0.000 000 000 000 000 000 008 543 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 543 5(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
0.000 000 000 000 000 000 008 543 5(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: 0.000 000 000 000 000 000 008 543 5.

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) 0.000 000 000 000 000 000 008 543 5 × 2 = 0 + 0.000 000 000 000 000 000 017 087;
  • 2) 0.000 000 000 000 000 000 017 087 × 2 = 0 + 0.000 000 000 000 000 000 034 174;
  • 3) 0.000 000 000 000 000 000 034 174 × 2 = 0 + 0.000 000 000 000 000 000 068 348;
  • 4) 0.000 000 000 000 000 000 068 348 × 2 = 0 + 0.000 000 000 000 000 000 136 696;
  • 5) 0.000 000 000 000 000 000 136 696 × 2 = 0 + 0.000 000 000 000 000 000 273 392;
  • 6) 0.000 000 000 000 000 000 273 392 × 2 = 0 + 0.000 000 000 000 000 000 546 784;
  • 7) 0.000 000 000 000 000 000 546 784 × 2 = 0 + 0.000 000 000 000 000 001 093 568;
  • 8) 0.000 000 000 000 000 001 093 568 × 2 = 0 + 0.000 000 000 000 000 002 187 136;
  • 9) 0.000 000 000 000 000 002 187 136 × 2 = 0 + 0.000 000 000 000 000 004 374 272;
  • 10) 0.000 000 000 000 000 004 374 272 × 2 = 0 + 0.000 000 000 000 000 008 748 544;
  • 11) 0.000 000 000 000 000 008 748 544 × 2 = 0 + 0.000 000 000 000 000 017 497 088;
  • 12) 0.000 000 000 000 000 017 497 088 × 2 = 0 + 0.000 000 000 000 000 034 994 176;
  • 13) 0.000 000 000 000 000 034 994 176 × 2 = 0 + 0.000 000 000 000 000 069 988 352;
  • 14) 0.000 000 000 000 000 069 988 352 × 2 = 0 + 0.000 000 000 000 000 139 976 704;
  • 15) 0.000 000 000 000 000 139 976 704 × 2 = 0 + 0.000 000 000 000 000 279 953 408;
  • 16) 0.000 000 000 000 000 279 953 408 × 2 = 0 + 0.000 000 000 000 000 559 906 816;
  • 17) 0.000 000 000 000 000 559 906 816 × 2 = 0 + 0.000 000 000 000 001 119 813 632;
  • 18) 0.000 000 000 000 001 119 813 632 × 2 = 0 + 0.000 000 000 000 002 239 627 264;
  • 19) 0.000 000 000 000 002 239 627 264 × 2 = 0 + 0.000 000 000 000 004 479 254 528;
  • 20) 0.000 000 000 000 004 479 254 528 × 2 = 0 + 0.000 000 000 000 008 958 509 056;
  • 21) 0.000 000 000 000 008 958 509 056 × 2 = 0 + 0.000 000 000 000 017 917 018 112;
  • 22) 0.000 000 000 000 017 917 018 112 × 2 = 0 + 0.000 000 000 000 035 834 036 224;
  • 23) 0.000 000 000 000 035 834 036 224 × 2 = 0 + 0.000 000 000 000 071 668 072 448;
  • 24) 0.000 000 000 000 071 668 072 448 × 2 = 0 + 0.000 000 000 000 143 336 144 896;
  • 25) 0.000 000 000 000 143 336 144 896 × 2 = 0 + 0.000 000 000 000 286 672 289 792;
  • 26) 0.000 000 000 000 286 672 289 792 × 2 = 0 + 0.000 000 000 000 573 344 579 584;
  • 27) 0.000 000 000 000 573 344 579 584 × 2 = 0 + 0.000 000 000 001 146 689 159 168;
  • 28) 0.000 000 000 001 146 689 159 168 × 2 = 0 + 0.000 000 000 002 293 378 318 336;
  • 29) 0.000 000 000 002 293 378 318 336 × 2 = 0 + 0.000 000 000 004 586 756 636 672;
  • 30) 0.000 000 000 004 586 756 636 672 × 2 = 0 + 0.000 000 000 009 173 513 273 344;
  • 31) 0.000 000 000 009 173 513 273 344 × 2 = 0 + 0.000 000 000 018 347 026 546 688;
  • 32) 0.000 000 000 018 347 026 546 688 × 2 = 0 + 0.000 000 000 036 694 053 093 376;
  • 33) 0.000 000 000 036 694 053 093 376 × 2 = 0 + 0.000 000 000 073 388 106 186 752;
  • 34) 0.000 000 000 073 388 106 186 752 × 2 = 0 + 0.000 000 000 146 776 212 373 504;
  • 35) 0.000 000 000 146 776 212 373 504 × 2 = 0 + 0.000 000 000 293 552 424 747 008;
  • 36) 0.000 000 000 293 552 424 747 008 × 2 = 0 + 0.000 000 000 587 104 849 494 016;
  • 37) 0.000 000 000 587 104 849 494 016 × 2 = 0 + 0.000 000 001 174 209 698 988 032;
  • 38) 0.000 000 001 174 209 698 988 032 × 2 = 0 + 0.000 000 002 348 419 397 976 064;
  • 39) 0.000 000 002 348 419 397 976 064 × 2 = 0 + 0.000 000 004 696 838 795 952 128;
  • 40) 0.000 000 004 696 838 795 952 128 × 2 = 0 + 0.000 000 009 393 677 591 904 256;
  • 41) 0.000 000 009 393 677 591 904 256 × 2 = 0 + 0.000 000 018 787 355 183 808 512;
  • 42) 0.000 000 018 787 355 183 808 512 × 2 = 0 + 0.000 000 037 574 710 367 617 024;
  • 43) 0.000 000 037 574 710 367 617 024 × 2 = 0 + 0.000 000 075 149 420 735 234 048;
  • 44) 0.000 000 075 149 420 735 234 048 × 2 = 0 + 0.000 000 150 298 841 470 468 096;
  • 45) 0.000 000 150 298 841 470 468 096 × 2 = 0 + 0.000 000 300 597 682 940 936 192;
  • 46) 0.000 000 300 597 682 940 936 192 × 2 = 0 + 0.000 000 601 195 365 881 872 384;
  • 47) 0.000 000 601 195 365 881 872 384 × 2 = 0 + 0.000 001 202 390 731 763 744 768;
  • 48) 0.000 001 202 390 731 763 744 768 × 2 = 0 + 0.000 002 404 781 463 527 489 536;
  • 49) 0.000 002 404 781 463 527 489 536 × 2 = 0 + 0.000 004 809 562 927 054 979 072;
  • 50) 0.000 004 809 562 927 054 979 072 × 2 = 0 + 0.000 009 619 125 854 109 958 144;
  • 51) 0.000 009 619 125 854 109 958 144 × 2 = 0 + 0.000 019 238 251 708 219 916 288;
  • 52) 0.000 019 238 251 708 219 916 288 × 2 = 0 + 0.000 038 476 503 416 439 832 576;
  • 53) 0.000 038 476 503 416 439 832 576 × 2 = 0 + 0.000 076 953 006 832 879 665 152;
  • 54) 0.000 076 953 006 832 879 665 152 × 2 = 0 + 0.000 153 906 013 665 759 330 304;
  • 55) 0.000 153 906 013 665 759 330 304 × 2 = 0 + 0.000 307 812 027 331 518 660 608;
  • 56) 0.000 307 812 027 331 518 660 608 × 2 = 0 + 0.000 615 624 054 663 037 321 216;
  • 57) 0.000 615 624 054 663 037 321 216 × 2 = 0 + 0.001 231 248 109 326 074 642 432;
  • 58) 0.001 231 248 109 326 074 642 432 × 2 = 0 + 0.002 462 496 218 652 149 284 864;
  • 59) 0.002 462 496 218 652 149 284 864 × 2 = 0 + 0.004 924 992 437 304 298 569 728;
  • 60) 0.004 924 992 437 304 298 569 728 × 2 = 0 + 0.009 849 984 874 608 597 139 456;
  • 61) 0.009 849 984 874 608 597 139 456 × 2 = 0 + 0.019 699 969 749 217 194 278 912;
  • 62) 0.019 699 969 749 217 194 278 912 × 2 = 0 + 0.039 399 939 498 434 388 557 824;
  • 63) 0.039 399 939 498 434 388 557 824 × 2 = 0 + 0.078 799 878 996 868 777 115 648;
  • 64) 0.078 799 878 996 868 777 115 648 × 2 = 0 + 0.157 599 757 993 737 554 231 296;
  • 65) 0.157 599 757 993 737 554 231 296 × 2 = 0 + 0.315 199 515 987 475 108 462 592;
  • 66) 0.315 199 515 987 475 108 462 592 × 2 = 0 + 0.630 399 031 974 950 216 925 184;
  • 67) 0.630 399 031 974 950 216 925 184 × 2 = 1 + 0.260 798 063 949 900 433 850 368;
  • 68) 0.260 798 063 949 900 433 850 368 × 2 = 0 + 0.521 596 127 899 800 867 700 736;
  • 69) 0.521 596 127 899 800 867 700 736 × 2 = 1 + 0.043 192 255 799 601 735 401 472;
  • 70) 0.043 192 255 799 601 735 401 472 × 2 = 0 + 0.086 384 511 599 203 470 802 944;
  • 71) 0.086 384 511 599 203 470 802 944 × 2 = 0 + 0.172 769 023 198 406 941 605 888;
  • 72) 0.172 769 023 198 406 941 605 888 × 2 = 0 + 0.345 538 046 396 813 883 211 776;
  • 73) 0.345 538 046 396 813 883 211 776 × 2 = 0 + 0.691 076 092 793 627 766 423 552;
  • 74) 0.691 076 092 793 627 766 423 552 × 2 = 1 + 0.382 152 185 587 255 532 847 104;
  • 75) 0.382 152 185 587 255 532 847 104 × 2 = 0 + 0.764 304 371 174 511 065 694 208;
  • 76) 0.764 304 371 174 511 065 694 208 × 2 = 1 + 0.528 608 742 349 022 131 388 416;
  • 77) 0.528 608 742 349 022 131 388 416 × 2 = 1 + 0.057 217 484 698 044 262 776 832;
  • 78) 0.057 217 484 698 044 262 776 832 × 2 = 0 + 0.114 434 969 396 088 525 553 664;
  • 79) 0.114 434 969 396 088 525 553 664 × 2 = 0 + 0.228 869 938 792 177 051 107 328;
  • 80) 0.228 869 938 792 177 051 107 328 × 2 = 0 + 0.457 739 877 584 354 102 214 656;
  • 81) 0.457 739 877 584 354 102 214 656 × 2 = 0 + 0.915 479 755 168 708 204 429 312;
  • 82) 0.915 479 755 168 708 204 429 312 × 2 = 1 + 0.830 959 510 337 416 408 858 624;
  • 83) 0.830 959 510 337 416 408 858 624 × 2 = 1 + 0.661 919 020 674 832 817 717 248;
  • 84) 0.661 919 020 674 832 817 717 248 × 2 = 1 + 0.323 838 041 349 665 635 434 496;
  • 85) 0.323 838 041 349 665 635 434 496 × 2 = 0 + 0.647 676 082 699 331 270 868 992;
  • 86) 0.647 676 082 699 331 270 868 992 × 2 = 1 + 0.295 352 165 398 662 541 737 984;
  • 87) 0.295 352 165 398 662 541 737 984 × 2 = 0 + 0.590 704 330 797 325 083 475 968;
  • 88) 0.590 704 330 797 325 083 475 968 × 2 = 1 + 0.181 408 661 594 650 166 951 936;
  • 89) 0.181 408 661 594 650 166 951 936 × 2 = 0 + 0.362 817 323 189 300 333 903 872;
  • 90) 0.362 817 323 189 300 333 903 872 × 2 = 0 + 0.725 634 646 378 600 667 807 744;
  • 91) 0.725 634 646 378 600 667 807 744 × 2 = 1 + 0.451 269 292 757 201 335 615 488;
  • 92) 0.451 269 292 757 201 335 615 488 × 2 = 0 + 0.902 538 585 514 402 671 230 976;
  • 93) 0.902 538 585 514 402 671 230 976 × 2 = 1 + 0.805 077 171 028 805 342 461 952;
  • 94) 0.805 077 171 028 805 342 461 952 × 2 = 1 + 0.610 154 342 057 610 684 923 904;
  • 95) 0.610 154 342 057 610 684 923 904 × 2 = 1 + 0.220 308 684 115 221 369 847 808;
  • 96) 0.220 308 684 115 221 369 847 808 × 2 = 0 + 0.440 617 368 230 442 739 695 616;
  • 97) 0.440 617 368 230 442 739 695 616 × 2 = 0 + 0.881 234 736 460 885 479 391 232;
  • 98) 0.881 234 736 460 885 479 391 232 × 2 = 1 + 0.762 469 472 921 770 958 782 464;
  • 99) 0.762 469 472 921 770 958 782 464 × 2 = 1 + 0.524 938 945 843 541 917 564 928;
  • 100) 0.524 938 945 843 541 917 564 928 × 2 = 1 + 0.049 877 891 687 083 835 129 856;
  • 101) 0.049 877 891 687 083 835 129 856 × 2 = 0 + 0.099 755 783 374 167 670 259 712;
  • 102) 0.099 755 783 374 167 670 259 712 × 2 = 0 + 0.199 511 566 748 335 340 519 424;
  • 103) 0.199 511 566 748 335 340 519 424 × 2 = 0 + 0.399 023 133 496 670 681 038 848;
  • 104) 0.399 023 133 496 670 681 038 848 × 2 = 0 + 0.798 046 266 993 341 362 077 696;
  • 105) 0.798 046 266 993 341 362 077 696 × 2 = 1 + 0.596 092 533 986 682 724 155 392;
  • 106) 0.596 092 533 986 682 724 155 392 × 2 = 1 + 0.192 185 067 973 365 448 310 784;
  • 107) 0.192 185 067 973 365 448 310 784 × 2 = 0 + 0.384 370 135 946 730 896 621 568;
  • 108) 0.384 370 135 946 730 896 621 568 × 2 = 0 + 0.768 740 271 893 461 793 243 136;
  • 109) 0.768 740 271 893 461 793 243 136 × 2 = 1 + 0.537 480 543 786 923 586 486 272;
  • 110) 0.537 480 543 786 923 586 486 272 × 2 = 1 + 0.074 961 087 573 847 172 972 544;
  • 111) 0.074 961 087 573 847 172 972 544 × 2 = 0 + 0.149 922 175 147 694 345 945 088;
  • 112) 0.149 922 175 147 694 345 945 088 × 2 = 0 + 0.299 844 350 295 388 691 890 176;
  • 113) 0.299 844 350 295 388 691 890 176 × 2 = 0 + 0.599 688 700 590 777 383 780 352;
  • 114) 0.599 688 700 590 777 383 780 352 × 2 = 1 + 0.199 377 401 181 554 767 560 704;
  • 115) 0.199 377 401 181 554 767 560 704 × 2 = 0 + 0.398 754 802 363 109 535 121 408;
  • 116) 0.398 754 802 363 109 535 121 408 × 2 = 0 + 0.797 509 604 726 219 070 242 816;
  • 117) 0.797 509 604 726 219 070 242 816 × 2 = 1 + 0.595 019 209 452 438 140 485 632;
  • 118) 0.595 019 209 452 438 140 485 632 × 2 = 1 + 0.190 038 418 904 876 280 971 264;
  • 119) 0.190 038 418 904 876 280 971 264 × 2 = 0 + 0.380 076 837 809 752 561 942 528;

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:


0.000 000 000 000 000 000 008 543 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 0111 0101 0010 1110 0111 0000 1100 1100 0100 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 543 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 0111 0101 0010 1110 0111 0000 1100 1100 0100 110(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:


0.000 000 000 000 000 000 008 543 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 0111 0101 0010 1110 0111 0000 1100 1100 0100 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1000 0111 0101 0010 1110 0111 0000 1100 1100 0100 110(2) × 20 =

1.0100 0010 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110(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 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110


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 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110 =

0100 0010 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110


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 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110


Decimal number 0.000 000 000 000 000 000 008 543 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1100 0011 1010 1001 0111 0011 1000 0110 0110 0010 0110

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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