 496 485 260 001 28;
  • 77) 0.092 639 868 650 496 485 260 001 28 × 2 = 0 + 0.185 279 737 300 992 970 520 002 56;
  • 78) 0.185 279 737 300 992 970 520 002 56 × 2 = 0 + 0.370 559 474 601 985 941 040 005 12;
  • 79) 0.370 559 474 601 985 941 040 005 12 × 2 = 0 + 0.741 118 949 203 971 882 080 010 24;
  • 80) 0.741 118 949 203 971 882 080 010 24 × 2 = 1 + 0.482 237 898 407 943 764 160 020 48;
  • 81) 0.482 237 898 407 943 764 160 020 48 × 2 = 0 + 0.964 475 796 815 887 528 320 040 96;
  • 82) 0.964 475 796 815 887 528 320 040 96 × 2 = 1 + 0.928 951 593 631 775 056 640 081 92;
  • 83) 0.928 951 593 631 775 056 640 081 92 × 2 = 1 + 0.857 903 187 263 550 113 280 163 84;
  • 84) 0.857 903 187 263 550 113 280 163 84 × 2 = 1 + 0.715 806 374 527 100 226 560 327 68;
  • 85) 0.715 806 374 527 100 226 560 327 68 × 2 = 1 + 0.431 612 749 054 200 453 120 655 36;
  • 86) 0.431 612 749 054 200 453 120 655 36 × 2 = 0 + 0.863 225 498 108 400 906 241 310 72;
  • 87) 0.863 225 498 108 400 906 241 310 72 × 2 = 1 + 0.726 450 996 216 801 812 482 621 44;
  • 88) 0.726 450 996 216 801 812 482 621 44 × 2 = 1 + 0.452 901 992 433 603 624 965 242 88;
  • 89) 0.452 901 992 433 603 624 965 242 88 × 2 = 0 + 0.905 803 984 867 207 249 930 485 76;
  • 90) 0.905 803 984 867 207 249 930 485 76 × 2 = 1 + 0.811 607 969 734 414 499 860 971 52;
  • 91) 0.811 607 969 734 414 499 860 971 52 × 2 = 1 + 0.623 215 939 468 828 999 721 943 04;
  • 92) 0.623 215 939 468 828 999 721 943 04 × 2 = 1 + 0.246 431 878 937 657 999 443 886 08;
  • 93) 0.246 431 878 937 657 999 443 886 08 × 2 = 0 + 0.492 863 757 875 315 998 887 772 16;
  • 94) 0.492 863 757 875 315 998 887 772 16 × 2 = 0 + 0.985 727 515 750 631 997 775 544 32;
  • 95) 0.985 727 515 750 631 997 775 544 32 × 2 = 1 + 0.971 455 031 501 263 995 551 088 64;
  • 96) 0.971 455 031 501 263 995 551 088 64 × 2 = 1 + 0.942 910 063 002 527 991 102 177 28;
  • 97) 0.942 910 063 002 527 991 102 177 28 × 2 = 1 + 0.885 820 126 005 055 982 204 354 56;
  • 98) 0.885 820 126 005 055 982 204 354 56 × 2 = 1 + 0.771 640 252 010 111 964 408 709 12;
  • 99) 0.771 640 252 010 111 964 408 709 12 × 2 = 1 + 0.543 280 504 020 223 928 817 418 24;
  • 100) 0.543 280 504 020 223 928 817 418 24 × 2 = 1 + 0.086 561 008 040 447 857 634 836 48;
  • 101) 0.086 561 008 040 447 857 634 836 48 × 2 = 0 + 0.173 122 016 080 895 715 269 672 96;
  • 102) 0.173 122 016 080 895 715 269 672 96 × 2 = 0 + 0.346 244 032 161 791 430 539 345 92;
  • 103) 0.346 244 032 161 791 430 539 345 92 × 2 = 0 + 0.692 488 064 323 582 861 078 691 84;
  • 104) 0.692 488 064 323 582 861 078 691 84 × 2 = 1 + 0.384 976 128 647 165 722 157 383 68;
  • 105) 0.384 976 128 647 165 722 157 383 68 × 2 = 0 + 0.769 952 257 294 331 444 314 767 36;
  • 106) 0.769 952 257 294 331 444 314 767 36 × 2 = 1 + 0.539 904 514 588 662 888 629 534 72;
  • 107) 0.539 904 514 588 662 888 629 534 72 × 2 = 1 + 0.079 809 029 177 325 777 259 069 44;
  • 108) 0.079 809 029 177 325 777 259 069 44 × 2 = 0 + 0.159 618 058 354 651 554 518 138 88;
  • 109) 0.159 618 058 354 651 554 518 138 88 × 2 = 0 + 0.319 236 116 709 303 109 036 277 76;
  • 110) 0.319 236 116 709 303 109 036 277 76 × 2 = 0 + 0.638 472 233 418 606 218 072 555 52;
  • 111) 0.638 472 233 418 606 218 072 555 52 × 2 = 1 + 0.276 944 466 837 212 436 145 111 04;
  • 112) 0.276 944 466 837 212 436 145 111 04 × 2 = 0 + 0.553 888 933 674 424 872 290 222 08;
  • 113) 0.553 888 933 674 424 872 290 222 08 × 2 = 1 + 0.107 777 867 348 849 744 580 444 16;
  • 114) 0.107 777 867 348 849 744 580 444 16 × 2 = 0 + 0.215 555 734 697 699 489 160 888 32;
  • 115) 0.215 555 734 697 699 489 160 888 32 × 2 = 0 + 0.431 111 469 395 398 978 321 776 64;
  • 116) 0.431 111 469 395 398 978 321 776 64 × 2 = 0 + 0.862 222 938 790 797 956 643 553 28;
  • 117) 0.862 222 938 790 797 956 643 553 28 × 2 = 1 + 0.724 445 877 581 595 913 287 106 56;
  • 118) 0.724 445 877 581 595 913 287 106 56 × 2 = 1 + 0.448 891 755 163 191 826 574 213 12;
  • 119) 0.448 891 755 163 191 826 574 213 12 × 2 = 0 + 0.897 783 510 326 383 653 148 426 24;

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:


0.000 000 000 000 000 000 008 537 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0111 1011 0111 0011 1111 0001 0110 0010 1000 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0111 1011 0111 0011 1111 0001 0110 0010 1000 110(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:


0.000 000 000 000 000 000 008 537 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0111 1011 0111 0011 1111 0001 0110 0010 1000 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0111 1011 0111 0011 1111 0001 0110 0010 1000 110(2) × 20 =

1.0100 0010 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110(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 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110


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 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110 =

0100 0010 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110


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 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110


Decimal number 0.000 000 000 000 000 000 008 537 73 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1011 1101 1011 1001 1111 1000 1011 0001 0100 0110

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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