0.000 000 000 000 000 000 008 534 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534(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
0.000 000 000 000 000 000 008 534(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: 0.000 000 000 000 000 000 008 534.

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) 0.000 000 000 000 000 000 008 534 × 2 = 0 + 0.000 000 000 000 000 000 017 068;
  • 2) 0.000 000 000 000 000 000 017 068 × 2 = 0 + 0.000 000 000 000 000 000 034 136;
  • 3) 0.000 000 000 000 000 000 034 136 × 2 = 0 + 0.000 000 000 000 000 000 068 272;
  • 4) 0.000 000 000 000 000 000 068 272 × 2 = 0 + 0.000 000 000 000 000 000 136 544;
  • 5) 0.000 000 000 000 000 000 136 544 × 2 = 0 + 0.000 000 000 000 000 000 273 088;
  • 6) 0.000 000 000 000 000 000 273 088 × 2 = 0 + 0.000 000 000 000 000 000 546 176;
  • 7) 0.000 000 000 000 000 000 546 176 × 2 = 0 + 0.000 000 000 000 000 001 092 352;
  • 8) 0.000 000 000 000 000 001 092 352 × 2 = 0 + 0.000 000 000 000 000 002 184 704;
  • 9) 0.000 000 000 000 000 002 184 704 × 2 = 0 + 0.000 000 000 000 000 004 369 408;
  • 10) 0.000 000 000 000 000 004 369 408 × 2 = 0 + 0.000 000 000 000 000 008 738 816;
  • 11) 0.000 000 000 000 000 008 738 816 × 2 = 0 + 0.000 000 000 000 000 017 477 632;
  • 12) 0.000 000 000 000 000 017 477 632 × 2 = 0 + 0.000 000 000 000 000 034 955 264;
  • 13) 0.000 000 000 000 000 034 955 264 × 2 = 0 + 0.000 000 000 000 000 069 910 528;
  • 14) 0.000 000 000 000 000 069 910 528 × 2 = 0 + 0.000 000 000 000 000 139 821 056;
  • 15) 0.000 000 000 000 000 139 821 056 × 2 = 0 + 0.000 000 000 000 000 279 642 112;
  • 16) 0.000 000 000 000 000 279 642 112 × 2 = 0 + 0.000 000 000 000 000 559 284 224;
  • 17) 0.000 000 000 000 000 559 284 224 × 2 = 0 + 0.000 000 000 000 001 118 568 448;
  • 18) 0.000 000 000 000 001 118 568 448 × 2 = 0 + 0.000 000 000 000 002 237 136 896;
  • 19) 0.000 000 000 000 002 237 136 896 × 2 = 0 + 0.000 000 000 000 004 474 273 792;
  • 20) 0.000 000 000 000 004 474 273 792 × 2 = 0 + 0.000 000 000 000 008 948 547 584;
  • 21) 0.000 000 000 000 008 948 547 584 × 2 = 0 + 0.000 000 000 000 017 897 095 168;
  • 22) 0.000 000 000 000 017 897 095 168 × 2 = 0 + 0.000 000 000 000 035 794 190 336;
  • 23) 0.000 000 000 000 035 794 190 336 × 2 = 0 + 0.000 000 000 000 071 588 380 672;
  • 24) 0.000 000 000 000 071 588 380 672 × 2 = 0 + 0.000 000 000 000 143 176 761 344;
  • 25) 0.000 000 000 000 143 176 761 344 × 2 = 0 + 0.000 000 000 000 286 353 522 688;
  • 26) 0.000 000 000 000 286 353 522 688 × 2 = 0 + 0.000 000 000 000 572 707 045 376;
  • 27) 0.000 000 000 000 572 707 045 376 × 2 = 0 + 0.000 000 000 001 145 414 090 752;
  • 28) 0.000 000 000 001 145 414 090 752 × 2 = 0 + 0.000 000 000 002 290 828 181 504;
  • 29) 0.000 000 000 002 290 828 181 504 × 2 = 0 + 0.000 000 000 004 581 656 363 008;
  • 30) 0.000 000 000 004 581 656 363 008 × 2 = 0 + 0.000 000 000 009 163 312 726 016;
  • 31) 0.000 000 000 009 163 312 726 016 × 2 = 0 + 0.000 000 000 018 326 625 452 032;
  • 32) 0.000 000 000 018 326 625 452 032 × 2 = 0 + 0.000 000 000 036 653 250 904 064;
  • 33) 0.000 000 000 036 653 250 904 064 × 2 = 0 + 0.000 000 000 073 306 501 808 128;
  • 34) 0.000 000 000 073 306 501 808 128 × 2 = 0 + 0.000 000 000 146 613 003 616 256;
  • 35) 0.000 000 000 146 613 003 616 256 × 2 = 0 + 0.000 000 000 293 226 007 232 512;
  • 36) 0.000 000 000 293 226 007 232 512 × 2 = 0 + 0.000 000 000 586 452 014 465 024;
  • 37) 0.000 000 000 586 452 014 465 024 × 2 = 0 + 0.000 000 001 172 904 028 930 048;
  • 38) 0.000 000 001 172 904 028 930 048 × 2 = 0 + 0.000 000 002 345 808 057 860 096;
  • 39) 0.000 000 002 345 808 057 860 096 × 2 = 0 + 0.000 000 004 691 616 115 720 192;
  • 40) 0.000 000 004 691 616 115 720 192 × 2 = 0 + 0.000 000 009 383 232 231 440 384;
  • 41) 0.000 000 009 383 232 231 440 384 × 2 = 0 + 0.000 000 018 766 464 462 880 768;
  • 42) 0.000 000 018 766 464 462 880 768 × 2 = 0 + 0.000 000 037 532 928 925 761 536;
  • 43) 0.000 000 037 532 928 925 761 536 × 2 = 0 + 0.000 000 075 065 857 851 523 072;
  • 44) 0.000 000 075 065 857 851 523 072 × 2 = 0 + 0.000 000 150 131 715 703 046 144;
  • 45) 0.000 000 150 131 715 703 046 144 × 2 = 0 + 0.000 000 300 263 431 406 092 288;
  • 46) 0.000 000 300 263 431 406 092 288 × 2 = 0 + 0.000 000 600 526 862 812 184 576;
  • 47) 0.000 000 600 526 862 812 184 576 × 2 = 0 + 0.000 001 201 053 725 624 369 152;
  • 48) 0.000 001 201 053 725 624 369 152 × 2 = 0 + 0.000 002 402 107 451 248 738 304;
  • 49) 0.000 002 402 107 451 248 738 304 × 2 = 0 + 0.000 004 804 214 902 497 476 608;
  • 50) 0.000 004 804 214 902 497 476 608 × 2 = 0 + 0.000 009 608 429 804 994 953 216;
  • 51) 0.000 009 608 429 804 994 953 216 × 2 = 0 + 0.000 019 216 859 609 989 906 432;
  • 52) 0.000 019 216 859 609 989 906 432 × 2 = 0 + 0.000 038 433 719 219 979 812 864;
  • 53) 0.000 038 433 719 219 979 812 864 × 2 = 0 + 0.000 076 867 438 439 959 625 728;
  • 54) 0.000 076 867 438 439 959 625 728 × 2 = 0 + 0.000 153 734 876 879 919 251 456;
  • 55) 0.000 153 734 876 879 919 251 456 × 2 = 0 + 0.000 307 469 753 759 838 502 912;
  • 56) 0.000 307 469 753 759 838 502 912 × 2 = 0 + 0.000 614 939 507 519 677 005 824;
  • 57) 0.000 614 939 507 519 677 005 824 × 2 = 0 + 0.001 229 879 015 039 354 011 648;
  • 58) 0.001 229 879 015 039 354 011 648 × 2 = 0 + 0.002 459 758 030 078 708 023 296;
  • 59) 0.002 459 758 030 078 708 023 296 × 2 = 0 + 0.004 919 516 060 157 416 046 592;
  • 60) 0.004 919 516 060 157 416 046 592 × 2 = 0 + 0.009 839 032 120 314 832 093 184;
  • 61) 0.009 839 032 120 314 832 093 184 × 2 = 0 + 0.019 678 064 240 629 664 186 368;
  • 62) 0.019 678 064 240 629 664 186 368 × 2 = 0 + 0.039 356 128 481 259 328 372 736;
  • 63) 0.039 356 128 481 259 328 372 736 × 2 = 0 + 0.078 712 256 962 518 656 745 472;
  • 64) 0.078 712 256 962 518 656 745 472 × 2 = 0 + 0.157 424 513 925 037 313 490 944;
  • 65) 0.157 424 513 925 037 313 490 944 × 2 = 0 + 0.314 849 027 850 074 626 981 888;
  • 66) 0.314 849 027 850 074 626 981 888 × 2 = 0 + 0.629 698 055 700 149 253 963 776;
  • 67) 0.629 698 055 700 149 253 963 776 × 2 = 1 + 0.259 396 111 400 298 507 927 552;
  • 68) 0.259 396 111 400 298 507 927 552 × 2 = 0 + 0.518 792 222 800 597 015 855 104;
  • 69) 0.518 792 222 800 597 015 855 104 × 2 = 1 + 0.037 584 445 601 194 031 710 208;
  • 70) 0.037 584 445 601 194 031 710 208 × 2 = 0 + 0.075 168 891 202 388 063 420 416;
  • 71) 0.075 168 891 202 388 063 420 416 × 2 = 0 + 0.150 337 782 404 776 126 840 832;
  • 72) 0.150 337 782 404 776 126 840 832 × 2 = 0 + 0.300 675 564 809 552 253 681 664;
  • 73) 0.300 675 564 809 552 253 681 664 × 2 = 0 + 0.601 351 129 619 104 507 363 328;
  • 74) 0.601 351 129 619 104 507 363 328 × 2 = 1 + 0.202 702 259 238 209 014 726 656;
  • 75) 0.202 702 259 238 209 014 726 656 × 2 = 0 + 0.405 404 518 476 418 029 453 312;
  • 76) 0.405 404 518 476 418 029 453 312 × 2 = 0 + 0.810 809 036 952 836 058 906 624;
  • 77) 0.810 809 036 952 836 058 906 624 × 2 = 1 + 0.621 618 073 905 672 117 813 248;
  • 78) 0.621 618 073 905 672 117 813 248 × 2 = 1 + 0.243 236 147 811 344 235 626 496;
  • 79) 0.243 236 147 811 344 235 626 496 × 2 = 0 + 0.486 472 295 622 688 471 252 992;
  • 80) 0.486 472 295 622 688 471 252 992 × 2 = 0 + 0.972 944 591 245 376 942 505 984;
  • 81) 0.972 944 591 245 376 942 505 984 × 2 = 1 + 0.945 889 182 490 753 885 011 968;
  • 82) 0.945 889 182 490 753 885 011 968 × 2 = 1 + 0.891 778 364 981 507 770 023 936;
  • 83) 0.891 778 364 981 507 770 023 936 × 2 = 1 + 0.783 556 729 963 015 540 047 872;
  • 84) 0.783 556 729 963 015 540 047 872 × 2 = 1 + 0.567 113 459 926 031 080 095 744;
  • 85) 0.567 113 459 926 031 080 095 744 × 2 = 1 + 0.134 226 919 852 062 160 191 488;
  • 86) 0.134 226 919 852 062 160 191 488 × 2 = 0 + 0.268 453 839 704 124 320 382 976;
  • 87) 0.268 453 839 704 124 320 382 976 × 2 = 0 + 0.536 907 679 408 248 640 765 952;
  • 88) 0.536 907 679 408 248 640 765 952 × 2 = 1 + 0.073 815 358 816 497 281 531 904;
  • 89) 0.073 815 358 816 497 281 531 904 × 2 = 0 + 0.147 630 717 632 994 563 063 808;
  • 90) 0.147 630 717 632 994 563 063 808 × 2 = 0 + 0.295 261 435 265 989 126 127 616;
  • 91) 0.295 261 435 265 989 126 127 616 × 2 = 0 + 0.590 522 870 531 978 252 255 232;
  • 92) 0.590 522 870 531 978 252 255 232 × 2 = 1 + 0.181 045 741 063 956 504 510 464;
  • 93) 0.181 045 741 063 956 504 510 464 × 2 = 0 + 0.362 091 482 127 913 009 020 928;
  • 94) 0.362 091 482 127 913 009 020 928 × 2 = 0 + 0.724 182 964 255 826 018 041 856;
  • 95) 0.724 182 964 255 826 018 041 856 × 2 = 1 + 0.448 365 928 511 652 036 083 712;
  • 96) 0.448 365 928 511 652 036 083 712 × 2 = 0 + 0.896 731 857 023 304 072 167 424;
  • 97) 0.896 731 857 023 304 072 167 424 × 2 = 1 + 0.793 463 714 046 608 144 334 848;
  • 98) 0.793 463 714 046 608 144 334 848 × 2 = 1 + 0.586 927 428 093 216 288 669 696;
  • 99) 0.586 927 428 093 216 288 669 696 × 2 = 1 + 0.173 854 856 186 432 577 339 392;
  • 100) 0.173 854 856 186 432 577 339 392 × 2 = 0 + 0.347 709 712 372 865 154 678 784;
  • 101) 0.347 709 712 372 865 154 678 784 × 2 = 0 + 0.695 419 424 745 730 309 357 568;
  • 102) 0.695 419 424 745 730 309 357 568 × 2 = 1 + 0.390 838 849 491 460 618 715 136;
  • 103) 0.390 838 849 491 460 618 715 136 × 2 = 0 + 0.781 677 698 982 921 237 430 272;
  • 104) 0.781 677 698 982 921 237 430 272 × 2 = 1 + 0.563 355 397 965 842 474 860 544;
  • 105) 0.563 355 397 965 842 474 860 544 × 2 = 1 + 0.126 710 795 931 684 949 721 088;
  • 106) 0.126 710 795 931 684 949 721 088 × 2 = 0 + 0.253 421 591 863 369 899 442 176;
  • 107) 0.253 421 591 863 369 899 442 176 × 2 = 0 + 0.506 843 183 726 739 798 884 352;
  • 108) 0.506 843 183 726 739 798 884 352 × 2 = 1 + 0.013 686 367 453 479 597 768 704;
  • 109) 0.013 686 367 453 479 597 768 704 × 2 = 0 + 0.027 372 734 906 959 195 537 408;
  • 110) 0.027 372 734 906 959 195 537 408 × 2 = 0 + 0.054 745 469 813 918 391 074 816;
  • 111) 0.054 745 469 813 918 391 074 816 × 2 = 0 + 0.109 490 939 627 836 782 149 632;
  • 112) 0.109 490 939 627 836 782 149 632 × 2 = 0 + 0.218 981 879 255 673 564 299 264;
  • 113) 0.218 981 879 255 673 564 299 264 × 2 = 0 + 0.437 963 758 511 347 128 598 528;
  • 114) 0.437 963 758 511 347 128 598 528 × 2 = 0 + 0.875 927 517 022 694 257 197 056;
  • 115) 0.875 927 517 022 694 257 197 056 × 2 = 1 + 0.751 855 034 045 388 514 394 112;
  • 116) 0.751 855 034 045 388 514 394 112 × 2 = 1 + 0.503 710 068 090 777 028 788 224;
  • 117) 0.503 710 068 090 777 028 788 224 × 2 = 1 + 0.007 420 136 181 554 057 576 448;
  • 118) 0.007 420 136 181 554 057 576 448 × 2 = 0 + 0.014 840 272 363 108 115 152 896;
  • 119) 0.014 840 272 363 108 115 152 896 × 2 = 0 + 0.029 680 544 726 216 230 305 792;

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:


0.000 000 000 000 000 000 008 534(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 1111 1001 0001 0010 1110 0101 1001 0000 0011 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 1111 1001 0001 0010 1110 0101 1001 0000 0011 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:


0.000 000 000 000 000 000 008 534(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 1111 1001 0001 0010 1110 0101 1001 0000 0011 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 1111 1001 0001 0010 1110 0101 1001 0000 0011 100(2) × 20 =

1.0100 0010 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100(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 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100


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 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100 =

0100 0010 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100


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 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100


Decimal number 0.000 000 000 000 000 000 008 534 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 0111 1100 1000 1001 0111 0010 1100 1000 0001 1100

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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