-0.000 000 000 000 000 004 573 690 045 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 004 573 690 045 8(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
-0.000 000 000 000 000 004 573 690 045 8(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. Start with the positive version of the number:

|-0.000 000 000 000 000 004 573 690 045 8| = 0.000 000 000 000 000 004 573 690 045 8


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 004 573 690 045 8.

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) 0.000 000 000 000 000 004 573 690 045 8 × 2 = 0 + 0.000 000 000 000 000 009 147 380 091 6;
  • 2) 0.000 000 000 000 000 009 147 380 091 6 × 2 = 0 + 0.000 000 000 000 000 018 294 760 183 2;
  • 3) 0.000 000 000 000 000 018 294 760 183 2 × 2 = 0 + 0.000 000 000 000 000 036 589 520 366 4;
  • 4) 0.000 000 000 000 000 036 589 520 366 4 × 2 = 0 + 0.000 000 000 000 000 073 179 040 732 8;
  • 5) 0.000 000 000 000 000 073 179 040 732 8 × 2 = 0 + 0.000 000 000 000 000 146 358 081 465 6;
  • 6) 0.000 000 000 000 000 146 358 081 465 6 × 2 = 0 + 0.000 000 000 000 000 292 716 162 931 2;
  • 7) 0.000 000 000 000 000 292 716 162 931 2 × 2 = 0 + 0.000 000 000 000 000 585 432 325 862 4;
  • 8) 0.000 000 000 000 000 585 432 325 862 4 × 2 = 0 + 0.000 000 000 000 001 170 864 651 724 8;
  • 9) 0.000 000 000 000 001 170 864 651 724 8 × 2 = 0 + 0.000 000 000 000 002 341 729 303 449 6;
  • 10) 0.000 000 000 000 002 341 729 303 449 6 × 2 = 0 + 0.000 000 000 000 004 683 458 606 899 2;
  • 11) 0.000 000 000 000 004 683 458 606 899 2 × 2 = 0 + 0.000 000 000 000 009 366 917 213 798 4;
  • 12) 0.000 000 000 000 009 366 917 213 798 4 × 2 = 0 + 0.000 000 000 000 018 733 834 427 596 8;
  • 13) 0.000 000 000 000 018 733 834 427 596 8 × 2 = 0 + 0.000 000 000 000 037 467 668 855 193 6;
  • 14) 0.000 000 000 000 037 467 668 855 193 6 × 2 = 0 + 0.000 000 000 000 074 935 337 710 387 2;
  • 15) 0.000 000 000 000 074 935 337 710 387 2 × 2 = 0 + 0.000 000 000 000 149 870 675 420 774 4;
  • 16) 0.000 000 000 000 149 870 675 420 774 4 × 2 = 0 + 0.000 000 000 000 299 741 350 841 548 8;
  • 17) 0.000 000 000 000 299 741 350 841 548 8 × 2 = 0 + 0.000 000 000 000 599 482 701 683 097 6;
  • 18) 0.000 000 000 000 599 482 701 683 097 6 × 2 = 0 + 0.000 000 000 001 198 965 403 366 195 2;
  • 19) 0.000 000 000 001 198 965 403 366 195 2 × 2 = 0 + 0.000 000 000 002 397 930 806 732 390 4;
  • 20) 0.000 000 000 002 397 930 806 732 390 4 × 2 = 0 + 0.000 000 000 004 795 861 613 464 780 8;
  • 21) 0.000 000 000 004 795 861 613 464 780 8 × 2 = 0 + 0.000 000 000 009 591 723 226 929 561 6;
  • 22) 0.000 000 000 009 591 723 226 929 561 6 × 2 = 0 + 0.000 000 000 019 183 446 453 859 123 2;
  • 23) 0.000 000 000 019 183 446 453 859 123 2 × 2 = 0 + 0.000 000 000 038 366 892 907 718 246 4;
  • 24) 0.000 000 000 038 366 892 907 718 246 4 × 2 = 0 + 0.000 000 000 076 733 785 815 436 492 8;
  • 25) 0.000 000 000 076 733 785 815 436 492 8 × 2 = 0 + 0.000 000 000 153 467 571 630 872 985 6;
  • 26) 0.000 000 000 153 467 571 630 872 985 6 × 2 = 0 + 0.000 000 000 306 935 143 261 745 971 2;
  • 27) 0.000 000 000 306 935 143 261 745 971 2 × 2 = 0 + 0.000 000 000 613 870 286 523 491 942 4;
  • 28) 0.000 000 000 613 870 286 523 491 942 4 × 2 = 0 + 0.000 000 001 227 740 573 046 983 884 8;
  • 29) 0.000 000 001 227 740 573 046 983 884 8 × 2 = 0 + 0.000 000 002 455 481 146 093 967 769 6;
  • 30) 0.000 000 002 455 481 146 093 967 769 6 × 2 = 0 + 0.000 000 004 910 962 292 187 935 539 2;
  • 31) 0.000 000 004 910 962 292 187 935 539 2 × 2 = 0 + 0.000 000 009 821 924 584 375 871 078 4;
  • 32) 0.000 000 009 821 924 584 375 871 078 4 × 2 = 0 + 0.000 000 019 643 849 168 751 742 156 8;
  • 33) 0.000 000 019 643 849 168 751 742 156 8 × 2 = 0 + 0.000 000 039 287 698 337 503 484 313 6;
  • 34) 0.000 000 039 287 698 337 503 484 313 6 × 2 = 0 + 0.000 000 078 575 396 675 006 968 627 2;
  • 35) 0.000 000 078 575 396 675 006 968 627 2 × 2 = 0 + 0.000 000 157 150 793 350 013 937 254 4;
  • 36) 0.000 000 157 150 793 350 013 937 254 4 × 2 = 0 + 0.000 000 314 301 586 700 027 874 508 8;
  • 37) 0.000 000 314 301 586 700 027 874 508 8 × 2 = 0 + 0.000 000 628 603 173 400 055 749 017 6;
  • 38) 0.000 000 628 603 173 400 055 749 017 6 × 2 = 0 + 0.000 001 257 206 346 800 111 498 035 2;
  • 39) 0.000 001 257 206 346 800 111 498 035 2 × 2 = 0 + 0.000 002 514 412 693 600 222 996 070 4;
  • 40) 0.000 002 514 412 693 600 222 996 070 4 × 2 = 0 + 0.000 005 028 825 387 200 445 992 140 8;
  • 41) 0.000 005 028 825 387 200 445 992 140 8 × 2 = 0 + 0.000 010 057 650 774 400 891 984 281 6;
  • 42) 0.000 010 057 650 774 400 891 984 281 6 × 2 = 0 + 0.000 020 115 301 548 801 783 968 563 2;
  • 43) 0.000 020 115 301 548 801 783 968 563 2 × 2 = 0 + 0.000 040 230 603 097 603 567 937 126 4;
  • 44) 0.000 040 230 603 097 603 567 937 126 4 × 2 = 0 + 0.000 080 461 206 195 207 135 874 252 8;
  • 45) 0.000 080 461 206 195 207 135 874 252 8 × 2 = 0 + 0.000 160 922 412 390 414 271 748 505 6;
  • 46) 0.000 160 922 412 390 414 271 748 505 6 × 2 = 0 + 0.000 321 844 824 780 828 543 497 011 2;
  • 47) 0.000 321 844 824 780 828 543 497 011 2 × 2 = 0 + 0.000 643 689 649 561 657 086 994 022 4;
  • 48) 0.000 643 689 649 561 657 086 994 022 4 × 2 = 0 + 0.001 287 379 299 123 314 173 988 044 8;
  • 49) 0.001 287 379 299 123 314 173 988 044 8 × 2 = 0 + 0.002 574 758 598 246 628 347 976 089 6;
  • 50) 0.002 574 758 598 246 628 347 976 089 6 × 2 = 0 + 0.005 149 517 196 493 256 695 952 179 2;
  • 51) 0.005 149 517 196 493 256 695 952 179 2 × 2 = 0 + 0.010 299 034 392 986 513 391 904 358 4;
  • 52) 0.010 299 034 392 986 513 391 904 358 4 × 2 = 0 + 0.020 598 068 785 973 026 783 808 716 8;
  • 53) 0.020 598 068 785 973 026 783 808 716 8 × 2 = 0 + 0.041 196 137 571 946 053 567 617 433 6;
  • 54) 0.041 196 137 571 946 053 567 617 433 6 × 2 = 0 + 0.082 392 275 143 892 107 135 234 867 2;
  • 55) 0.082 392 275 143 892 107 135 234 867 2 × 2 = 0 + 0.164 784 550 287 784 214 270 469 734 4;
  • 56) 0.164 784 550 287 784 214 270 469 734 4 × 2 = 0 + 0.329 569 100 575 568 428 540 939 468 8;
  • 57) 0.329 569 100 575 568 428 540 939 468 8 × 2 = 0 + 0.659 138 201 151 136 857 081 878 937 6;
  • 58) 0.659 138 201 151 136 857 081 878 937 6 × 2 = 1 + 0.318 276 402 302 273 714 163 757 875 2;
  • 59) 0.318 276 402 302 273 714 163 757 875 2 × 2 = 0 + 0.636 552 804 604 547 428 327 515 750 4;
  • 60) 0.636 552 804 604 547 428 327 515 750 4 × 2 = 1 + 0.273 105 609 209 094 856 655 031 500 8;
  • 61) 0.273 105 609 209 094 856 655 031 500 8 × 2 = 0 + 0.546 211 218 418 189 713 310 063 001 6;
  • 62) 0.546 211 218 418 189 713 310 063 001 6 × 2 = 1 + 0.092 422 436 836 379 426 620 126 003 2;
  • 63) 0.092 422 436 836 379 426 620 126 003 2 × 2 = 0 + 0.184 844 873 672 758 853 240 252 006 4;
  • 64) 0.184 844 873 672 758 853 240 252 006 4 × 2 = 0 + 0.369 689 747 345 517 706 480 504 012 8;
  • 65) 0.369 689 747 345 517 706 480 504 012 8 × 2 = 0 + 0.739 379 494 691 035 412 961 008 025 6;
  • 66) 0.739 379 494 691 035 412 961 008 025 6 × 2 = 1 + 0.478 758 989 382 070 825 922 016 051 2;
  • 67) 0.478 758 989 382 070 825 922 016 051 2 × 2 = 0 + 0.957 517 978 764 141 651 844 032 102 4;
  • 68) 0.957 517 978 764 141 651 844 032 102 4 × 2 = 1 + 0.915 035 957 528 283 303 688 064 204 8;
  • 69) 0.915 035 957 528 283 303 688 064 204 8 × 2 = 1 + 0.830 071 915 056 566 607 376 128 409 6;
  • 70) 0.830 071 915 056 566 607 376 128 409 6 × 2 = 1 + 0.660 143 830 113 133 214 752 256 819 2;
  • 71) 0.660 143 830 113 133 214 752 256 819 2 × 2 = 1 + 0.320 287 660 226 266 429 504 513 638 4;
  • 72) 0.320 287 660 226 266 429 504 513 638 4 × 2 = 0 + 0.640 575 320 452 532 859 009 027 276 8;
  • 73) 0.640 575 320 452 532 859 009 027 276 8 × 2 = 1 + 0.281 150 640 905 065 718 018 054 553 6;
  • 74) 0.281 150 640 905 065 718 018 054 553 6 × 2 = 0 + 0.562 301 281 810 131 436 036 109 107 2;
  • 75) 0.562 301 281 810 131 436 036 109 107 2 × 2 = 1 + 0.124 602 563 620 262 872 072 218 214 4;
  • 76) 0.124 602 563 620 262 872 072 218 214 4 × 2 = 0 + 0.249 205 127 240 525 744 144 436 428 8;
  • 77) 0.249 205 127 240 525 744 144 436 428 8 × 2 = 0 + 0.498 410 254 481 051 488 288 872 857 6;
  • 78) 0.498 410 254 481 051 488 288 872 857 6 × 2 = 0 + 0.996 820 508 962 102 976 577 745 715 2;
  • 79) 0.996 820 508 962 102 976 577 745 715 2 × 2 = 1 + 0.993 641 017 924 205 953 155 491 430 4;
  • 80) 0.993 641 017 924 205 953 155 491 430 4 × 2 = 1 + 0.987 282 035 848 411 906 310 982 860 8;
  • 81) 0.987 282 035 848 411 906 310 982 860 8 × 2 = 1 + 0.974 564 071 696 823 812 621 965 721 6;
  • 82) 0.974 564 071 696 823 812 621 965 721 6 × 2 = 1 + 0.949 128 143 393 647 625 243 931 443 2;
  • 83) 0.949 128 143 393 647 625 243 931 443 2 × 2 = 1 + 0.898 256 286 787 295 250 487 862 886 4;
  • 84) 0.898 256 286 787 295 250 487 862 886 4 × 2 = 1 + 0.796 512 573 574 590 500 975 725 772 8;
  • 85) 0.796 512 573 574 590 500 975 725 772 8 × 2 = 1 + 0.593 025 147 149 181 001 951 451 545 6;
  • 86) 0.593 025 147 149 181 001 951 451 545 6 × 2 = 1 + 0.186 050 294 298 362 003 902 903 091 2;
  • 87) 0.186 050 294 298 362 003 902 903 091 2 × 2 = 0 + 0.372 100 588 596 724 007 805 806 182 4;
  • 88) 0.372 100 588 596 724 007 805 806 182 4 × 2 = 0 + 0.744 201 177 193 448 015 611 612 364 8;
  • 89) 0.744 201 177 193 448 015 611 612 364 8 × 2 = 1 + 0.488 402 354 386 896 031 223 224 729 6;
  • 90) 0.488 402 354 386 896 031 223 224 729 6 × 2 = 0 + 0.976 804 708 773 792 062 446 449 459 2;
  • 91) 0.976 804 708 773 792 062 446 449 459 2 × 2 = 1 + 0.953 609 417 547 584 124 892 898 918 4;
  • 92) 0.953 609 417 547 584 124 892 898 918 4 × 2 = 1 + 0.907 218 835 095 168 249 785 797 836 8;
  • 93) 0.907 218 835 095 168 249 785 797 836 8 × 2 = 1 + 0.814 437 670 190 336 499 571 595 673 6;
  • 94) 0.814 437 670 190 336 499 571 595 673 6 × 2 = 1 + 0.628 875 340 380 672 999 143 191 347 2;
  • 95) 0.628 875 340 380 672 999 143 191 347 2 × 2 = 1 + 0.257 750 680 761 345 998 286 382 694 4;
  • 96) 0.257 750 680 761 345 998 286 382 694 4 × 2 = 0 + 0.515 501 361 522 691 996 572 765 388 8;
  • 97) 0.515 501 361 522 691 996 572 765 388 8 × 2 = 1 + 0.031 002 723 045 383 993 145 530 777 6;
  • 98) 0.031 002 723 045 383 993 145 530 777 6 × 2 = 0 + 0.062 005 446 090 767 986 291 061 555 2;
  • 99) 0.062 005 446 090 767 986 291 061 555 2 × 2 = 0 + 0.124 010 892 181 535 972 582 123 110 4;
  • 100) 0.124 010 892 181 535 972 582 123 110 4 × 2 = 0 + 0.248 021 784 363 071 945 164 246 220 8;
  • 101) 0.248 021 784 363 071 945 164 246 220 8 × 2 = 0 + 0.496 043 568 726 143 890 328 492 441 6;
  • 102) 0.496 043 568 726 143 890 328 492 441 6 × 2 = 0 + 0.992 087 137 452 287 780 656 984 883 2;
  • 103) 0.992 087 137 452 287 780 656 984 883 2 × 2 = 1 + 0.984 174 274 904 575 561 313 969 766 4;
  • 104) 0.984 174 274 904 575 561 313 969 766 4 × 2 = 1 + 0.968 348 549 809 151 122 627 939 532 8;
  • 105) 0.968 348 549 809 151 122 627 939 532 8 × 2 = 1 + 0.936 697 099 618 302 245 255 879 065 6;
  • 106) 0.936 697 099 618 302 245 255 879 065 6 × 2 = 1 + 0.873 394 199 236 604 490 511 758 131 2;
  • 107) 0.873 394 199 236 604 490 511 758 131 2 × 2 = 1 + 0.746 788 398 473 208 981 023 516 262 4;
  • 108) 0.746 788 398 473 208 981 023 516 262 4 × 2 = 1 + 0.493 576 796 946 417 962 047 032 524 8;
  • 109) 0.493 576 796 946 417 962 047 032 524 8 × 2 = 0 + 0.987 153 593 892 835 924 094 065 049 6;
  • 110) 0.987 153 593 892 835 924 094 065 049 6 × 2 = 1 + 0.974 307 187 785 671 848 188 130 099 2;

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).


5. 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:


0.000 000 000 000 000 004 573 690 045 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1100 1011 1110 1000 0011 1111 01(2)

6. Positive number before normalization:

0.000 000 000 000 000 004 573 690 045 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1100 1011 1110 1000 0011 1111 01(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 58 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 004 573 690 045 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1100 1011 1110 1000 0011 1111 01(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1100 1011 1110 1000 0011 1111 01(2) × 20 =

1.0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101(2) × 2-58


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): -58

Mantissa (not normalized):
1.0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-58 + 2(11-1) - 1 =

(-58 + 1 023)(10) =

965(10)


10. 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;
  • 965 ÷ 2 = 482 + 1;
  • 482 ÷ 2 = 241 + 0;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

965(10) =

011 1100 0101(2)


12. 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. 0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101 =

0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1100 0101

Mantissa (52 bits) =
0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101


Decimal number -0.000 000 000 000 000 004 573 690 045 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 0101 - 0101 0001 0111 1010 1000 1111 1111 0010 1111 1010 0000 1111 1101

» Convert decimal -0.000 000 000 000 000 004 573 690 050 7 to 64 bit double precision IEEE 754 binary floating point representation standard

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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