0.000 000 000 000 000 000 008 554 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 554(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
0.000 000 000 000 000 000 008 554(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: 0.000 000 000 000 000 000 008 554.

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) 0.000 000 000 000 000 000 008 554 × 2 = 0 + 0.000 000 000 000 000 000 017 108;
  • 2) 0.000 000 000 000 000 000 017 108 × 2 = 0 + 0.000 000 000 000 000 000 034 216;
  • 3) 0.000 000 000 000 000 000 034 216 × 2 = 0 + 0.000 000 000 000 000 000 068 432;
  • 4) 0.000 000 000 000 000 000 068 432 × 2 = 0 + 0.000 000 000 000 000 000 136 864;
  • 5) 0.000 000 000 000 000 000 136 864 × 2 = 0 + 0.000 000 000 000 000 000 273 728;
  • 6) 0.000 000 000 000 000 000 273 728 × 2 = 0 + 0.000 000 000 000 000 000 547 456;
  • 7) 0.000 000 000 000 000 000 547 456 × 2 = 0 + 0.000 000 000 000 000 001 094 912;
  • 8) 0.000 000 000 000 000 001 094 912 × 2 = 0 + 0.000 000 000 000 000 002 189 824;
  • 9) 0.000 000 000 000 000 002 189 824 × 2 = 0 + 0.000 000 000 000 000 004 379 648;
  • 10) 0.000 000 000 000 000 004 379 648 × 2 = 0 + 0.000 000 000 000 000 008 759 296;
  • 11) 0.000 000 000 000 000 008 759 296 × 2 = 0 + 0.000 000 000 000 000 017 518 592;
  • 12) 0.000 000 000 000 000 017 518 592 × 2 = 0 + 0.000 000 000 000 000 035 037 184;
  • 13) 0.000 000 000 000 000 035 037 184 × 2 = 0 + 0.000 000 000 000 000 070 074 368;
  • 14) 0.000 000 000 000 000 070 074 368 × 2 = 0 + 0.000 000 000 000 000 140 148 736;
  • 15) 0.000 000 000 000 000 140 148 736 × 2 = 0 + 0.000 000 000 000 000 280 297 472;
  • 16) 0.000 000 000 000 000 280 297 472 × 2 = 0 + 0.000 000 000 000 000 560 594 944;
  • 17) 0.000 000 000 000 000 560 594 944 × 2 = 0 + 0.000 000 000 000 001 121 189 888;
  • 18) 0.000 000 000 000 001 121 189 888 × 2 = 0 + 0.000 000 000 000 002 242 379 776;
  • 19) 0.000 000 000 000 002 242 379 776 × 2 = 0 + 0.000 000 000 000 004 484 759 552;
  • 20) 0.000 000 000 000 004 484 759 552 × 2 = 0 + 0.000 000 000 000 008 969 519 104;
  • 21) 0.000 000 000 000 008 969 519 104 × 2 = 0 + 0.000 000 000 000 017 939 038 208;
  • 22) 0.000 000 000 000 017 939 038 208 × 2 = 0 + 0.000 000 000 000 035 878 076 416;
  • 23) 0.000 000 000 000 035 878 076 416 × 2 = 0 + 0.000 000 000 000 071 756 152 832;
  • 24) 0.000 000 000 000 071 756 152 832 × 2 = 0 + 0.000 000 000 000 143 512 305 664;
  • 25) 0.000 000 000 000 143 512 305 664 × 2 = 0 + 0.000 000 000 000 287 024 611 328;
  • 26) 0.000 000 000 000 287 024 611 328 × 2 = 0 + 0.000 000 000 000 574 049 222 656;
  • 27) 0.000 000 000 000 574 049 222 656 × 2 = 0 + 0.000 000 000 001 148 098 445 312;
  • 28) 0.000 000 000 001 148 098 445 312 × 2 = 0 + 0.000 000 000 002 296 196 890 624;
  • 29) 0.000 000 000 002 296 196 890 624 × 2 = 0 + 0.000 000 000 004 592 393 781 248;
  • 30) 0.000 000 000 004 592 393 781 248 × 2 = 0 + 0.000 000 000 009 184 787 562 496;
  • 31) 0.000 000 000 009 184 787 562 496 × 2 = 0 + 0.000 000 000 018 369 575 124 992;
  • 32) 0.000 000 000 018 369 575 124 992 × 2 = 0 + 0.000 000 000 036 739 150 249 984;
  • 33) 0.000 000 000 036 739 150 249 984 × 2 = 0 + 0.000 000 000 073 478 300 499 968;
  • 34) 0.000 000 000 073 478 300 499 968 × 2 = 0 + 0.000 000 000 146 956 600 999 936;
  • 35) 0.000 000 000 146 956 600 999 936 × 2 = 0 + 0.000 000 000 293 913 201 999 872;
  • 36) 0.000 000 000 293 913 201 999 872 × 2 = 0 + 0.000 000 000 587 826 403 999 744;
  • 37) 0.000 000 000 587 826 403 999 744 × 2 = 0 + 0.000 000 001 175 652 807 999 488;
  • 38) 0.000 000 001 175 652 807 999 488 × 2 = 0 + 0.000 000 002 351 305 615 998 976;
  • 39) 0.000 000 002 351 305 615 998 976 × 2 = 0 + 0.000 000 004 702 611 231 997 952;
  • 40) 0.000 000 004 702 611 231 997 952 × 2 = 0 + 0.000 000 009 405 222 463 995 904;
  • 41) 0.000 000 009 405 222 463 995 904 × 2 = 0 + 0.000 000 018 810 444 927 991 808;
  • 42) 0.000 000 018 810 444 927 991 808 × 2 = 0 + 0.000 000 037 620 889 855 983 616;
  • 43) 0.000 000 037 620 889 855 983 616 × 2 = 0 + 0.000 000 075 241 779 711 967 232;
  • 44) 0.000 000 075 241 779 711 967 232 × 2 = 0 + 0.000 000 150 483 559 423 934 464;
  • 45) 0.000 000 150 483 559 423 934 464 × 2 = 0 + 0.000 000 300 967 118 847 868 928;
  • 46) 0.000 000 300 967 118 847 868 928 × 2 = 0 + 0.000 000 601 934 237 695 737 856;
  • 47) 0.000 000 601 934 237 695 737 856 × 2 = 0 + 0.000 001 203 868 475 391 475 712;
  • 48) 0.000 001 203 868 475 391 475 712 × 2 = 0 + 0.000 002 407 736 950 782 951 424;
  • 49) 0.000 002 407 736 950 782 951 424 × 2 = 0 + 0.000 004 815 473 901 565 902 848;
  • 50) 0.000 004 815 473 901 565 902 848 × 2 = 0 + 0.000 009 630 947 803 131 805 696;
  • 51) 0.000 009 630 947 803 131 805 696 × 2 = 0 + 0.000 019 261 895 606 263 611 392;
  • 52) 0.000 019 261 895 606 263 611 392 × 2 = 0 + 0.000 038 523 791 212 527 222 784;
  • 53) 0.000 038 523 791 212 527 222 784 × 2 = 0 + 0.000 077 047 582 425 054 445 568;
  • 54) 0.000 077 047 582 425 054 445 568 × 2 = 0 + 0.000 154 095 164 850 108 891 136;
  • 55) 0.000 154 095 164 850 108 891 136 × 2 = 0 + 0.000 308 190 329 700 217 782 272;
  • 56) 0.000 308 190 329 700 217 782 272 × 2 = 0 + 0.000 616 380 659 400 435 564 544;
  • 57) 0.000 616 380 659 400 435 564 544 × 2 = 0 + 0.001 232 761 318 800 871 129 088;
  • 58) 0.001 232 761 318 800 871 129 088 × 2 = 0 + 0.002 465 522 637 601 742 258 176;
  • 59) 0.002 465 522 637 601 742 258 176 × 2 = 0 + 0.004 931 045 275 203 484 516 352;
  • 60) 0.004 931 045 275 203 484 516 352 × 2 = 0 + 0.009 862 090 550 406 969 032 704;
  • 61) 0.009 862 090 550 406 969 032 704 × 2 = 0 + 0.019 724 181 100 813 938 065 408;
  • 62) 0.019 724 181 100 813 938 065 408 × 2 = 0 + 0.039 448 362 201 627 876 130 816;
  • 63) 0.039 448 362 201 627 876 130 816 × 2 = 0 + 0.078 896 724 403 255 752 261 632;
  • 64) 0.078 896 724 403 255 752 261 632 × 2 = 0 + 0.157 793 448 806 511 504 523 264;
  • 65) 0.157 793 448 806 511 504 523 264 × 2 = 0 + 0.315 586 897 613 023 009 046 528;
  • 66) 0.315 586 897 613 023 009 046 528 × 2 = 0 + 0.631 173 795 226 046 018 093 056;
  • 67) 0.631 173 795 226 046 018 093 056 × 2 = 1 + 0.262 347 590 452 092 036 186 112;
  • 68) 0.262 347 590 452 092 036 186 112 × 2 = 0 + 0.524 695 180 904 184 072 372 224;
  • 69) 0.524 695 180 904 184 072 372 224 × 2 = 1 + 0.049 390 361 808 368 144 744 448;
  • 70) 0.049 390 361 808 368 144 744 448 × 2 = 0 + 0.098 780 723 616 736 289 488 896;
  • 71) 0.098 780 723 616 736 289 488 896 × 2 = 0 + 0.197 561 447 233 472 578 977 792;
  • 72) 0.197 561 447 233 472 578 977 792 × 2 = 0 + 0.395 122 894 466 945 157 955 584;
  • 73) 0.395 122 894 466 945 157 955 584 × 2 = 0 + 0.790 245 788 933 890 315 911 168;
  • 74) 0.790 245 788 933 890 315 911 168 × 2 = 1 + 0.580 491 577 867 780 631 822 336;
  • 75) 0.580 491 577 867 780 631 822 336 × 2 = 1 + 0.160 983 155 735 561 263 644 672;
  • 76) 0.160 983 155 735 561 263 644 672 × 2 = 0 + 0.321 966 311 471 122 527 289 344;
  • 77) 0.321 966 311 471 122 527 289 344 × 2 = 0 + 0.643 932 622 942 245 054 578 688;
  • 78) 0.643 932 622 942 245 054 578 688 × 2 = 1 + 0.287 865 245 884 490 109 157 376;
  • 79) 0.287 865 245 884 490 109 157 376 × 2 = 0 + 0.575 730 491 768 980 218 314 752;
  • 80) 0.575 730 491 768 980 218 314 752 × 2 = 1 + 0.151 460 983 537 960 436 629 504;
  • 81) 0.151 460 983 537 960 436 629 504 × 2 = 0 + 0.302 921 967 075 920 873 259 008;
  • 82) 0.302 921 967 075 920 873 259 008 × 2 = 0 + 0.605 843 934 151 841 746 518 016;
  • 83) 0.605 843 934 151 841 746 518 016 × 2 = 1 + 0.211 687 868 303 683 493 036 032;
  • 84) 0.211 687 868 303 683 493 036 032 × 2 = 0 + 0.423 375 736 607 366 986 072 064;
  • 85) 0.423 375 736 607 366 986 072 064 × 2 = 0 + 0.846 751 473 214 733 972 144 128;
  • 86) 0.846 751 473 214 733 972 144 128 × 2 = 1 + 0.693 502 946 429 467 944 288 256;
  • 87) 0.693 502 946 429 467 944 288 256 × 2 = 1 + 0.387 005 892 858 935 888 576 512;
  • 88) 0.387 005 892 858 935 888 576 512 × 2 = 0 + 0.774 011 785 717 871 777 153 024;
  • 89) 0.774 011 785 717 871 777 153 024 × 2 = 1 + 0.548 023 571 435 743 554 306 048;
  • 90) 0.548 023 571 435 743 554 306 048 × 2 = 1 + 0.096 047 142 871 487 108 612 096;
  • 91) 0.096 047 142 871 487 108 612 096 × 2 = 0 + 0.192 094 285 742 974 217 224 192;
  • 92) 0.192 094 285 742 974 217 224 192 × 2 = 0 + 0.384 188 571 485 948 434 448 384;
  • 93) 0.384 188 571 485 948 434 448 384 × 2 = 0 + 0.768 377 142 971 896 868 896 768;
  • 94) 0.768 377 142 971 896 868 896 768 × 2 = 1 + 0.536 754 285 943 793 737 793 536;
  • 95) 0.536 754 285 943 793 737 793 536 × 2 = 1 + 0.073 508 571 887 587 475 587 072;
  • 96) 0.073 508 571 887 587 475 587 072 × 2 = 0 + 0.147 017 143 775 174 951 174 144;
  • 97) 0.147 017 143 775 174 951 174 144 × 2 = 0 + 0.294 034 287 550 349 902 348 288;
  • 98) 0.294 034 287 550 349 902 348 288 × 2 = 0 + 0.588 068 575 100 699 804 696 576;
  • 99) 0.588 068 575 100 699 804 696 576 × 2 = 1 + 0.176 137 150 201 399 609 393 152;
  • 100) 0.176 137 150 201 399 609 393 152 × 2 = 0 + 0.352 274 300 402 799 218 786 304;
  • 101) 0.352 274 300 402 799 218 786 304 × 2 = 0 + 0.704 548 600 805 598 437 572 608;
  • 102) 0.704 548 600 805 598 437 572 608 × 2 = 1 + 0.409 097 201 611 196 875 145 216;
  • 103) 0.409 097 201 611 196 875 145 216 × 2 = 0 + 0.818 194 403 222 393 750 290 432;
  • 104) 0.818 194 403 222 393 750 290 432 × 2 = 1 + 0.636 388 806 444 787 500 580 864;
  • 105) 0.636 388 806 444 787 500 580 864 × 2 = 1 + 0.272 777 612 889 575 001 161 728;
  • 106) 0.272 777 612 889 575 001 161 728 × 2 = 0 + 0.545 555 225 779 150 002 323 456;
  • 107) 0.545 555 225 779 150 002 323 456 × 2 = 1 + 0.091 110 451 558 300 004 646 912;
  • 108) 0.091 110 451 558 300 004 646 912 × 2 = 0 + 0.182 220 903 116 600 009 293 824;
  • 109) 0.182 220 903 116 600 009 293 824 × 2 = 0 + 0.364 441 806 233 200 018 587 648;
  • 110) 0.364 441 806 233 200 018 587 648 × 2 = 0 + 0.728 883 612 466 400 037 175 296;
  • 111) 0.728 883 612 466 400 037 175 296 × 2 = 1 + 0.457 767 224 932 800 074 350 592;
  • 112) 0.457 767 224 932 800 074 350 592 × 2 = 0 + 0.915 534 449 865 600 148 701 184;
  • 113) 0.915 534 449 865 600 148 701 184 × 2 = 1 + 0.831 068 899 731 200 297 402 368;
  • 114) 0.831 068 899 731 200 297 402 368 × 2 = 1 + 0.662 137 799 462 400 594 804 736;
  • 115) 0.662 137 799 462 400 594 804 736 × 2 = 1 + 0.324 275 598 924 801 189 609 472;
  • 116) 0.324 275 598 924 801 189 609 472 × 2 = 0 + 0.648 551 197 849 602 379 218 944;
  • 117) 0.648 551 197 849 602 379 218 944 × 2 = 1 + 0.297 102 395 699 204 758 437 888;
  • 118) 0.297 102 395 699 204 758 437 888 × 2 = 0 + 0.594 204 791 398 409 516 875 776;
  • 119) 0.594 204 791 398 409 516 875 776 × 2 = 1 + 0.188 409 582 796 819 033 751 552;

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:


0.000 000 000 000 000 000 008 554(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0101 0010 0110 1100 0110 0010 0101 1010 0010 1110 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 554(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0101 0010 0110 1100 0110 0010 0101 1010 0010 1110 101(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:


0.000 000 000 000 000 000 008 554(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0101 0010 0110 1100 0110 0010 0101 1010 0010 1110 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0101 0010 0110 1100 0110 0010 0101 1010 0010 1110 101(2) × 20 =

1.0100 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101(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 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101


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 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101 =

0100 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101


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 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101


Decimal number 0.000 000 000 000 000 000 008 554 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0010 1001 0011 0110 0011 0001 0010 1101 0001 0111 0101

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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