0.000 000 000 000 000 000 008 521 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 521(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
0.000 000 000 000 000 000 008 521(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: 0.000 000 000 000 000 000 008 521.

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) 0.000 000 000 000 000 000 008 521 × 2 = 0 + 0.000 000 000 000 000 000 017 042;
  • 2) 0.000 000 000 000 000 000 017 042 × 2 = 0 + 0.000 000 000 000 000 000 034 084;
  • 3) 0.000 000 000 000 000 000 034 084 × 2 = 0 + 0.000 000 000 000 000 000 068 168;
  • 4) 0.000 000 000 000 000 000 068 168 × 2 = 0 + 0.000 000 000 000 000 000 136 336;
  • 5) 0.000 000 000 000 000 000 136 336 × 2 = 0 + 0.000 000 000 000 000 000 272 672;
  • 6) 0.000 000 000 000 000 000 272 672 × 2 = 0 + 0.000 000 000 000 000 000 545 344;
  • 7) 0.000 000 000 000 000 000 545 344 × 2 = 0 + 0.000 000 000 000 000 001 090 688;
  • 8) 0.000 000 000 000 000 001 090 688 × 2 = 0 + 0.000 000 000 000 000 002 181 376;
  • 9) 0.000 000 000 000 000 002 181 376 × 2 = 0 + 0.000 000 000 000 000 004 362 752;
  • 10) 0.000 000 000 000 000 004 362 752 × 2 = 0 + 0.000 000 000 000 000 008 725 504;
  • 11) 0.000 000 000 000 000 008 725 504 × 2 = 0 + 0.000 000 000 000 000 017 451 008;
  • 12) 0.000 000 000 000 000 017 451 008 × 2 = 0 + 0.000 000 000 000 000 034 902 016;
  • 13) 0.000 000 000 000 000 034 902 016 × 2 = 0 + 0.000 000 000 000 000 069 804 032;
  • 14) 0.000 000 000 000 000 069 804 032 × 2 = 0 + 0.000 000 000 000 000 139 608 064;
  • 15) 0.000 000 000 000 000 139 608 064 × 2 = 0 + 0.000 000 000 000 000 279 216 128;
  • 16) 0.000 000 000 000 000 279 216 128 × 2 = 0 + 0.000 000 000 000 000 558 432 256;
  • 17) 0.000 000 000 000 000 558 432 256 × 2 = 0 + 0.000 000 000 000 001 116 864 512;
  • 18) 0.000 000 000 000 001 116 864 512 × 2 = 0 + 0.000 000 000 000 002 233 729 024;
  • 19) 0.000 000 000 000 002 233 729 024 × 2 = 0 + 0.000 000 000 000 004 467 458 048;
  • 20) 0.000 000 000 000 004 467 458 048 × 2 = 0 + 0.000 000 000 000 008 934 916 096;
  • 21) 0.000 000 000 000 008 934 916 096 × 2 = 0 + 0.000 000 000 000 017 869 832 192;
  • 22) 0.000 000 000 000 017 869 832 192 × 2 = 0 + 0.000 000 000 000 035 739 664 384;
  • 23) 0.000 000 000 000 035 739 664 384 × 2 = 0 + 0.000 000 000 000 071 479 328 768;
  • 24) 0.000 000 000 000 071 479 328 768 × 2 = 0 + 0.000 000 000 000 142 958 657 536;
  • 25) 0.000 000 000 000 142 958 657 536 × 2 = 0 + 0.000 000 000 000 285 917 315 072;
  • 26) 0.000 000 000 000 285 917 315 072 × 2 = 0 + 0.000 000 000 000 571 834 630 144;
  • 27) 0.000 000 000 000 571 834 630 144 × 2 = 0 + 0.000 000 000 001 143 669 260 288;
  • 28) 0.000 000 000 001 143 669 260 288 × 2 = 0 + 0.000 000 000 002 287 338 520 576;
  • 29) 0.000 000 000 002 287 338 520 576 × 2 = 0 + 0.000 000 000 004 574 677 041 152;
  • 30) 0.000 000 000 004 574 677 041 152 × 2 = 0 + 0.000 000 000 009 149 354 082 304;
  • 31) 0.000 000 000 009 149 354 082 304 × 2 = 0 + 0.000 000 000 018 298 708 164 608;
  • 32) 0.000 000 000 018 298 708 164 608 × 2 = 0 + 0.000 000 000 036 597 416 329 216;
  • 33) 0.000 000 000 036 597 416 329 216 × 2 = 0 + 0.000 000 000 073 194 832 658 432;
  • 34) 0.000 000 000 073 194 832 658 432 × 2 = 0 + 0.000 000 000 146 389 665 316 864;
  • 35) 0.000 000 000 146 389 665 316 864 × 2 = 0 + 0.000 000 000 292 779 330 633 728;
  • 36) 0.000 000 000 292 779 330 633 728 × 2 = 0 + 0.000 000 000 585 558 661 267 456;
  • 37) 0.000 000 000 585 558 661 267 456 × 2 = 0 + 0.000 000 001 171 117 322 534 912;
  • 38) 0.000 000 001 171 117 322 534 912 × 2 = 0 + 0.000 000 002 342 234 645 069 824;
  • 39) 0.000 000 002 342 234 645 069 824 × 2 = 0 + 0.000 000 004 684 469 290 139 648;
  • 40) 0.000 000 004 684 469 290 139 648 × 2 = 0 + 0.000 000 009 368 938 580 279 296;
  • 41) 0.000 000 009 368 938 580 279 296 × 2 = 0 + 0.000 000 018 737 877 160 558 592;
  • 42) 0.000 000 018 737 877 160 558 592 × 2 = 0 + 0.000 000 037 475 754 321 117 184;
  • 43) 0.000 000 037 475 754 321 117 184 × 2 = 0 + 0.000 000 074 951 508 642 234 368;
  • 44) 0.000 000 074 951 508 642 234 368 × 2 = 0 + 0.000 000 149 903 017 284 468 736;
  • 45) 0.000 000 149 903 017 284 468 736 × 2 = 0 + 0.000 000 299 806 034 568 937 472;
  • 46) 0.000 000 299 806 034 568 937 472 × 2 = 0 + 0.000 000 599 612 069 137 874 944;
  • 47) 0.000 000 599 612 069 137 874 944 × 2 = 0 + 0.000 001 199 224 138 275 749 888;
  • 48) 0.000 001 199 224 138 275 749 888 × 2 = 0 + 0.000 002 398 448 276 551 499 776;
  • 49) 0.000 002 398 448 276 551 499 776 × 2 = 0 + 0.000 004 796 896 553 102 999 552;
  • 50) 0.000 004 796 896 553 102 999 552 × 2 = 0 + 0.000 009 593 793 106 205 999 104;
  • 51) 0.000 009 593 793 106 205 999 104 × 2 = 0 + 0.000 019 187 586 212 411 998 208;
  • 52) 0.000 019 187 586 212 411 998 208 × 2 = 0 + 0.000 038 375 172 424 823 996 416;
  • 53) 0.000 038 375 172 424 823 996 416 × 2 = 0 + 0.000 076 750 344 849 647 992 832;
  • 54) 0.000 076 750 344 849 647 992 832 × 2 = 0 + 0.000 153 500 689 699 295 985 664;
  • 55) 0.000 153 500 689 699 295 985 664 × 2 = 0 + 0.000 307 001 379 398 591 971 328;
  • 56) 0.000 307 001 379 398 591 971 328 × 2 = 0 + 0.000 614 002 758 797 183 942 656;
  • 57) 0.000 614 002 758 797 183 942 656 × 2 = 0 + 0.001 228 005 517 594 367 885 312;
  • 58) 0.001 228 005 517 594 367 885 312 × 2 = 0 + 0.002 456 011 035 188 735 770 624;
  • 59) 0.002 456 011 035 188 735 770 624 × 2 = 0 + 0.004 912 022 070 377 471 541 248;
  • 60) 0.004 912 022 070 377 471 541 248 × 2 = 0 + 0.009 824 044 140 754 943 082 496;
  • 61) 0.009 824 044 140 754 943 082 496 × 2 = 0 + 0.019 648 088 281 509 886 164 992;
  • 62) 0.019 648 088 281 509 886 164 992 × 2 = 0 + 0.039 296 176 563 019 772 329 984;
  • 63) 0.039 296 176 563 019 772 329 984 × 2 = 0 + 0.078 592 353 126 039 544 659 968;
  • 64) 0.078 592 353 126 039 544 659 968 × 2 = 0 + 0.157 184 706 252 079 089 319 936;
  • 65) 0.157 184 706 252 079 089 319 936 × 2 = 0 + 0.314 369 412 504 158 178 639 872;
  • 66) 0.314 369 412 504 158 178 639 872 × 2 = 0 + 0.628 738 825 008 316 357 279 744;
  • 67) 0.628 738 825 008 316 357 279 744 × 2 = 1 + 0.257 477 650 016 632 714 559 488;
  • 68) 0.257 477 650 016 632 714 559 488 × 2 = 0 + 0.514 955 300 033 265 429 118 976;
  • 69) 0.514 955 300 033 265 429 118 976 × 2 = 1 + 0.029 910 600 066 530 858 237 952;
  • 70) 0.029 910 600 066 530 858 237 952 × 2 = 0 + 0.059 821 200 133 061 716 475 904;
  • 71) 0.059 821 200 133 061 716 475 904 × 2 = 0 + 0.119 642 400 266 123 432 951 808;
  • 72) 0.119 642 400 266 123 432 951 808 × 2 = 0 + 0.239 284 800 532 246 865 903 616;
  • 73) 0.239 284 800 532 246 865 903 616 × 2 = 0 + 0.478 569 601 064 493 731 807 232;
  • 74) 0.478 569 601 064 493 731 807 232 × 2 = 0 + 0.957 139 202 128 987 463 614 464;
  • 75) 0.957 139 202 128 987 463 614 464 × 2 = 1 + 0.914 278 404 257 974 927 228 928;
  • 76) 0.914 278 404 257 974 927 228 928 × 2 = 1 + 0.828 556 808 515 949 854 457 856;
  • 77) 0.828 556 808 515 949 854 457 856 × 2 = 1 + 0.657 113 617 031 899 708 915 712;
  • 78) 0.657 113 617 031 899 708 915 712 × 2 = 1 + 0.314 227 234 063 799 417 831 424;
  • 79) 0.314 227 234 063 799 417 831 424 × 2 = 0 + 0.628 454 468 127 598 835 662 848;
  • 80) 0.628 454 468 127 598 835 662 848 × 2 = 1 + 0.256 908 936 255 197 671 325 696;
  • 81) 0.256 908 936 255 197 671 325 696 × 2 = 0 + 0.513 817 872 510 395 342 651 392;
  • 82) 0.513 817 872 510 395 342 651 392 × 2 = 1 + 0.027 635 745 020 790 685 302 784;
  • 83) 0.027 635 745 020 790 685 302 784 × 2 = 0 + 0.055 271 490 041 581 370 605 568;
  • 84) 0.055 271 490 041 581 370 605 568 × 2 = 0 + 0.110 542 980 083 162 741 211 136;
  • 85) 0.110 542 980 083 162 741 211 136 × 2 = 0 + 0.221 085 960 166 325 482 422 272;
  • 86) 0.221 085 960 166 325 482 422 272 × 2 = 0 + 0.442 171 920 332 650 964 844 544;
  • 87) 0.442 171 920 332 650 964 844 544 × 2 = 0 + 0.884 343 840 665 301 929 689 088;
  • 88) 0.884 343 840 665 301 929 689 088 × 2 = 1 + 0.768 687 681 330 603 859 378 176;
  • 89) 0.768 687 681 330 603 859 378 176 × 2 = 1 + 0.537 375 362 661 207 718 756 352;
  • 90) 0.537 375 362 661 207 718 756 352 × 2 = 1 + 0.074 750 725 322 415 437 512 704;
  • 91) 0.074 750 725 322 415 437 512 704 × 2 = 0 + 0.149 501 450 644 830 875 025 408;
  • 92) 0.149 501 450 644 830 875 025 408 × 2 = 0 + 0.299 002 901 289 661 750 050 816;
  • 93) 0.299 002 901 289 661 750 050 816 × 2 = 0 + 0.598 005 802 579 323 500 101 632;
  • 94) 0.598 005 802 579 323 500 101 632 × 2 = 1 + 0.196 011 605 158 647 000 203 264;
  • 95) 0.196 011 605 158 647 000 203 264 × 2 = 0 + 0.392 023 210 317 294 000 406 528;
  • 96) 0.392 023 210 317 294 000 406 528 × 2 = 0 + 0.784 046 420 634 588 000 813 056;
  • 97) 0.784 046 420 634 588 000 813 056 × 2 = 1 + 0.568 092 841 269 176 001 626 112;
  • 98) 0.568 092 841 269 176 001 626 112 × 2 = 1 + 0.136 185 682 538 352 003 252 224;
  • 99) 0.136 185 682 538 352 003 252 224 × 2 = 0 + 0.272 371 365 076 704 006 504 448;
  • 100) 0.272 371 365 076 704 006 504 448 × 2 = 0 + 0.544 742 730 153 408 013 008 896;
  • 101) 0.544 742 730 153 408 013 008 896 × 2 = 1 + 0.089 485 460 306 816 026 017 792;
  • 102) 0.089 485 460 306 816 026 017 792 × 2 = 0 + 0.178 970 920 613 632 052 035 584;
  • 103) 0.178 970 920 613 632 052 035 584 × 2 = 0 + 0.357 941 841 227 264 104 071 168;
  • 104) 0.357 941 841 227 264 104 071 168 × 2 = 0 + 0.715 883 682 454 528 208 142 336;
  • 105) 0.715 883 682 454 528 208 142 336 × 2 = 1 + 0.431 767 364 909 056 416 284 672;
  • 106) 0.431 767 364 909 056 416 284 672 × 2 = 0 + 0.863 534 729 818 112 832 569 344;
  • 107) 0.863 534 729 818 112 832 569 344 × 2 = 1 + 0.727 069 459 636 225 665 138 688;
  • 108) 0.727 069 459 636 225 665 138 688 × 2 = 1 + 0.454 138 919 272 451 330 277 376;
  • 109) 0.454 138 919 272 451 330 277 376 × 2 = 0 + 0.908 277 838 544 902 660 554 752;
  • 110) 0.908 277 838 544 902 660 554 752 × 2 = 1 + 0.816 555 677 089 805 321 109 504;
  • 111) 0.816 555 677 089 805 321 109 504 × 2 = 1 + 0.633 111 354 179 610 642 219 008;
  • 112) 0.633 111 354 179 610 642 219 008 × 2 = 1 + 0.266 222 708 359 221 284 438 016;
  • 113) 0.266 222 708 359 221 284 438 016 × 2 = 0 + 0.532 445 416 718 442 568 876 032;
  • 114) 0.532 445 416 718 442 568 876 032 × 2 = 1 + 0.064 890 833 436 885 137 752 064;
  • 115) 0.064 890 833 436 885 137 752 064 × 2 = 0 + 0.129 781 666 873 770 275 504 128;
  • 116) 0.129 781 666 873 770 275 504 128 × 2 = 0 + 0.259 563 333 747 540 551 008 256;
  • 117) 0.259 563 333 747 540 551 008 256 × 2 = 0 + 0.519 126 667 495 081 102 016 512;
  • 118) 0.519 126 667 495 081 102 016 512 × 2 = 1 + 0.038 253 334 990 162 204 033 024;
  • 119) 0.038 253 334 990 162 204 033 024 × 2 = 0 + 0.076 506 669 980 324 408 066 048;

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:


0.000 000 000 000 000 000 008 521(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1101 0100 0001 1100 0100 1100 1000 1011 0111 0100 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 521(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1101 0100 0001 1100 0100 1100 1000 1011 0111 0100 010(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:


0.000 000 000 000 000 000 008 521(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1101 0100 0001 1100 0100 1100 1000 1011 0111 0100 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 1101 0100 0001 1100 0100 1100 1000 1011 0111 0100 010(2) × 20 =

1.0100 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010(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 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010


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 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010 =

0100 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010


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 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010


Decimal number 0.000 000 000 000 000 000 008 521 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0001 1110 1010 0000 1110 0010 0110 0100 0101 1011 1010 0010

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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