0.000 000 000 000 000 000 008 531 43 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 531 43(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
0.000 000 000 000 000 000 008 531 43(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: 0.000 000 000 000 000 000 008 531 43.

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) 0.000 000 000 000 000 000 008 531 43 × 2 = 0 + 0.000 000 000 000 000 000 017 062 86;
  • 2) 0.000 000 000 000 000 000 017 062 86 × 2 = 0 + 0.000 000 000 000 000 000 034 125 72;
  • 3) 0.000 000 000 000 000 000 034 125 72 × 2 = 0 + 0.000 000 000 000 000 000 068 251 44;
  • 4) 0.000 000 000 000 000 000 068 251 44 × 2 = 0 + 0.000 000 000 000 000 000 136 502 88;
  • 5) 0.000 000 000 000 000 000 136 502 88 × 2 = 0 + 0.000 000 000 000 000 000 273 005 76;
  • 6) 0.000 000 000 000 000 000 273 005 76 × 2 = 0 + 0.000 000 000 000 000 000 546 011 52;
  • 7) 0.000 000 000 000 000 000 546 011 52 × 2 = 0 + 0.000 000 000 000 000 001 092 023 04;
  • 8) 0.000 000 000 000 000 001 092 023 04 × 2 = 0 + 0.000 000 000 000 000 002 184 046 08;
  • 9) 0.000 000 000 000 000 002 184 046 08 × 2 = 0 + 0.000 000 000 000 000 004 368 092 16;
  • 10) 0.000 000 000 000 000 004 368 092 16 × 2 = 0 + 0.000 000 000 000 000 008 736 184 32;
  • 11) 0.000 000 000 000 000 008 736 184 32 × 2 = 0 + 0.000 000 000 000 000 017 472 368 64;
  • 12) 0.000 000 000 000 000 017 472 368 64 × 2 = 0 + 0.000 000 000 000 000 034 944 737 28;
  • 13) 0.000 000 000 000 000 034 944 737 28 × 2 = 0 + 0.000 000 000 000 000 069 889 474 56;
  • 14) 0.000 000 000 000 000 069 889 474 56 × 2 = 0 + 0.000 000 000 000 000 139 778 949 12;
  • 15) 0.000 000 000 000 000 139 778 949 12 × 2 = 0 + 0.000 000 000 000 000 279 557 898 24;
  • 16) 0.000 000 000 000 000 279 557 898 24 × 2 = 0 + 0.000 000 000 000 000 559 115 796 48;
  • 17) 0.000 000 000 000 000 559 115 796 48 × 2 = 0 + 0.000 000 000 000 001 118 231 592 96;
  • 18) 0.000 000 000 000 001 118 231 592 96 × 2 = 0 + 0.000 000 000 000 002 236 463 185 92;
  • 19) 0.000 000 000 000 002 236 463 185 92 × 2 = 0 + 0.000 000 000 000 004 472 926 371 84;
  • 20) 0.000 000 000 000 004 472 926 371 84 × 2 = 0 + 0.000 000 000 000 008 945 852 743 68;
  • 21) 0.000 000 000 000 008 945 852 743 68 × 2 = 0 + 0.000 000 000 000 017 891 705 487 36;
  • 22) 0.000 000 000 000 017 891 705 487 36 × 2 = 0 + 0.000 000 000 000 035 783 410 974 72;
  • 23) 0.000 000 000 000 035 783 410 974 72 × 2 = 0 + 0.000 000 000 000 071 566 821 949 44;
  • 24) 0.000 000 000 000 071 566 821 949 44 × 2 = 0 + 0.000 000 000 000 143 133 643 898 88;
  • 25) 0.000 000 000 000 143 133 643 898 88 × 2 = 0 + 0.000 000 000 000 286 267 287 797 76;
  • 26) 0.000 000 000 000 286 267 287 797 76 × 2 = 0 + 0.000 000 000 000 572 534 575 595 52;
  • 27) 0.000 000 000 000 572 534 575 595 52 × 2 = 0 + 0.000 000 000 001 145 069 151 191 04;
  • 28) 0.000 000 000 001 145 069 151 191 04 × 2 = 0 + 0.000 000 000 002 290 138 302 382 08;
  • 29) 0.000 000 000 002 290 138 302 382 08 × 2 = 0 + 0.000 000 000 004 580 276 604 764 16;
  • 30) 0.000 000 000 004 580 276 604 764 16 × 2 = 0 + 0.000 000 000 009 160 553 209 528 32;
  • 31) 0.000 000 000 009 160 553 209 528 32 × 2 = 0 + 0.000 000 000 018 321 106 419 056 64;
  • 32) 0.000 000 000 018 321 106 419 056 64 × 2 = 0 + 0.000 000 000 036 642 212 838 113 28;
  • 33) 0.000 000 000 036 642 212 838 113 28 × 2 = 0 + 0.000 000 000 073 284 425 676 226 56;
  • 34) 0.000 000 000 073 284 425 676 226 56 × 2 = 0 + 0.000 000 000 146 568 851 352 453 12;
  • 35) 0.000 000 000 146 568 851 352 453 12 × 2 = 0 + 0.000 000 000 293 137 702 704 906 24;
  • 36) 0.000 000 000 293 137 702 704 906 24 × 2 = 0 + 0.000 000 000 586 275 405 409 812 48;
  • 37) 0.000 000 000 586 275 405 409 812 48 × 2 = 0 + 0.000 000 001 172 550 810 819 624 96;
  • 38) 0.000 000 001 172 550 810 819 624 96 × 2 = 0 + 0.000 000 002 345 101 621 639 249 92;
  • 39) 0.000 000 002 345 101 621 639 249 92 × 2 = 0 + 0.000 000 004 690 203 243 278 499 84;
  • 40) 0.000 000 004 690 203 243 278 499 84 × 2 = 0 + 0.000 000 009 380 406 486 556 999 68;
  • 41) 0.000 000 009 380 406 486 556 999 68 × 2 = 0 + 0.000 000 018 760 812 973 113 999 36;
  • 42) 0.000 000 018 760 812 973 113 999 36 × 2 = 0 + 0.000 000 037 521 625 946 227 998 72;
  • 43) 0.000 000 037 521 625 946 227 998 72 × 2 = 0 + 0.000 000 075 043 251 892 455 997 44;
  • 44) 0.000 000 075 043 251 892 455 997 44 × 2 = 0 + 0.000 000 150 086 503 784 911 994 88;
  • 45) 0.000 000 150 086 503 784 911 994 88 × 2 = 0 + 0.000 000 300 173 007 569 823 989 76;
  • 46) 0.000 000 300 173 007 569 823 989 76 × 2 = 0 + 0.000 000 600 346 015 139 647 979 52;
  • 47) 0.000 000 600 346 015 139 647 979 52 × 2 = 0 + 0.000 001 200 692 030 279 295 959 04;
  • 48) 0.000 001 200 692 030 279 295 959 04 × 2 = 0 + 0.000 002 401 384 060 558 591 918 08;
  • 49) 0.000 002 401 384 060 558 591 918 08 × 2 = 0 + 0.000 004 802 768 121 117 183 836 16;
  • 50) 0.000 004 802 768 121 117 183 836 16 × 2 = 0 + 0.000 009 605 536 242 234 367 672 32;
  • 51) 0.000 009 605 536 242 234 367 672 32 × 2 = 0 + 0.000 019 211 072 484 468 735 344 64;
  • 52) 0.000 019 211 072 484 468 735 344 64 × 2 = 0 + 0.000 038 422 144 968 937 470 689 28;
  • 53) 0.000 038 422 144 968 937 470 689 28 × 2 = 0 + 0.000 076 844 289 937 874 941 378 56;
  • 54) 0.000 076 844 289 937 874 941 378 56 × 2 = 0 + 0.000 153 688 579 875 749 882 757 12;
  • 55) 0.000 153 688 579 875 749 882 757 12 × 2 = 0 + 0.000 307 377 159 751 499 765 514 24;
  • 56) 0.000 307 377 159 751 499 765 514 24 × 2 = 0 + 0.000 614 754 319 502 999 531 028 48;
  • 57) 0.000 614 754 319 502 999 531 028 48 × 2 = 0 + 0.001 229 508 639 005 999 062 056 96;
  • 58) 0.001 229 508 639 005 999 062 056 96 × 2 = 0 + 0.002 459 017 278 011 998 124 113 92;
  • 59) 0.002 459 017 278 011 998 124 113 92 × 2 = 0 + 0.004 918 034 556 023 996 248 227 84;
  • 60) 0.004 918 034 556 023 996 248 227 84 × 2 = 0 + 0.009 836 069 112 047 992 496 455 68;
  • 61) 0.009 836 069 112 047 992 496 455 68 × 2 = 0 + 0.019 672 138 224 095 984 992 911 36;
  • 62) 0.019 672 138 224 095 984 992 911 36 × 2 = 0 + 0.039 344 276 448 191 969 985 822 72;
  • 63) 0.039 344 276 448 191 969 985 822 72 × 2 = 0 + 0.078 688 552 896 383 939 971 645 44;
  • 64) 0.078 688 552 896 383 939 971 645 44 × 2 = 0 + 0.157 377 105 792 767 879 943 290 88;
  • 65) 0.157 377 105 792 767 879 943 290 88 × 2 = 0 + 0.314 754 211 585 535 759 886 581 76;
  • 66) 0.314 754 211 585 535 759 886 581 76 × 2 = 0 + 0.629 508 423 171 071 519 773 163 52;
  • 67) 0.629 508 423 171 071 519 773 163 52 × 2 = 1 + 0.259 016 846 342 143 039 546 327 04;
  • 68) 0.259 016 846 342 143 039 546 327 04 × 2 = 0 + 0.518 033 692 684 286 079 092 654 08;
  • 69) 0.518 033 692 684 286 079 092 654 08 × 2 = 1 + 0.036 067 385 368 572 158 185 308 16;
  • 70) 0.036 067 385 368 572 158 185 308 16 × 2 = 0 + 0.072 134 770 737 144 316 370 616 32;
  • 71) 0.072 134 770 737 144 316 370 616 32 × 2 = 0 + 0.144 269 541 474 288 632 741 232 64;
  • 72) 0.144 269 541 474 288 632 741 232 64 × 2 = 0 + 0.288 539 082 948 577 265 482 465 28;
  • 73) 0.288 539 082 948 577 265 482 465 28 × 2 = 0 + 0.577 078 165 897 154 530 964 930 56;
  • 74) 0.577 078 165 897 154 530 964 930 56 × 2 = 1 + 0.154 156 331 794 309 061 929 861 12;
  • 75) 0.154 156 331 794 309 061 929 861 12 × 2 = 0 + 0.308 312 663 588 618 123 859 722 24;
  • 76) 0.308 312 663 588 618 123 859 722 24 × 2 = 0 + 0.616 625 327 177 236 247 719 444 48;
  • 77) 0.616 625 327 177 236 247 719 444 48 × 2 = 1 + 0.233 250 654 354 472 495 438 888 96;
  • 78) 0.233 250 654 354 472 495 438 888 96 × 2 = 0 + 0.466 501 308 708 944 990 877 777 92;
  • 79) 0.466 501 308 708 944 990 877 777 92 × 2 = 0 + 0.933 002 617 417 889 981 755 555 84;
  • 80) 0.933 002 617 417 889 981 755 555 84 × 2 = 1 + 0.866 005 234 835 779 963 511 111 68;
  • 81) 0.866 005 234 835 779 963 511 111 68 × 2 = 1 + 0.732 010 469 671 559 927 022 223 36;
  • 82) 0.732 010 469 671 559 927 022 223 36 × 2 = 1 + 0.464 020 939 343 119 854 044 446 72;
  • 83) 0.464 020 939 343 119 854 044 446 72 × 2 = 0 + 0.928 041 878 686 239 708 088 893 44;
  • 84) 0.928 041 878 686 239 708 088 893 44 × 2 = 1 + 0.856 083 757 372 479 416 177 786 88;
  • 85) 0.856 083 757 372 479 416 177 786 88 × 2 = 1 + 0.712 167 514 744 958 832 355 573 76;
  • 86) 0.712 167 514 744 958 832 355 573 76 × 2 = 1 + 0.424 335 029 489 917 664 711 147 52;
  • 87) 0.424 335 029 489 917 664 711 147 52 × 2 = 0 + 0.848 670 058 979 835 329 422 295 04;
  • 88) 0.848 670 058 979 835 329 422 295 04 × 2 = 1 + 0.697 340 117 959 670 658 844 590 08;
  • 89) 0.697 340 117 959 670 658 844 590 08 × 2 = 1 + 0.394 680 235 919 341 317 689 180 16;
  • 90) 0.394 680 235 919 341 317 689 180 16 × 2 = 0 + 0.789 360 471 838 682 635 378 360 32;
  • 91) 0.789 360 471 838 682 635 378 360 32 × 2 = 1 + 0.578 720 943 677 365 270 756 720 64;
  • 92) 0.578 720 943 677 365 270 756 720 64 × 2 = 1 + 0.157 441 887 354 730 541 513 441 28;
  • 93) 0.157 441 887 354 730 541 513 441 28 × 2 = 0 + 0.314 883 774 709 461 083 026 882 56;
  • 94) 0.314 883 774 709 461 083 026 882 56 × 2 = 0 + 0.629 767 549 418 922 166 053 765 12;
  • 95) 0.629 767 549 418 922 166 053 765 12 × 2 = 1 + 0.259 535 098 837 844 332 107 530 24;
  • 96) 0.259 535 098 837 844 332 107 530 24 × 2 = 0 + 0.519 070 197 675 688 664 215 060 48;
  • 97) 0.519 070 197 675 688 664 215 060 48 × 2 = 1 + 0.038 140 395 351 377 328 430 120 96;
  • 98) 0.038 140 395 351 377 328 430 120 96 × 2 = 0 + 0.076 280 790 702 754 656 860 241 92;
  • 99) 0.076 280 790 702 754 656 860 241 92 × 2 = 0 + 0.152 561 581 405 509 313 720 483 84;
  • 100) 0.152 561 581 405 509 313 720 483 84 × 2 = 0 + 0.305 123 162 811 018 627 440 967 68;
  • 101) 0.305 123 162 811 018 627 440 967 68 × 2 = 0 + 0.610 246 325 622 037 254 881 935 36;
  • 102) 0.610 246 325 622 037 254 881 935 36 × 2 = 1 + 0.220 492 651 244 074 509 763 870 72;
  • 103) 0.220 492 651 244 074 509 763 870 72 × 2 = 0 + 0.440 985 302 488 149 019 527 741 44;
  • 104) 0.440 985 302 488 149 019 527 741 44 × 2 = 0 + 0.881 970 604 976 298 039 055 482 88;
  • 105) 0.881 970 604 976 298 039 055 482 88 × 2 = 1 + 0.763 941 209 952 596 078 110 965 76;
  • 106) 0.763 941 209 952 596 078 110 965 76 × 2 = 1 + 0.527 882 419 905 192 156 221 931 52;
  • 107) 0.527 882 419 905 192 156 221 931 52 × 2 = 1 + 0.055 764 839 810 384 312 443 863 04;
  • 108) 0.055 764 839 810 384 312 443 863 04 × 2 = 0 + 0.111 529 679 620 768 624 887 726 08;
  • 109) 0.111 529 679 620 768 624 887 726 08 × 2 = 0 + 0.223 059 359 241 537 249 775 452 16;
  • 110) 0.223 059 359 241 537 249 775 452 16 × 2 = 0 + 0.446 118 718 483 074 499 550 904 32;
  • 111) 0.446 118 718 483 074 499 550 904 32 × 2 = 0 + 0.892 237 436 966 148 999 101 808 64;
  • 112) 0.892 237 436 966 148 999 101 808 64 × 2 = 1 + 0.784 474 873 932 297 998 203 617 28;
  • 113) 0.784 474 873 932 297 998 203 617 28 × 2 = 1 + 0.568 949 747 864 595 996 407 234 56;
  • 114) 0.568 949 747 864 595 996 407 234 56 × 2 = 1 + 0.137 899 495 729 191 992 814 469 12;
  • 115) 0.137 899 495 729 191 992 814 469 12 × 2 = 0 + 0.275 798 991 458 383 985 628 938 24;
  • 116) 0.275 798 991 458 383 985 628 938 24 × 2 = 0 + 0.551 597 982 916 767 971 257 876 48;
  • 117) 0.551 597 982 916 767 971 257 876 48 × 2 = 1 + 0.103 195 965 833 535 942 515 752 96;
  • 118) 0.103 195 965 833 535 942 515 752 96 × 2 = 0 + 0.206 391 931 667 071 885 031 505 92;
  • 119) 0.206 391 931 667 071 885 031 505 92 × 2 = 0 + 0.412 783 863 334 143 770 063 011 84;

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:


0.000 000 000 000 000 000 008 531 43(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1101 1101 1011 0010 1000 0100 1110 0001 1100 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 531 43(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1101 1101 1011 0010 1000 0100 1110 0001 1100 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:


0.000 000 000 000 000 000 008 531 43(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1101 1101 1011 0010 1000 0100 1110 0001 1100 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1101 1101 1011 0010 1000 0100 1110 0001 1100 100(2) × 20 =

1.0100 0010 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100(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 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100


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 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100 =

0100 0010 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100


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 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100


Decimal number 0.000 000 000 000 000 000 008 531 43 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0100 1110 1110 1101 1001 0100 0010 0111 0000 1110 0100

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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