64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 9 223 372 036 854 775 810.151 382 799 620 931 041 140 181 605 442 194 268 107 4 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 9 223 372 036 854 775 810.151 382 799 620 931 041 140 181 605 442 194 268 107 4(10) converted and written in 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: 9 223 372 036 854 775 810.
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;
  • 9 223 372 036 854 775 810 ÷ 2 = 4 611 686 018 427 387 905 + 0;
  • 4 611 686 018 427 387 905 ÷ 2 = 2 305 843 009 213 693 952 + 1;
  • 2 305 843 009 213 693 952 ÷ 2 = 1 152 921 504 606 846 976 + 0;
  • 1 152 921 504 606 846 976 ÷ 2 = 576 460 752 303 423 488 + 0;
  • 576 460 752 303 423 488 ÷ 2 = 288 230 376 151 711 744 + 0;
  • 288 230 376 151 711 744 ÷ 2 = 144 115 188 075 855 872 + 0;
  • 144 115 188 075 855 872 ÷ 2 = 72 057 594 037 927 936 + 0;
  • 72 057 594 037 927 936 ÷ 2 = 36 028 797 018 963 968 + 0;
  • 36 028 797 018 963 968 ÷ 2 = 18 014 398 509 481 984 + 0;
  • 18 014 398 509 481 984 ÷ 2 = 9 007 199 254 740 992 + 0;
  • 9 007 199 254 740 992 ÷ 2 = 4 503 599 627 370 496 + 0;
  • 4 503 599 627 370 496 ÷ 2 = 2 251 799 813 685 248 + 0;
  • 2 251 799 813 685 248 ÷ 2 = 1 125 899 906 842 624 + 0;
  • 1 125 899 906 842 624 ÷ 2 = 562 949 953 421 312 + 0;
  • 562 949 953 421 312 ÷ 2 = 281 474 976 710 656 + 0;
  • 281 474 976 710 656 ÷ 2 = 140 737 488 355 328 + 0;
  • 140 737 488 355 328 ÷ 2 = 70 368 744 177 664 + 0;
  • 70 368 744 177 664 ÷ 2 = 35 184 372 088 832 + 0;
  • 35 184 372 088 832 ÷ 2 = 17 592 186 044 416 + 0;
  • 17 592 186 044 416 ÷ 2 = 8 796 093 022 208 + 0;
  • 8 796 093 022 208 ÷ 2 = 4 398 046 511 104 + 0;
  • 4 398 046 511 104 ÷ 2 = 2 199 023 255 552 + 0;
  • 2 199 023 255 552 ÷ 2 = 1 099 511 627 776 + 0;
  • 1 099 511 627 776 ÷ 2 = 549 755 813 888 + 0;
  • 549 755 813 888 ÷ 2 = 274 877 906 944 + 0;
  • 274 877 906 944 ÷ 2 = 137 438 953 472 + 0;
  • 137 438 953 472 ÷ 2 = 68 719 476 736 + 0;
  • 68 719 476 736 ÷ 2 = 34 359 738 368 + 0;
  • 34 359 738 368 ÷ 2 = 17 179 869 184 + 0;
  • 17 179 869 184 ÷ 2 = 8 589 934 592 + 0;
  • 8 589 934 592 ÷ 2 = 4 294 967 296 + 0;
  • 4 294 967 296 ÷ 2 = 2 147 483 648 + 0;
  • 2 147 483 648 ÷ 2 = 1 073 741 824 + 0;
  • 1 073 741 824 ÷ 2 = 536 870 912 + 0;
  • 536 870 912 ÷ 2 = 268 435 456 + 0;
  • 268 435 456 ÷ 2 = 134 217 728 + 0;
  • 134 217 728 ÷ 2 = 67 108 864 + 0;
  • 67 108 864 ÷ 2 = 33 554 432 + 0;
  • 33 554 432 ÷ 2 = 16 777 216 + 0;
  • 16 777 216 ÷ 2 = 8 388 608 + 0;
  • 8 388 608 ÷ 2 = 4 194 304 + 0;
  • 4 194 304 ÷ 2 = 2 097 152 + 0;
  • 2 097 152 ÷ 2 = 1 048 576 + 0;
  • 1 048 576 ÷ 2 = 524 288 + 0;
  • 524 288 ÷ 2 = 262 144 + 0;
  • 262 144 ÷ 2 = 131 072 + 0;
  • 131 072 ÷ 2 = 65 536 + 0;
  • 65 536 ÷ 2 = 32 768 + 0;
  • 32 768 ÷ 2 = 16 384 + 0;
  • 16 384 ÷ 2 = 8 192 + 0;
  • 8 192 ÷ 2 = 4 096 + 0;
  • 4 096 ÷ 2 = 2 048 + 0;
  • 2 048 ÷ 2 = 1 024 + 0;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.


9 223 372 036 854 775 810(10) =


1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2)


3. Convert to binary (base 2) the fractional part: 0.151 382 799 620 931 041 140 181 605 442 194 268 107 4.

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) 0.151 382 799 620 931 041 140 181 605 442 194 268 107 4 × 2 = 0 + 0.302 765 599 241 862 082 280 363 210 884 388 536 214 8;
  • 2) 0.302 765 599 241 862 082 280 363 210 884 388 536 214 8 × 2 = 0 + 0.605 531 198 483 724 164 560 726 421 768 777 072 429 6;
  • 3) 0.605 531 198 483 724 164 560 726 421 768 777 072 429 6 × 2 = 1 + 0.211 062 396 967 448 329 121 452 843 537 554 144 859 2;
  • 4) 0.211 062 396 967 448 329 121 452 843 537 554 144 859 2 × 2 = 0 + 0.422 124 793 934 896 658 242 905 687 075 108 289 718 4;
  • 5) 0.422 124 793 934 896 658 242 905 687 075 108 289 718 4 × 2 = 0 + 0.844 249 587 869 793 316 485 811 374 150 216 579 436 8;
  • 6) 0.844 249 587 869 793 316 485 811 374 150 216 579 436 8 × 2 = 1 + 0.688 499 175 739 586 632 971 622 748 300 433 158 873 6;
  • 7) 0.688 499 175 739 586 632 971 622 748 300 433 158 873 6 × 2 = 1 + 0.376 998 351 479 173 265 943 245 496 600 866 317 747 2;
  • 8) 0.376 998 351 479 173 265 943 245 496 600 866 317 747 2 × 2 = 0 + 0.753 996 702 958 346 531 886 490 993 201 732 635 494 4;
  • 9) 0.753 996 702 958 346 531 886 490 993 201 732 635 494 4 × 2 = 1 + 0.507 993 405 916 693 063 772 981 986 403 465 270 988 8;
  • 10) 0.507 993 405 916 693 063 772 981 986 403 465 270 988 8 × 2 = 1 + 0.015 986 811 833 386 127 545 963 972 806 930 541 977 6;
  • 11) 0.015 986 811 833 386 127 545 963 972 806 930 541 977 6 × 2 = 0 + 0.031 973 623 666 772 255 091 927 945 613 861 083 955 2;
  • 12) 0.031 973 623 666 772 255 091 927 945 613 861 083 955 2 × 2 = 0 + 0.063 947 247 333 544 510 183 855 891 227 722 167 910 4;
  • 13) 0.063 947 247 333 544 510 183 855 891 227 722 167 910 4 × 2 = 0 + 0.127 894 494 667 089 020 367 711 782 455 444 335 820 8;
  • 14) 0.127 894 494 667 089 020 367 711 782 455 444 335 820 8 × 2 = 0 + 0.255 788 989 334 178 040 735 423 564 910 888 671 641 6;
  • 15) 0.255 788 989 334 178 040 735 423 564 910 888 671 641 6 × 2 = 0 + 0.511 577 978 668 356 081 470 847 129 821 777 343 283 2;
  • 16) 0.511 577 978 668 356 081 470 847 129 821 777 343 283 2 × 2 = 1 + 0.023 155 957 336 712 162 941 694 259 643 554 686 566 4;
  • 17) 0.023 155 957 336 712 162 941 694 259 643 554 686 566 4 × 2 = 0 + 0.046 311 914 673 424 325 883 388 519 287 109 373 132 8;
  • 18) 0.046 311 914 673 424 325 883 388 519 287 109 373 132 8 × 2 = 0 + 0.092 623 829 346 848 651 766 777 038 574 218 746 265 6;
  • 19) 0.092 623 829 346 848 651 766 777 038 574 218 746 265 6 × 2 = 0 + 0.185 247 658 693 697 303 533 554 077 148 437 492 531 2;
  • 20) 0.185 247 658 693 697 303 533 554 077 148 437 492 531 2 × 2 = 0 + 0.370 495 317 387 394 607 067 108 154 296 874 985 062 4;
  • 21) 0.370 495 317 387 394 607 067 108 154 296 874 985 062 4 × 2 = 0 + 0.740 990 634 774 789 214 134 216 308 593 749 970 124 8;
  • 22) 0.740 990 634 774 789 214 134 216 308 593 749 970 124 8 × 2 = 1 + 0.481 981 269 549 578 428 268 432 617 187 499 940 249 6;
  • 23) 0.481 981 269 549 578 428 268 432 617 187 499 940 249 6 × 2 = 0 + 0.963 962 539 099 156 856 536 865 234 374 999 880 499 2;
  • 24) 0.963 962 539 099 156 856 536 865 234 374 999 880 499 2 × 2 = 1 + 0.927 925 078 198 313 713 073 730 468 749 999 760 998 4;
  • 25) 0.927 925 078 198 313 713 073 730 468 749 999 760 998 4 × 2 = 1 + 0.855 850 156 396 627 426 147 460 937 499 999 521 996 8;
  • 26) 0.855 850 156 396 627 426 147 460 937 499 999 521 996 8 × 2 = 1 + 0.711 700 312 793 254 852 294 921 874 999 999 043 993 6;
  • 27) 0.711 700 312 793 254 852 294 921 874 999 999 043 993 6 × 2 = 1 + 0.423 400 625 586 509 704 589 843 749 999 998 087 987 2;
  • 28) 0.423 400 625 586 509 704 589 843 749 999 998 087 987 2 × 2 = 0 + 0.846 801 251 173 019 409 179 687 499 999 996 175 974 4;
  • 29) 0.846 801 251 173 019 409 179 687 499 999 996 175 974 4 × 2 = 1 + 0.693 602 502 346 038 818 359 374 999 999 992 351 948 8;
  • 30) 0.693 602 502 346 038 818 359 374 999 999 992 351 948 8 × 2 = 1 + 0.387 205 004 692 077 636 718 749 999 999 984 703 897 6;
  • 31) 0.387 205 004 692 077 636 718 749 999 999 984 703 897 6 × 2 = 0 + 0.774 410 009 384 155 273 437 499 999 999 969 407 795 2;
  • 32) 0.774 410 009 384 155 273 437 499 999 999 969 407 795 2 × 2 = 1 + 0.548 820 018 768 310 546 874 999 999 999 938 815 590 4;
  • 33) 0.548 820 018 768 310 546 874 999 999 999 938 815 590 4 × 2 = 1 + 0.097 640 037 536 621 093 749 999 999 999 877 631 180 8;
  • 34) 0.097 640 037 536 621 093 749 999 999 999 877 631 180 8 × 2 = 0 + 0.195 280 075 073 242 187 499 999 999 999 755 262 361 6;
  • 35) 0.195 280 075 073 242 187 499 999 999 999 755 262 361 6 × 2 = 0 + 0.390 560 150 146 484 374 999 999 999 999 510 524 723 2;
  • 36) 0.390 560 150 146 484 374 999 999 999 999 510 524 723 2 × 2 = 0 + 0.781 120 300 292 968 749 999 999 999 999 021 049 446 4;
  • 37) 0.781 120 300 292 968 749 999 999 999 999 021 049 446 4 × 2 = 1 + 0.562 240 600 585 937 499 999 999 999 998 042 098 892 8;
  • 38) 0.562 240 600 585 937 499 999 999 999 998 042 098 892 8 × 2 = 1 + 0.124 481 201 171 874 999 999 999 999 996 084 197 785 6;
  • 39) 0.124 481 201 171 874 999 999 999 999 996 084 197 785 6 × 2 = 0 + 0.248 962 402 343 749 999 999 999 999 992 168 395 571 2;
  • 40) 0.248 962 402 343 749 999 999 999 999 992 168 395 571 2 × 2 = 0 + 0.497 924 804 687 499 999 999 999 999 984 336 791 142 4;
  • 41) 0.497 924 804 687 499 999 999 999 999 984 336 791 142 4 × 2 = 0 + 0.995 849 609 374 999 999 999 999 999 968 673 582 284 8;
  • 42) 0.995 849 609 374 999 999 999 999 999 968 673 582 284 8 × 2 = 1 + 0.991 699 218 749 999 999 999 999 999 937 347 164 569 6;
  • 43) 0.991 699 218 749 999 999 999 999 999 937 347 164 569 6 × 2 = 1 + 0.983 398 437 499 999 999 999 999 999 874 694 329 139 2;
  • 44) 0.983 398 437 499 999 999 999 999 999 874 694 329 139 2 × 2 = 1 + 0.966 796 874 999 999 999 999 999 999 749 388 658 278 4;
  • 45) 0.966 796 874 999 999 999 999 999 999 749 388 658 278 4 × 2 = 1 + 0.933 593 749 999 999 999 999 999 999 498 777 316 556 8;
  • 46) 0.933 593 749 999 999 999 999 999 999 498 777 316 556 8 × 2 = 1 + 0.867 187 499 999 999 999 999 999 998 997 554 633 113 6;
  • 47) 0.867 187 499 999 999 999 999 999 998 997 554 633 113 6 × 2 = 1 + 0.734 374 999 999 999 999 999 999 997 995 109 266 227 2;
  • 48) 0.734 374 999 999 999 999 999 999 997 995 109 266 227 2 × 2 = 1 + 0.468 749 999 999 999 999 999 999 995 990 218 532 454 4;
  • 49) 0.468 749 999 999 999 999 999 999 995 990 218 532 454 4 × 2 = 0 + 0.937 499 999 999 999 999 999 999 991 980 437 064 908 8;
  • 50) 0.937 499 999 999 999 999 999 999 991 980 437 064 908 8 × 2 = 1 + 0.874 999 999 999 999 999 999 999 983 960 874 129 817 6;
  • 51) 0.874 999 999 999 999 999 999 999 983 960 874 129 817 6 × 2 = 1 + 0.749 999 999 999 999 999 999 999 967 921 748 259 635 2;
  • 52) 0.749 999 999 999 999 999 999 999 967 921 748 259 635 2 × 2 = 1 + 0.499 999 999 999 999 999 999 999 935 843 496 519 270 4;
  • 53) 0.499 999 999 999 999 999 999 999 935 843 496 519 270 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 871 686 993 038 540 8;

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...)


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:


0.151 382 799 620 931 041 140 181 605 442 194 268 107 4(10) =


0.0010 0110 1100 0001 0000 0101 1110 1101 1000 1100 0111 1111 0111 0(2)


5. Positive number before normalization:

9 223 372 036 854 775 810.151 382 799 620 931 041 140 181 605 442 194 268 107 4(10) =


1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010.0010 0110 1100 0001 0000 0101 1110 1101 1000 1100 0111 1111 0111 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 63 positions to the left, so that only one non zero digit remains to the left of it:


9 223 372 036 854 775 810.151 382 799 620 931 041 140 181 605 442 194 268 107 4(10) =


1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010.0010 0110 1100 0001 0000 0101 1110 1101 1000 1100 0111 1111 0111 0(2) =


1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010.0010 0110 1100 0001 0000 0101 1110 1101 1000 1100 0111 1111 0111 0(2) × 20 =


1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0100 1101 1000 0010 0000 1011 1101 1011 0001 1000 1111 1110 1110(2) × 263


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): 63


Mantissa (not normalized):
1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0100 1101 1000 0010 0000 1011 1101 1011 0001 1000 1111 1110 1110


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


63 + 2(11-1) - 1 =


(63 + 1 023)(10) =


1 086(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;
  • 1 086 ÷ 2 = 543 + 0;
  • 543 ÷ 2 = 271 + 1;
  • 271 ÷ 2 = 135 + 1;
  • 135 ÷ 2 = 67 + 1;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =


1086(10) =


100 0011 1110(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0100 1101 1000 0010 0000 1011 1101 1011 0001 1000 1111 1110 1110 =


0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000


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) =
100 0011 1110


Mantissa (52 bits) =
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000


The base ten decimal number 9 223 372 036 854 775 810.151 382 799 620 931 041 140 181 605 442 194 268 107 4 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0011 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

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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