0.000 000 000 000 000 000 008 534 24 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 24(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
0.000 000 000 000 000 000 008 534 24(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: 0.000 000 000 000 000 000 008 534 24.

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) 0.000 000 000 000 000 000 008 534 24 × 2 = 0 + 0.000 000 000 000 000 000 017 068 48;
  • 2) 0.000 000 000 000 000 000 017 068 48 × 2 = 0 + 0.000 000 000 000 000 000 034 136 96;
  • 3) 0.000 000 000 000 000 000 034 136 96 × 2 = 0 + 0.000 000 000 000 000 000 068 273 92;
  • 4) 0.000 000 000 000 000 000 068 273 92 × 2 = 0 + 0.000 000 000 000 000 000 136 547 84;
  • 5) 0.000 000 000 000 000 000 136 547 84 × 2 = 0 + 0.000 000 000 000 000 000 273 095 68;
  • 6) 0.000 000 000 000 000 000 273 095 68 × 2 = 0 + 0.000 000 000 000 000 000 546 191 36;
  • 7) 0.000 000 000 000 000 000 546 191 36 × 2 = 0 + 0.000 000 000 000 000 001 092 382 72;
  • 8) 0.000 000 000 000 000 001 092 382 72 × 2 = 0 + 0.000 000 000 000 000 002 184 765 44;
  • 9) 0.000 000 000 000 000 002 184 765 44 × 2 = 0 + 0.000 000 000 000 000 004 369 530 88;
  • 10) 0.000 000 000 000 000 004 369 530 88 × 2 = 0 + 0.000 000 000 000 000 008 739 061 76;
  • 11) 0.000 000 000 000 000 008 739 061 76 × 2 = 0 + 0.000 000 000 000 000 017 478 123 52;
  • 12) 0.000 000 000 000 000 017 478 123 52 × 2 = 0 + 0.000 000 000 000 000 034 956 247 04;
  • 13) 0.000 000 000 000 000 034 956 247 04 × 2 = 0 + 0.000 000 000 000 000 069 912 494 08;
  • 14) 0.000 000 000 000 000 069 912 494 08 × 2 = 0 + 0.000 000 000 000 000 139 824 988 16;
  • 15) 0.000 000 000 000 000 139 824 988 16 × 2 = 0 + 0.000 000 000 000 000 279 649 976 32;
  • 16) 0.000 000 000 000 000 279 649 976 32 × 2 = 0 + 0.000 000 000 000 000 559 299 952 64;
  • 17) 0.000 000 000 000 000 559 299 952 64 × 2 = 0 + 0.000 000 000 000 001 118 599 905 28;
  • 18) 0.000 000 000 000 001 118 599 905 28 × 2 = 0 + 0.000 000 000 000 002 237 199 810 56;
  • 19) 0.000 000 000 000 002 237 199 810 56 × 2 = 0 + 0.000 000 000 000 004 474 399 621 12;
  • 20) 0.000 000 000 000 004 474 399 621 12 × 2 = 0 + 0.000 000 000 000 008 948 799 242 24;
  • 21) 0.000 000 000 000 008 948 799 242 24 × 2 = 0 + 0.000 000 000 000 017 897 598 484 48;
  • 22) 0.000 000 000 000 017 897 598 484 48 × 2 = 0 + 0.000 000 000 000 035 795 196 968 96;
  • 23) 0.000 000 000 000 035 795 196 968 96 × 2 = 0 + 0.000 000 000 000 071 590 393 937 92;
  • 24) 0.000 000 000 000 071 590 393 937 92 × 2 = 0 + 0.000 000 000 000 143 180 787 875 84;
  • 25) 0.000 000 000 000 143 180 787 875 84 × 2 = 0 + 0.000 000 000 000 286 361 575 751 68;
  • 26) 0.000 000 000 000 286 361 575 751 68 × 2 = 0 + 0.000 000 000 000 572 723 151 503 36;
  • 27) 0.000 000 000 000 572 723 151 503 36 × 2 = 0 + 0.000 000 000 001 145 446 303 006 72;
  • 28) 0.000 000 000 001 145 446 303 006 72 × 2 = 0 + 0.000 000 000 002 290 892 606 013 44;
  • 29) 0.000 000 000 002 290 892 606 013 44 × 2 = 0 + 0.000 000 000 004 581 785 212 026 88;
  • 30) 0.000 000 000 004 581 785 212 026 88 × 2 = 0 + 0.000 000 000 009 163 570 424 053 76;
  • 31) 0.000 000 000 009 163 570 424 053 76 × 2 = 0 + 0.000 000 000 018 327 140 848 107 52;
  • 32) 0.000 000 000 018 327 140 848 107 52 × 2 = 0 + 0.000 000 000 036 654 281 696 215 04;
  • 33) 0.000 000 000 036 654 281 696 215 04 × 2 = 0 + 0.000 000 000 073 308 563 392 430 08;
  • 34) 0.000 000 000 073 308 563 392 430 08 × 2 = 0 + 0.000 000 000 146 617 126 784 860 16;
  • 35) 0.000 000 000 146 617 126 784 860 16 × 2 = 0 + 0.000 000 000 293 234 253 569 720 32;
  • 36) 0.000 000 000 293 234 253 569 720 32 × 2 = 0 + 0.000 000 000 586 468 507 139 440 64;
  • 37) 0.000 000 000 586 468 507 139 440 64 × 2 = 0 + 0.000 000 001 172 937 014 278 881 28;
  • 38) 0.000 000 001 172 937 014 278 881 28 × 2 = 0 + 0.000 000 002 345 874 028 557 762 56;
  • 39) 0.000 000 002 345 874 028 557 762 56 × 2 = 0 + 0.000 000 004 691 748 057 115 525 12;
  • 40) 0.000 000 004 691 748 057 115 525 12 × 2 = 0 + 0.000 000 009 383 496 114 231 050 24;
  • 41) 0.000 000 009 383 496 114 231 050 24 × 2 = 0 + 0.000 000 018 766 992 228 462 100 48;
  • 42) 0.000 000 018 766 992 228 462 100 48 × 2 = 0 + 0.000 000 037 533 984 456 924 200 96;
  • 43) 0.000 000 037 533 984 456 924 200 96 × 2 = 0 + 0.000 000 075 067 968 913 848 401 92;
  • 44) 0.000 000 075 067 968 913 848 401 92 × 2 = 0 + 0.000 000 150 135 937 827 696 803 84;
  • 45) 0.000 000 150 135 937 827 696 803 84 × 2 = 0 + 0.000 000 300 271 875 655 393 607 68;
  • 46) 0.000 000 300 271 875 655 393 607 68 × 2 = 0 + 0.000 000 600 543 751 310 787 215 36;
  • 47) 0.000 000 600 543 751 310 787 215 36 × 2 = 0 + 0.000 001 201 087 502 621 574 430 72;
  • 48) 0.000 001 201 087 502 621 574 430 72 × 2 = 0 + 0.000 002 402 175 005 243 148 861 44;
  • 49) 0.000 002 402 175 005 243 148 861 44 × 2 = 0 + 0.000 004 804 350 010 486 297 722 88;
  • 50) 0.000 004 804 350 010 486 297 722 88 × 2 = 0 + 0.000 009 608 700 020 972 595 445 76;
  • 51) 0.000 009 608 700 020 972 595 445 76 × 2 = 0 + 0.000 019 217 400 041 945 190 891 52;
  • 52) 0.000 019 217 400 041 945 190 891 52 × 2 = 0 + 0.000 038 434 800 083 890 381 783 04;
  • 53) 0.000 038 434 800 083 890 381 783 04 × 2 = 0 + 0.000 076 869 600 167 780 763 566 08;
  • 54) 0.000 076 869 600 167 780 763 566 08 × 2 = 0 + 0.000 153 739 200 335 561 527 132 16;
  • 55) 0.000 153 739 200 335 561 527 132 16 × 2 = 0 + 0.000 307 478 400 671 123 054 264 32;
  • 56) 0.000 307 478 400 671 123 054 264 32 × 2 = 0 + 0.000 614 956 801 342 246 108 528 64;
  • 57) 0.000 614 956 801 342 246 108 528 64 × 2 = 0 + 0.001 229 913 602 684 492 217 057 28;
  • 58) 0.001 229 913 602 684 492 217 057 28 × 2 = 0 + 0.002 459 827 205 368 984 434 114 56;
  • 59) 0.002 459 827 205 368 984 434 114 56 × 2 = 0 + 0.004 919 654 410 737 968 868 229 12;
  • 60) 0.004 919 654 410 737 968 868 229 12 × 2 = 0 + 0.009 839 308 821 475 937 736 458 24;
  • 61) 0.009 839 308 821 475 937 736 458 24 × 2 = 0 + 0.019 678 617 642 951 875 472 916 48;
  • 62) 0.019 678 617 642 951 875 472 916 48 × 2 = 0 + 0.039 357 235 285 903 750 945 832 96;
  • 63) 0.039 357 235 285 903 750 945 832 96 × 2 = 0 + 0.078 714 470 571 807 501 891 665 92;
  • 64) 0.078 714 470 571 807 501 891 665 92 × 2 = 0 + 0.157 428 941 143 615 003 783 331 84;
  • 65) 0.157 428 941 143 615 003 783 331 84 × 2 = 0 + 0.314 857 882 287 230 007 566 663 68;
  • 66) 0.314 857 882 287 230 007 566 663 68 × 2 = 0 + 0.629 715 764 574 460 015 133 327 36;
  • 67) 0.629 715 764 574 460 015 133 327 36 × 2 = 1 + 0.259 431 529 148 920 030 266 654 72;
  • 68) 0.259 431 529 148 920 030 266 654 72 × 2 = 0 + 0.518 863 058 297 840 060 533 309 44;
  • 69) 0.518 863 058 297 840 060 533 309 44 × 2 = 1 + 0.037 726 116 595 680 121 066 618 88;
  • 70) 0.037 726 116 595 680 121 066 618 88 × 2 = 0 + 0.075 452 233 191 360 242 133 237 76;
  • 71) 0.075 452 233 191 360 242 133 237 76 × 2 = 0 + 0.150 904 466 382 720 484 266 475 52;
  • 72) 0.150 904 466 382 720 484 266 475 52 × 2 = 0 + 0.301 808 932 765 440 968 532 951 04;
  • 73) 0.301 808 932 765 440 968 532 951 04 × 2 = 0 + 0.603 617 865 530 881 937 065 902 08;
  • 74) 0.603 617 865 530 881 937 065 902 08 × 2 = 1 + 0.207 235 731 061 763 874 131 804 16;
  • 75) 0.207 235 731 061 763 874 131 804 16 × 2 = 0 + 0.414 471 462 123 527 748 263 608 32;
  • 76) 0.414 471 462 123 527 748 263 608 32 × 2 = 0 + 0.828 942 924 247 055 496 527 216 64;
  • 77) 0.828 942 924 247 055 496 527 216 64 × 2 = 1 + 0.657 885 848 494 110 993 054 433 28;
  • 78) 0.657 885 848 494 110 993 054 433 28 × 2 = 1 + 0.315 771 696 988 221 986 108 866 56;
  • 79) 0.315 771 696 988 221 986 108 866 56 × 2 = 0 + 0.631 543 393 976 443 972 217 733 12;
  • 80) 0.631 543 393 976 443 972 217 733 12 × 2 = 1 + 0.263 086 787 952 887 944 435 466 24;
  • 81) 0.263 086 787 952 887 944 435 466 24 × 2 = 0 + 0.526 173 575 905 775 888 870 932 48;
  • 82) 0.526 173 575 905 775 888 870 932 48 × 2 = 1 + 0.052 347 151 811 551 777 741 864 96;
  • 83) 0.052 347 151 811 551 777 741 864 96 × 2 = 0 + 0.104 694 303 623 103 555 483 729 92;
  • 84) 0.104 694 303 623 103 555 483 729 92 × 2 = 0 + 0.209 388 607 246 207 110 967 459 84;
  • 85) 0.209 388 607 246 207 110 967 459 84 × 2 = 0 + 0.418 777 214 492 414 221 934 919 68;
  • 86) 0.418 777 214 492 414 221 934 919 68 × 2 = 0 + 0.837 554 428 984 828 443 869 839 36;
  • 87) 0.837 554 428 984 828 443 869 839 36 × 2 = 1 + 0.675 108 857 969 656 887 739 678 72;
  • 88) 0.675 108 857 969 656 887 739 678 72 × 2 = 1 + 0.350 217 715 939 313 775 479 357 44;
  • 89) 0.350 217 715 939 313 775 479 357 44 × 2 = 0 + 0.700 435 431 878 627 550 958 714 88;
  • 90) 0.700 435 431 878 627 550 958 714 88 × 2 = 1 + 0.400 870 863 757 255 101 917 429 76;
  • 91) 0.400 870 863 757 255 101 917 429 76 × 2 = 0 + 0.801 741 727 514 510 203 834 859 52;
  • 92) 0.801 741 727 514 510 203 834 859 52 × 2 = 1 + 0.603 483 455 029 020 407 669 719 04;
  • 93) 0.603 483 455 029 020 407 669 719 04 × 2 = 1 + 0.206 966 910 058 040 815 339 438 08;
  • 94) 0.206 966 910 058 040 815 339 438 08 × 2 = 0 + 0.413 933 820 116 081 630 678 876 16;
  • 95) 0.413 933 820 116 081 630 678 876 16 × 2 = 0 + 0.827 867 640 232 163 261 357 752 32;
  • 96) 0.827 867 640 232 163 261 357 752 32 × 2 = 1 + 0.655 735 280 464 326 522 715 504 64;
  • 97) 0.655 735 280 464 326 522 715 504 64 × 2 = 1 + 0.311 470 560 928 653 045 431 009 28;
  • 98) 0.311 470 560 928 653 045 431 009 28 × 2 = 0 + 0.622 941 121 857 306 090 862 018 56;
  • 99) 0.622 941 121 857 306 090 862 018 56 × 2 = 1 + 0.245 882 243 714 612 181 724 037 12;
  • 100) 0.245 882 243 714 612 181 724 037 12 × 2 = 0 + 0.491 764 487 429 224 363 448 074 24;
  • 101) 0.491 764 487 429 224 363 448 074 24 × 2 = 0 + 0.983 528 974 858 448 726 896 148 48;
  • 102) 0.983 528 974 858 448 726 896 148 48 × 2 = 1 + 0.967 057 949 716 897 453 792 296 96;
  • 103) 0.967 057 949 716 897 453 792 296 96 × 2 = 1 + 0.934 115 899 433 794 907 584 593 92;
  • 104) 0.934 115 899 433 794 907 584 593 92 × 2 = 1 + 0.868 231 798 867 589 815 169 187 84;
  • 105) 0.868 231 798 867 589 815 169 187 84 × 2 = 1 + 0.736 463 597 735 179 630 338 375 68;
  • 106) 0.736 463 597 735 179 630 338 375 68 × 2 = 1 + 0.472 927 195 470 359 260 676 751 36;
  • 107) 0.472 927 195 470 359 260 676 751 36 × 2 = 0 + 0.945 854 390 940 718 521 353 502 72;
  • 108) 0.945 854 390 940 718 521 353 502 72 × 2 = 1 + 0.891 708 781 881 437 042 707 005 44;
  • 109) 0.891 708 781 881 437 042 707 005 44 × 2 = 1 + 0.783 417 563 762 874 085 414 010 88;
  • 110) 0.783 417 563 762 874 085 414 010 88 × 2 = 1 + 0.566 835 127 525 748 170 828 021 76;
  • 111) 0.566 835 127 525 748 170 828 021 76 × 2 = 1 + 0.133 670 255 051 496 341 656 043 52;
  • 112) 0.133 670 255 051 496 341 656 043 52 × 2 = 0 + 0.267 340 510 102 992 683 312 087 04;
  • 113) 0.267 340 510 102 992 683 312 087 04 × 2 = 0 + 0.534 681 020 205 985 366 624 174 08;
  • 114) 0.534 681 020 205 985 366 624 174 08 × 2 = 1 + 0.069 362 040 411 970 733 248 348 16;
  • 115) 0.069 362 040 411 970 733 248 348 16 × 2 = 0 + 0.138 724 080 823 941 466 496 696 32;
  • 116) 0.138 724 080 823 941 466 496 696 32 × 2 = 0 + 0.277 448 161 647 882 932 993 392 64;
  • 117) 0.277 448 161 647 882 932 993 392 64 × 2 = 0 + 0.554 896 323 295 765 865 986 785 28;
  • 118) 0.554 896 323 295 765 865 986 785 28 × 2 = 1 + 0.109 792 646 591 531 731 973 570 56;
  • 119) 0.109 792 646 591 531 731 973 570 56 × 2 = 0 + 0.219 585 293 183 063 463 947 141 12;

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:


0.000 000 000 000 000 000 008 534 24(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0100 0011 0101 1001 1010 0111 1101 1110 0100 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 24(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0100 0011 0101 1001 1010 0111 1101 1110 0100 010(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:


0.000 000 000 000 000 000 008 534 24(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0100 0011 0101 1001 1010 0111 1101 1110 0100 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0100 0011 0101 1001 1010 0111 1101 1110 0100 010(2) × 20 =

1.0100 0010 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010(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 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010


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 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010 =

0100 0010 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010


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 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010


Decimal number 0.000 000 000 000 000 000 008 534 24 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1010 0001 1010 1100 1101 0011 1110 1111 0010 0010

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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