0.000 000 000 000 000 000 008 536 71 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 71(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
0.000 000 000 000 000 000 008 536 71(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: 0.000 000 000 000 000 000 008 536 71.

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) 0.000 000 000 000 000 000 008 536 71 × 2 = 0 + 0.000 000 000 000 000 000 017 073 42;
  • 2) 0.000 000 000 000 000 000 017 073 42 × 2 = 0 + 0.000 000 000 000 000 000 034 146 84;
  • 3) 0.000 000 000 000 000 000 034 146 84 × 2 = 0 + 0.000 000 000 000 000 000 068 293 68;
  • 4) 0.000 000 000 000 000 000 068 293 68 × 2 = 0 + 0.000 000 000 000 000 000 136 587 36;
  • 5) 0.000 000 000 000 000 000 136 587 36 × 2 = 0 + 0.000 000 000 000 000 000 273 174 72;
  • 6) 0.000 000 000 000 000 000 273 174 72 × 2 = 0 + 0.000 000 000 000 000 000 546 349 44;
  • 7) 0.000 000 000 000 000 000 546 349 44 × 2 = 0 + 0.000 000 000 000 000 001 092 698 88;
  • 8) 0.000 000 000 000 000 001 092 698 88 × 2 = 0 + 0.000 000 000 000 000 002 185 397 76;
  • 9) 0.000 000 000 000 000 002 185 397 76 × 2 = 0 + 0.000 000 000 000 000 004 370 795 52;
  • 10) 0.000 000 000 000 000 004 370 795 52 × 2 = 0 + 0.000 000 000 000 000 008 741 591 04;
  • 11) 0.000 000 000 000 000 008 741 591 04 × 2 = 0 + 0.000 000 000 000 000 017 483 182 08;
  • 12) 0.000 000 000 000 000 017 483 182 08 × 2 = 0 + 0.000 000 000 000 000 034 966 364 16;
  • 13) 0.000 000 000 000 000 034 966 364 16 × 2 = 0 + 0.000 000 000 000 000 069 932 728 32;
  • 14) 0.000 000 000 000 000 069 932 728 32 × 2 = 0 + 0.000 000 000 000 000 139 865 456 64;
  • 15) 0.000 000 000 000 000 139 865 456 64 × 2 = 0 + 0.000 000 000 000 000 279 730 913 28;
  • 16) 0.000 000 000 000 000 279 730 913 28 × 2 = 0 + 0.000 000 000 000 000 559 461 826 56;
  • 17) 0.000 000 000 000 000 559 461 826 56 × 2 = 0 + 0.000 000 000 000 001 118 923 653 12;
  • 18) 0.000 000 000 000 001 118 923 653 12 × 2 = 0 + 0.000 000 000 000 002 237 847 306 24;
  • 19) 0.000 000 000 000 002 237 847 306 24 × 2 = 0 + 0.000 000 000 000 004 475 694 612 48;
  • 20) 0.000 000 000 000 004 475 694 612 48 × 2 = 0 + 0.000 000 000 000 008 951 389 224 96;
  • 21) 0.000 000 000 000 008 951 389 224 96 × 2 = 0 + 0.000 000 000 000 017 902 778 449 92;
  • 22) 0.000 000 000 000 017 902 778 449 92 × 2 = 0 + 0.000 000 000 000 035 805 556 899 84;
  • 23) 0.000 000 000 000 035 805 556 899 84 × 2 = 0 + 0.000 000 000 000 071 611 113 799 68;
  • 24) 0.000 000 000 000 071 611 113 799 68 × 2 = 0 + 0.000 000 000 000 143 222 227 599 36;
  • 25) 0.000 000 000 000 143 222 227 599 36 × 2 = 0 + 0.000 000 000 000 286 444 455 198 72;
  • 26) 0.000 000 000 000 286 444 455 198 72 × 2 = 0 + 0.000 000 000 000 572 888 910 397 44;
  • 27) 0.000 000 000 000 572 888 910 397 44 × 2 = 0 + 0.000 000 000 001 145 777 820 794 88;
  • 28) 0.000 000 000 001 145 777 820 794 88 × 2 = 0 + 0.000 000 000 002 291 555 641 589 76;
  • 29) 0.000 000 000 002 291 555 641 589 76 × 2 = 0 + 0.000 000 000 004 583 111 283 179 52;
  • 30) 0.000 000 000 004 583 111 283 179 52 × 2 = 0 + 0.000 000 000 009 166 222 566 359 04;
  • 31) 0.000 000 000 009 166 222 566 359 04 × 2 = 0 + 0.000 000 000 018 332 445 132 718 08;
  • 32) 0.000 000 000 018 332 445 132 718 08 × 2 = 0 + 0.000 000 000 036 664 890 265 436 16;
  • 33) 0.000 000 000 036 664 890 265 436 16 × 2 = 0 + 0.000 000 000 073 329 780 530 872 32;
  • 34) 0.000 000 000 073 329 780 530 872 32 × 2 = 0 + 0.000 000 000 146 659 561 061 744 64;
  • 35) 0.000 000 000 146 659 561 061 744 64 × 2 = 0 + 0.000 000 000 293 319 122 123 489 28;
  • 36) 0.000 000 000 293 319 122 123 489 28 × 2 = 0 + 0.000 000 000 586 638 244 246 978 56;
  • 37) 0.000 000 000 586 638 244 246 978 56 × 2 = 0 + 0.000 000 001 173 276 488 493 957 12;
  • 38) 0.000 000 001 173 276 488 493 957 12 × 2 = 0 + 0.000 000 002 346 552 976 987 914 24;
  • 39) 0.000 000 002 346 552 976 987 914 24 × 2 = 0 + 0.000 000 004 693 105 953 975 828 48;
  • 40) 0.000 000 004 693 105 953 975 828 48 × 2 = 0 + 0.000 000 009 386 211 907 951 656 96;
  • 41) 0.000 000 009 386 211 907 951 656 96 × 2 = 0 + 0.000 000 018 772 423 815 903 313 92;
  • 42) 0.000 000 018 772 423 815 903 313 92 × 2 = 0 + 0.000 000 037 544 847 631 806 627 84;
  • 43) 0.000 000 037 544 847 631 806 627 84 × 2 = 0 + 0.000 000 075 089 695 263 613 255 68;
  • 44) 0.000 000 075 089 695 263 613 255 68 × 2 = 0 + 0.000 000 150 179 390 527 226 511 36;
  • 45) 0.000 000 150 179 390 527 226 511 36 × 2 = 0 + 0.000 000 300 358 781 054 453 022 72;
  • 46) 0.000 000 300 358 781 054 453 022 72 × 2 = 0 + 0.000 000 600 717 562 108 906 045 44;
  • 47) 0.000 000 600 717 562 108 906 045 44 × 2 = 0 + 0.000 001 201 435 124 217 812 090 88;
  • 48) 0.000 001 201 435 124 217 812 090 88 × 2 = 0 + 0.000 002 402 870 248 435 624 181 76;
  • 49) 0.000 002 402 870 248 435 624 181 76 × 2 = 0 + 0.000 004 805 740 496 871 248 363 52;
  • 50) 0.000 004 805 740 496 871 248 363 52 × 2 = 0 + 0.000 009 611 480 993 742 496 727 04;
  • 51) 0.000 009 611 480 993 742 496 727 04 × 2 = 0 + 0.000 019 222 961 987 484 993 454 08;
  • 52) 0.000 019 222 961 987 484 993 454 08 × 2 = 0 + 0.000 038 445 923 974 969 986 908 16;
  • 53) 0.000 038 445 923 974 969 986 908 16 × 2 = 0 + 0.000 076 891 847 949 939 973 816 32;
  • 54) 0.000 076 891 847 949 939 973 816 32 × 2 = 0 + 0.000 153 783 695 899 879 947 632 64;
  • 55) 0.000 153 783 695 899 879 947 632 64 × 2 = 0 + 0.000 307 567 391 799 759 895 265 28;
  • 56) 0.000 307 567 391 799 759 895 265 28 × 2 = 0 + 0.000 615 134 783 599 519 790 530 56;
  • 57) 0.000 615 134 783 599 519 790 530 56 × 2 = 0 + 0.001 230 269 567 199 039 581 061 12;
  • 58) 0.001 230 269 567 199 039 581 061 12 × 2 = 0 + 0.002 460 539 134 398 079 162 122 24;
  • 59) 0.002 460 539 134 398 079 162 122 24 × 2 = 0 + 0.004 921 078 268 796 158 324 244 48;
  • 60) 0.004 921 078 268 796 158 324 244 48 × 2 = 0 + 0.009 842 156 537 592 316 648 488 96;
  • 61) 0.009 842 156 537 592 316 648 488 96 × 2 = 0 + 0.019 684 313 075 184 633 296 977 92;
  • 62) 0.019 684 313 075 184 633 296 977 92 × 2 = 0 + 0.039 368 626 150 369 266 593 955 84;
  • 63) 0.039 368 626 150 369 266 593 955 84 × 2 = 0 + 0.078 737 252 300 738 533 187 911 68;
  • 64) 0.078 737 252 300 738 533 187 911 68 × 2 = 0 + 0.157 474 504 601 477 066 375 823 36;
  • 65) 0.157 474 504 601 477 066 375 823 36 × 2 = 0 + 0.314 949 009 202 954 132 751 646 72;
  • 66) 0.314 949 009 202 954 132 751 646 72 × 2 = 0 + 0.629 898 018 405 908 265 503 293 44;
  • 67) 0.629 898 018 405 908 265 503 293 44 × 2 = 1 + 0.259 796 036 811 816 531 006 586 88;
  • 68) 0.259 796 036 811 816 531 006 586 88 × 2 = 0 + 0.519 592 073 623 633 062 013 173 76;
  • 69) 0.519 592 073 623 633 062 013 173 76 × 2 = 1 + 0.039 184 147 247 266 124 026 347 52;
  • 70) 0.039 184 147 247 266 124 026 347 52 × 2 = 0 + 0.078 368 294 494 532 248 052 695 04;
  • 71) 0.078 368 294 494 532 248 052 695 04 × 2 = 0 + 0.156 736 588 989 064 496 105 390 08;
  • 72) 0.156 736 588 989 064 496 105 390 08 × 2 = 0 + 0.313 473 177 978 128 992 210 780 16;
  • 73) 0.313 473 177 978 128 992 210 780 16 × 2 = 0 + 0.626 946 355 956 257 984 421 560 32;
  • 74) 0.626 946 355 956 257 984 421 560 32 × 2 = 1 + 0.253 892 711 912 515 968 843 120 64;
  • 75) 0.253 892 711 912 515 968 843 120 64 × 2 = 0 + 0.507 785 423 825 031 937 686 241 28;
  • 76) 0.507 785 423 825 031 937 686 241 28 × 2 = 1 + 0.015 570 847 650 063 875 372 482 56;
  • 77) 0.015 570 847 650 063 875 372 482 56 × 2 = 0 + 0.031 141 695 300 127 750 744 965 12;
  • 78) 0.031 141 695 300 127 750 744 965 12 × 2 = 0 + 0.062 283 390 600 255 501 489 930 24;
  • 79) 0.062 283 390 600 255 501 489 930 24 × 2 = 0 + 0.124 566 781 200 511 002 979 860 48;
  • 80) 0.124 566 781 200 511 002 979 860 48 × 2 = 0 + 0.249 133 562 401 022 005 959 720 96;
  • 81) 0.249 133 562 401 022 005 959 720 96 × 2 = 0 + 0.498 267 124 802 044 011 919 441 92;
  • 82) 0.498 267 124 802 044 011 919 441 92 × 2 = 0 + 0.996 534 249 604 088 023 838 883 84;
  • 83) 0.996 534 249 604 088 023 838 883 84 × 2 = 1 + 0.993 068 499 208 176 047 677 767 68;
  • 84) 0.993 068 499 208 176 047 677 767 68 × 2 = 1 + 0.986 136 998 416 352 095 355 535 36;
  • 85) 0.986 136 998 416 352 095 355 535 36 × 2 = 1 + 0.972 273 996 832 704 190 711 070 72;
  • 86) 0.972 273 996 832 704 190 711 070 72 × 2 = 1 + 0.944 547 993 665 408 381 422 141 44;
  • 87) 0.944 547 993 665 408 381 422 141 44 × 2 = 1 + 0.889 095 987 330 816 762 844 282 88;
  • 88) 0.889 095 987 330 816 762 844 282 88 × 2 = 1 + 0.778 191 974 661 633 525 688 565 76;
  • 89) 0.778 191 974 661 633 525 688 565 76 × 2 = 1 + 0.556 383 949 323 267 051 377 131 52;
  • 90) 0.556 383 949 323 267 051 377 131 52 × 2 = 1 + 0.112 767 898 646 534 102 754 263 04;
  • 91) 0.112 767 898 646 534 102 754 263 04 × 2 = 0 + 0.225 535 797 293 068 205 508 526 08;
  • 92) 0.225 535 797 293 068 205 508 526 08 × 2 = 0 + 0.451 071 594 586 136 411 017 052 16;
  • 93) 0.451 071 594 586 136 411 017 052 16 × 2 = 0 + 0.902 143 189 172 272 822 034 104 32;
  • 94) 0.902 143 189 172 272 822 034 104 32 × 2 = 1 + 0.804 286 378 344 545 644 068 208 64;
  • 95) 0.804 286 378 344 545 644 068 208 64 × 2 = 1 + 0.608 572 756 689 091 288 136 417 28;
  • 96) 0.608 572 756 689 091 288 136 417 28 × 2 = 1 + 0.217 145 513 378 182 576 272 834 56;
  • 97) 0.217 145 513 378 182 576 272 834 56 × 2 = 0 + 0.434 291 026 756 365 152 545 669 12;
  • 98) 0.434 291 026 756 365 152 545 669 12 × 2 = 0 + 0.868 582 053 512 730 305 091 338 24;
  • 99) 0.868 582 053 512 730 305 091 338 24 × 2 = 1 + 0.737 164 107 025 460 610 182 676 48;
  • 100) 0.737 164 107 025 460 610 182 676 48 × 2 = 1 + 0.474 328 214 050 921 220 365 352 96;
  • 101) 0.474 328 214 050 921 220 365 352 96 × 2 = 0 + 0.948 656 428 101 842 440 730 705 92;
  • 102) 0.948 656 428 101 842 440 730 705 92 × 2 = 1 + 0.897 312 856 203 684 881 461 411 84;
  • 103) 0.897 312 856 203 684 881 461 411 84 × 2 = 1 + 0.794 625 712 407 369 762 922 823 68;
  • 104) 0.794 625 712 407 369 762 922 823 68 × 2 = 1 + 0.589 251 424 814 739 525 845 647 36;
  • 105) 0.589 251 424 814 739 525 845 647 36 × 2 = 1 + 0.178 502 849 629 479 051 691 294 72;
  • 106) 0.178 502 849 629 479 051 691 294 72 × 2 = 0 + 0.357 005 699 258 958 103 382 589 44;
  • 107) 0.357 005 699 258 958 103 382 589 44 × 2 = 0 + 0.714 011 398 517 916 206 765 178 88;
  • 108) 0.714 011 398 517 916 206 765 178 88 × 2 = 1 + 0.428 022 797 035 832 413 530 357 76;
  • 109) 0.428 022 797 035 832 413 530 357 76 × 2 = 0 + 0.856 045 594 071 664 827 060 715 52;
  • 110) 0.856 045 594 071 664 827 060 715 52 × 2 = 1 + 0.712 091 188 143 329 654 121 431 04;
  • 111) 0.712 091 188 143 329 654 121 431 04 × 2 = 1 + 0.424 182 376 286 659 308 242 862 08;
  • 112) 0.424 182 376 286 659 308 242 862 08 × 2 = 0 + 0.848 364 752 573 318 616 485 724 16;
  • 113) 0.848 364 752 573 318 616 485 724 16 × 2 = 1 + 0.696 729 505 146 637 232 971 448 32;
  • 114) 0.696 729 505 146 637 232 971 448 32 × 2 = 1 + 0.393 459 010 293 274 465 942 896 64;
  • 115) 0.393 459 010 293 274 465 942 896 64 × 2 = 0 + 0.786 918 020 586 548 931 885 793 28;
  • 116) 0.786 918 020 586 548 931 885 793 28 × 2 = 1 + 0.573 836 041 173 097 863 771 586 56;
  • 117) 0.573 836 041 173 097 863 771 586 56 × 2 = 1 + 0.147 672 082 346 195 727 543 173 12;
  • 118) 0.147 672 082 346 195 727 543 173 12 × 2 = 0 + 0.295 344 164 692 391 455 086 346 24;
  • 119) 0.295 344 164 692 391 455 086 346 24 × 2 = 0 + 0.590 688 329 384 782 910 172 692 48;

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:


0.000 000 000 000 000 000 008 536 71(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0011 1111 1100 0111 0011 0111 1001 0110 1101 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 71(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0011 1111 1100 0111 0011 0111 1001 0110 1101 100(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:


0.000 000 000 000 000 000 008 536 71(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0011 1111 1100 0111 0011 0111 1001 0110 1101 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0011 1111 1100 0111 0011 0111 1001 0110 1101 100(2) × 20 =

1.0100 0010 1000 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100(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 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100


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 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100 =

0100 0010 1000 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100


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 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100


Decimal number 0.000 000 000 000 000 000 008 536 71 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0001 1111 1110 0011 1001 1011 1100 1011 0110 1100

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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