Decimal to 64 Bit IEEE 754 Binary: Convert Number 0.000 000 000 000 000 000 000 09 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 0.000 000 000 000 000 000 000 09(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (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: 0.000 000 000 000 000 000 000 09.

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) 0.000 000 000 000 000 000 000 09 × 2 = 0 + 0.000 000 000 000 000 000 000 18;
  • 2) 0.000 000 000 000 000 000 000 18 × 2 = 0 + 0.000 000 000 000 000 000 000 36;
  • 3) 0.000 000 000 000 000 000 000 36 × 2 = 0 + 0.000 000 000 000 000 000 000 72;
  • 4) 0.000 000 000 000 000 000 000 72 × 2 = 0 + 0.000 000 000 000 000 000 001 44;
  • 5) 0.000 000 000 000 000 000 001 44 × 2 = 0 + 0.000 000 000 000 000 000 002 88;
  • 6) 0.000 000 000 000 000 000 002 88 × 2 = 0 + 0.000 000 000 000 000 000 005 76;
  • 7) 0.000 000 000 000 000 000 005 76 × 2 = 0 + 0.000 000 000 000 000 000 011 52;
  • 8) 0.000 000 000 000 000 000 011 52 × 2 = 0 + 0.000 000 000 000 000 000 023 04;
  • 9) 0.000 000 000 000 000 000 023 04 × 2 = 0 + 0.000 000 000 000 000 000 046 08;
  • 10) 0.000 000 000 000 000 000 046 08 × 2 = 0 + 0.000 000 000 000 000 000 092 16;
  • 11) 0.000 000 000 000 000 000 092 16 × 2 = 0 + 0.000 000 000 000 000 000 184 32;
  • 12) 0.000 000 000 000 000 000 184 32 × 2 = 0 + 0.000 000 000 000 000 000 368 64;
  • 13) 0.000 000 000 000 000 000 368 64 × 2 = 0 + 0.000 000 000 000 000 000 737 28;
  • 14) 0.000 000 000 000 000 000 737 28 × 2 = 0 + 0.000 000 000 000 000 001 474 56;
  • 15) 0.000 000 000 000 000 001 474 56 × 2 = 0 + 0.000 000 000 000 000 002 949 12;
  • 16) 0.000 000 000 000 000 002 949 12 × 2 = 0 + 0.000 000 000 000 000 005 898 24;
  • 17) 0.000 000 000 000 000 005 898 24 × 2 = 0 + 0.000 000 000 000 000 011 796 48;
  • 18) 0.000 000 000 000 000 011 796 48 × 2 = 0 + 0.000 000 000 000 000 023 592 96;
  • 19) 0.000 000 000 000 000 023 592 96 × 2 = 0 + 0.000 000 000 000 000 047 185 92;
  • 20) 0.000 000 000 000 000 047 185 92 × 2 = 0 + 0.000 000 000 000 000 094 371 84;
  • 21) 0.000 000 000 000 000 094 371 84 × 2 = 0 + 0.000 000 000 000 000 188 743 68;
  • 22) 0.000 000 000 000 000 188 743 68 × 2 = 0 + 0.000 000 000 000 000 377 487 36;
  • 23) 0.000 000 000 000 000 377 487 36 × 2 = 0 + 0.000 000 000 000 000 754 974 72;
  • 24) 0.000 000 000 000 000 754 974 72 × 2 = 0 + 0.000 000 000 000 001 509 949 44;
  • 25) 0.000 000 000 000 001 509 949 44 × 2 = 0 + 0.000 000 000 000 003 019 898 88;
  • 26) 0.000 000 000 000 003 019 898 88 × 2 = 0 + 0.000 000 000 000 006 039 797 76;
  • 27) 0.000 000 000 000 006 039 797 76 × 2 = 0 + 0.000 000 000 000 012 079 595 52;
  • 28) 0.000 000 000 000 012 079 595 52 × 2 = 0 + 0.000 000 000 000 024 159 191 04;
  • 29) 0.000 000 000 000 024 159 191 04 × 2 = 0 + 0.000 000 000 000 048 318 382 08;
  • 30) 0.000 000 000 000 048 318 382 08 × 2 = 0 + 0.000 000 000 000 096 636 764 16;
  • 31) 0.000 000 000 000 096 636 764 16 × 2 = 0 + 0.000 000 000 000 193 273 528 32;
  • 32) 0.000 000 000 000 193 273 528 32 × 2 = 0 + 0.000 000 000 000 386 547 056 64;
  • 33) 0.000 000 000 000 386 547 056 64 × 2 = 0 + 0.000 000 000 000 773 094 113 28;
  • 34) 0.000 000 000 000 773 094 113 28 × 2 = 0 + 0.000 000 000 001 546 188 226 56;
  • 35) 0.000 000 000 001 546 188 226 56 × 2 = 0 + 0.000 000 000 003 092 376 453 12;
  • 36) 0.000 000 000 003 092 376 453 12 × 2 = 0 + 0.000 000 000 006 184 752 906 24;
  • 37) 0.000 000 000 006 184 752 906 24 × 2 = 0 + 0.000 000 000 012 369 505 812 48;
  • 38) 0.000 000 000 012 369 505 812 48 × 2 = 0 + 0.000 000 000 024 739 011 624 96;
  • 39) 0.000 000 000 024 739 011 624 96 × 2 = 0 + 0.000 000 000 049 478 023 249 92;
  • 40) 0.000 000 000 049 478 023 249 92 × 2 = 0 + 0.000 000 000 098 956 046 499 84;
  • 41) 0.000 000 000 098 956 046 499 84 × 2 = 0 + 0.000 000 000 197 912 092 999 68;
  • 42) 0.000 000 000 197 912 092 999 68 × 2 = 0 + 0.000 000 000 395 824 185 999 36;
  • 43) 0.000 000 000 395 824 185 999 36 × 2 = 0 + 0.000 000 000 791 648 371 998 72;
  • 44) 0.000 000 000 791 648 371 998 72 × 2 = 0 + 0.000 000 001 583 296 743 997 44;
  • 45) 0.000 000 001 583 296 743 997 44 × 2 = 0 + 0.000 000 003 166 593 487 994 88;
  • 46) 0.000 000 003 166 593 487 994 88 × 2 = 0 + 0.000 000 006 333 186 975 989 76;
  • 47) 0.000 000 006 333 186 975 989 76 × 2 = 0 + 0.000 000 012 666 373 951 979 52;
  • 48) 0.000 000 012 666 373 951 979 52 × 2 = 0 + 0.000 000 025 332 747 903 959 04;
  • 49) 0.000 000 025 332 747 903 959 04 × 2 = 0 + 0.000 000 050 665 495 807 918 08;
  • 50) 0.000 000 050 665 495 807 918 08 × 2 = 0 + 0.000 000 101 330 991 615 836 16;
  • 51) 0.000 000 101 330 991 615 836 16 × 2 = 0 + 0.000 000 202 661 983 231 672 32;
  • 52) 0.000 000 202 661 983 231 672 32 × 2 = 0 + 0.000 000 405 323 966 463 344 64;
  • 53) 0.000 000 405 323 966 463 344 64 × 2 = 0 + 0.000 000 810 647 932 926 689 28;
  • 54) 0.000 000 810 647 932 926 689 28 × 2 = 0 + 0.000 001 621 295 865 853 378 56;
  • 55) 0.000 001 621 295 865 853 378 56 × 2 = 0 + 0.000 003 242 591 731 706 757 12;
  • 56) 0.000 003 242 591 731 706 757 12 × 2 = 0 + 0.000 006 485 183 463 413 514 24;
  • 57) 0.000 006 485 183 463 413 514 24 × 2 = 0 + 0.000 012 970 366 926 827 028 48;
  • 58) 0.000 012 970 366 926 827 028 48 × 2 = 0 + 0.000 025 940 733 853 654 056 96;
  • 59) 0.000 025 940 733 853 654 056 96 × 2 = 0 + 0.000 051 881 467 707 308 113 92;
  • 60) 0.000 051 881 467 707 308 113 92 × 2 = 0 + 0.000 103 762 935 414 616 227 84;
  • 61) 0.000 103 762 935 414 616 227 84 × 2 = 0 + 0.000 207 525 870 829 232 455 68;
  • 62) 0.000 207 525 870 829 232 455 68 × 2 = 0 + 0.000 415 051 741 658 464 911 36;
  • 63) 0.000 415 051 741 658 464 911 36 × 2 = 0 + 0.000 830 103 483 316 929 822 72;
  • 64) 0.000 830 103 483 316 929 822 72 × 2 = 0 + 0.001 660 206 966 633 859 645 44;
  • 65) 0.001 660 206 966 633 859 645 44 × 2 = 0 + 0.003 320 413 933 267 719 290 88;
  • 66) 0.003 320 413 933 267 719 290 88 × 2 = 0 + 0.006 640 827 866 535 438 581 76;
  • 67) 0.006 640 827 866 535 438 581 76 × 2 = 0 + 0.013 281 655 733 070 877 163 52;
  • 68) 0.013 281 655 733 070 877 163 52 × 2 = 0 + 0.026 563 311 466 141 754 327 04;
  • 69) 0.026 563 311 466 141 754 327 04 × 2 = 0 + 0.053 126 622 932 283 508 654 08;
  • 70) 0.053 126 622 932 283 508 654 08 × 2 = 0 + 0.106 253 245 864 567 017 308 16;
  • 71) 0.106 253 245 864 567 017 308 16 × 2 = 0 + 0.212 506 491 729 134 034 616 32;
  • 72) 0.212 506 491 729 134 034 616 32 × 2 = 0 + 0.425 012 983 458 268 069 232 64;
  • 73) 0.425 012 983 458 268 069 232 64 × 2 = 0 + 0.850 025 966 916 536 138 465 28;
  • 74) 0.850 025 966 916 536 138 465 28 × 2 = 1 + 0.700 051 933 833 072 276 930 56;
  • 75) 0.700 051 933 833 072 276 930 56 × 2 = 1 + 0.400 103 867 666 144 553 861 12;
  • 76) 0.400 103 867 666 144 553 861 12 × 2 = 0 + 0.800 207 735 332 289 107 722 24;
  • 77) 0.800 207 735 332 289 107 722 24 × 2 = 1 + 0.600 415 470 664 578 215 444 48;
  • 78) 0.600 415 470 664 578 215 444 48 × 2 = 1 + 0.200 830 941 329 156 430 888 96;
  • 79) 0.200 830 941 329 156 430 888 96 × 2 = 0 + 0.401 661 882 658 312 861 777 92;
  • 80) 0.401 661 882 658 312 861 777 92 × 2 = 0 + 0.803 323 765 316 625 723 555 84;
  • 81) 0.803 323 765 316 625 723 555 84 × 2 = 1 + 0.606 647 530 633 251 447 111 68;
  • 82) 0.606 647 530 633 251 447 111 68 × 2 = 1 + 0.213 295 061 266 502 894 223 36;
  • 83) 0.213 295 061 266 502 894 223 36 × 2 = 0 + 0.426 590 122 533 005 788 446 72;
  • 84) 0.426 590 122 533 005 788 446 72 × 2 = 0 + 0.853 180 245 066 011 576 893 44;
  • 85) 0.853 180 245 066 011 576 893 44 × 2 = 1 + 0.706 360 490 132 023 153 786 88;
  • 86) 0.706 360 490 132 023 153 786 88 × 2 = 1 + 0.412 720 980 264 046 307 573 76;
  • 87) 0.412 720 980 264 046 307 573 76 × 2 = 0 + 0.825 441 960 528 092 615 147 52;
  • 88) 0.825 441 960 528 092 615 147 52 × 2 = 1 + 0.650 883 921 056 185 230 295 04;
  • 89) 0.650 883 921 056 185 230 295 04 × 2 = 1 + 0.301 767 842 112 370 460 590 08;
  • 90) 0.301 767 842 112 370 460 590 08 × 2 = 0 + 0.603 535 684 224 740 921 180 16;
  • 91) 0.603 535 684 224 740 921 180 16 × 2 = 1 + 0.207 071 368 449 481 842 360 32;
  • 92) 0.207 071 368 449 481 842 360 32 × 2 = 0 + 0.414 142 736 898 963 684 720 64;
  • 93) 0.414 142 736 898 963 684 720 64 × 2 = 0 + 0.828 285 473 797 927 369 441 28;
  • 94) 0.828 285 473 797 927 369 441 28 × 2 = 1 + 0.656 570 947 595 854 738 882 56;
  • 95) 0.656 570 947 595 854 738 882 56 × 2 = 1 + 0.313 141 895 191 709 477 765 12;
  • 96) 0.313 141 895 191 709 477 765 12 × 2 = 0 + 0.626 283 790 383 418 955 530 24;
  • 97) 0.626 283 790 383 418 955 530 24 × 2 = 1 + 0.252 567 580 766 837 911 060 48;
  • 98) 0.252 567 580 766 837 911 060 48 × 2 = 0 + 0.505 135 161 533 675 822 120 96;
  • 99) 0.505 135 161 533 675 822 120 96 × 2 = 1 + 0.010 270 323 067 351 644 241 92;
  • 100) 0.010 270 323 067 351 644 241 92 × 2 = 0 + 0.020 540 646 134 703 288 483 84;
  • 101) 0.020 540 646 134 703 288 483 84 × 2 = 0 + 0.041 081 292 269 406 576 967 68;
  • 102) 0.041 081 292 269 406 576 967 68 × 2 = 0 + 0.082 162 584 538 813 153 935 36;
  • 103) 0.082 162 584 538 813 153 935 36 × 2 = 0 + 0.164 325 169 077 626 307 870 72;
  • 104) 0.164 325 169 077 626 307 870 72 × 2 = 0 + 0.328 650 338 155 252 615 741 44;
  • 105) 0.328 650 338 155 252 615 741 44 × 2 = 0 + 0.657 300 676 310 505 231 482 88;
  • 106) 0.657 300 676 310 505 231 482 88 × 2 = 1 + 0.314 601 352 621 010 462 965 76;
  • 107) 0.314 601 352 621 010 462 965 76 × 2 = 0 + 0.629 202 705 242 020 925 931 52;
  • 108) 0.629 202 705 242 020 925 931 52 × 2 = 1 + 0.258 405 410 484 041 851 863 04;
  • 109) 0.258 405 410 484 041 851 863 04 × 2 = 0 + 0.516 810 820 968 083 703 726 08;
  • 110) 0.516 810 820 968 083 703 726 08 × 2 = 1 + 0.033 621 641 936 167 407 452 16;
  • 111) 0.033 621 641 936 167 407 452 16 × 2 = 0 + 0.067 243 283 872 334 814 904 32;
  • 112) 0.067 243 283 872 334 814 904 32 × 2 = 0 + 0.134 486 567 744 669 629 808 64;
  • 113) 0.134 486 567 744 669 629 808 64 × 2 = 0 + 0.268 973 135 489 339 259 617 28;
  • 114) 0.268 973 135 489 339 259 617 28 × 2 = 0 + 0.537 946 270 978 678 519 234 56;
  • 115) 0.537 946 270 978 678 519 234 56 × 2 = 1 + 0.075 892 541 957 357 038 469 12;
  • 116) 0.075 892 541 957 357 038 469 12 × 2 = 0 + 0.151 785 083 914 714 076 938 24;
  • 117) 0.151 785 083 914 714 076 938 24 × 2 = 0 + 0.303 570 167 829 428 153 876 48;
  • 118) 0.303 570 167 829 428 153 876 48 × 2 = 0 + 0.607 140 335 658 856 307 752 96;
  • 119) 0.607 140 335 658 856 307 752 96 × 2 = 1 + 0.214 280 671 317 712 615 505 92;
  • 120) 0.214 280 671 317 712 615 505 92 × 2 = 0 + 0.428 561 342 635 425 231 011 84;
  • 121) 0.428 561 342 635 425 231 011 84 × 2 = 0 + 0.857 122 685 270 850 462 023 68;
  • 122) 0.857 122 685 270 850 462 023 68 × 2 = 1 + 0.714 245 370 541 700 924 047 36;
  • 123) 0.714 245 370 541 700 924 047 36 × 2 = 1 + 0.428 490 741 083 401 848 094 72;
  • 124) 0.428 490 741 083 401 848 094 72 × 2 = 0 + 0.856 981 482 166 803 696 189 44;
  • 125) 0.856 981 482 166 803 696 189 44 × 2 = 1 + 0.713 962 964 333 607 392 378 88;
  • 126) 0.713 962 964 333 607 392 378 88 × 2 = 1 + 0.427 925 928 667 214 784 757 76;

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:


0.000 000 000 000 000 000 000 09(10) =


0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 1100 1100 1101 1010 0110 1010 0000 0101 0100 0010 0010 0110 11(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 09(10) =


0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 1100 1100 1101 1010 0110 1010 0000 0101 0100 0010 0010 0110 11(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 74 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 09(10) =


0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 1100 1100 1101 1010 0110 1010 0000 0101 0100 0010 0010 0110 11(2) =


0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 1100 1100 1101 1010 0110 1010 0000 0101 0100 0010 0010 0110 11(2) × 20 =


1.1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011(2) × 2-74


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): -74


Mantissa (not normalized):
1.1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


-74 + 2(11-1) - 1 =


(-74 + 1 023)(10) =


949(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;
  • 949 ÷ 2 = 474 + 1;
  • 474 ÷ 2 = 237 + 0;
  • 237 ÷ 2 = 118 + 1;
  • 118 ÷ 2 = 59 + 0;
  • 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) =


949(10) =


011 1011 0101(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. 1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011 =


1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011


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 0101


Mantissa (52 bits) =
1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011


The base ten decimal number 0.000 000 000 000 000 000 000 09 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 0101 - 1011 0011 0011 0110 1001 1010 1000 0001 0101 0000 1000 1001 1011

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