0.000 000 000 000 000 000 008 528 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 528 1(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
0.000 000 000 000 000 000 008 528 1(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: 0.000 000 000 000 000 000 008 528 1.

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) 0.000 000 000 000 000 000 008 528 1 × 2 = 0 + 0.000 000 000 000 000 000 017 056 2;
  • 2) 0.000 000 000 000 000 000 017 056 2 × 2 = 0 + 0.000 000 000 000 000 000 034 112 4;
  • 3) 0.000 000 000 000 000 000 034 112 4 × 2 = 0 + 0.000 000 000 000 000 000 068 224 8;
  • 4) 0.000 000 000 000 000 000 068 224 8 × 2 = 0 + 0.000 000 000 000 000 000 136 449 6;
  • 5) 0.000 000 000 000 000 000 136 449 6 × 2 = 0 + 0.000 000 000 000 000 000 272 899 2;
  • 6) 0.000 000 000 000 000 000 272 899 2 × 2 = 0 + 0.000 000 000 000 000 000 545 798 4;
  • 7) 0.000 000 000 000 000 000 545 798 4 × 2 = 0 + 0.000 000 000 000 000 001 091 596 8;
  • 8) 0.000 000 000 000 000 001 091 596 8 × 2 = 0 + 0.000 000 000 000 000 002 183 193 6;
  • 9) 0.000 000 000 000 000 002 183 193 6 × 2 = 0 + 0.000 000 000 000 000 004 366 387 2;
  • 10) 0.000 000 000 000 000 004 366 387 2 × 2 = 0 + 0.000 000 000 000 000 008 732 774 4;
  • 11) 0.000 000 000 000 000 008 732 774 4 × 2 = 0 + 0.000 000 000 000 000 017 465 548 8;
  • 12) 0.000 000 000 000 000 017 465 548 8 × 2 = 0 + 0.000 000 000 000 000 034 931 097 6;
  • 13) 0.000 000 000 000 000 034 931 097 6 × 2 = 0 + 0.000 000 000 000 000 069 862 195 2;
  • 14) 0.000 000 000 000 000 069 862 195 2 × 2 = 0 + 0.000 000 000 000 000 139 724 390 4;
  • 15) 0.000 000 000 000 000 139 724 390 4 × 2 = 0 + 0.000 000 000 000 000 279 448 780 8;
  • 16) 0.000 000 000 000 000 279 448 780 8 × 2 = 0 + 0.000 000 000 000 000 558 897 561 6;
  • 17) 0.000 000 000 000 000 558 897 561 6 × 2 = 0 + 0.000 000 000 000 001 117 795 123 2;
  • 18) 0.000 000 000 000 001 117 795 123 2 × 2 = 0 + 0.000 000 000 000 002 235 590 246 4;
  • 19) 0.000 000 000 000 002 235 590 246 4 × 2 = 0 + 0.000 000 000 000 004 471 180 492 8;
  • 20) 0.000 000 000 000 004 471 180 492 8 × 2 = 0 + 0.000 000 000 000 008 942 360 985 6;
  • 21) 0.000 000 000 000 008 942 360 985 6 × 2 = 0 + 0.000 000 000 000 017 884 721 971 2;
  • 22) 0.000 000 000 000 017 884 721 971 2 × 2 = 0 + 0.000 000 000 000 035 769 443 942 4;
  • 23) 0.000 000 000 000 035 769 443 942 4 × 2 = 0 + 0.000 000 000 000 071 538 887 884 8;
  • 24) 0.000 000 000 000 071 538 887 884 8 × 2 = 0 + 0.000 000 000 000 143 077 775 769 6;
  • 25) 0.000 000 000 000 143 077 775 769 6 × 2 = 0 + 0.000 000 000 000 286 155 551 539 2;
  • 26) 0.000 000 000 000 286 155 551 539 2 × 2 = 0 + 0.000 000 000 000 572 311 103 078 4;
  • 27) 0.000 000 000 000 572 311 103 078 4 × 2 = 0 + 0.000 000 000 001 144 622 206 156 8;
  • 28) 0.000 000 000 001 144 622 206 156 8 × 2 = 0 + 0.000 000 000 002 289 244 412 313 6;
  • 29) 0.000 000 000 002 289 244 412 313 6 × 2 = 0 + 0.000 000 000 004 578 488 824 627 2;
  • 30) 0.000 000 000 004 578 488 824 627 2 × 2 = 0 + 0.000 000 000 009 156 977 649 254 4;
  • 31) 0.000 000 000 009 156 977 649 254 4 × 2 = 0 + 0.000 000 000 018 313 955 298 508 8;
  • 32) 0.000 000 000 018 313 955 298 508 8 × 2 = 0 + 0.000 000 000 036 627 910 597 017 6;
  • 33) 0.000 000 000 036 627 910 597 017 6 × 2 = 0 + 0.000 000 000 073 255 821 194 035 2;
  • 34) 0.000 000 000 073 255 821 194 035 2 × 2 = 0 + 0.000 000 000 146 511 642 388 070 4;
  • 35) 0.000 000 000 146 511 642 388 070 4 × 2 = 0 + 0.000 000 000 293 023 284 776 140 8;
  • 36) 0.000 000 000 293 023 284 776 140 8 × 2 = 0 + 0.000 000 000 586 046 569 552 281 6;
  • 37) 0.000 000 000 586 046 569 552 281 6 × 2 = 0 + 0.000 000 001 172 093 139 104 563 2;
  • 38) 0.000 000 001 172 093 139 104 563 2 × 2 = 0 + 0.000 000 002 344 186 278 209 126 4;
  • 39) 0.000 000 002 344 186 278 209 126 4 × 2 = 0 + 0.000 000 004 688 372 556 418 252 8;
  • 40) 0.000 000 004 688 372 556 418 252 8 × 2 = 0 + 0.000 000 009 376 745 112 836 505 6;
  • 41) 0.000 000 009 376 745 112 836 505 6 × 2 = 0 + 0.000 000 018 753 490 225 673 011 2;
  • 42) 0.000 000 018 753 490 225 673 011 2 × 2 = 0 + 0.000 000 037 506 980 451 346 022 4;
  • 43) 0.000 000 037 506 980 451 346 022 4 × 2 = 0 + 0.000 000 075 013 960 902 692 044 8;
  • 44) 0.000 000 075 013 960 902 692 044 8 × 2 = 0 + 0.000 000 150 027 921 805 384 089 6;
  • 45) 0.000 000 150 027 921 805 384 089 6 × 2 = 0 + 0.000 000 300 055 843 610 768 179 2;
  • 46) 0.000 000 300 055 843 610 768 179 2 × 2 = 0 + 0.000 000 600 111 687 221 536 358 4;
  • 47) 0.000 000 600 111 687 221 536 358 4 × 2 = 0 + 0.000 001 200 223 374 443 072 716 8;
  • 48) 0.000 001 200 223 374 443 072 716 8 × 2 = 0 + 0.000 002 400 446 748 886 145 433 6;
  • 49) 0.000 002 400 446 748 886 145 433 6 × 2 = 0 + 0.000 004 800 893 497 772 290 867 2;
  • 50) 0.000 004 800 893 497 772 290 867 2 × 2 = 0 + 0.000 009 601 786 995 544 581 734 4;
  • 51) 0.000 009 601 786 995 544 581 734 4 × 2 = 0 + 0.000 019 203 573 991 089 163 468 8;
  • 52) 0.000 019 203 573 991 089 163 468 8 × 2 = 0 + 0.000 038 407 147 982 178 326 937 6;
  • 53) 0.000 038 407 147 982 178 326 937 6 × 2 = 0 + 0.000 076 814 295 964 356 653 875 2;
  • 54) 0.000 076 814 295 964 356 653 875 2 × 2 = 0 + 0.000 153 628 591 928 713 307 750 4;
  • 55) 0.000 153 628 591 928 713 307 750 4 × 2 = 0 + 0.000 307 257 183 857 426 615 500 8;
  • 56) 0.000 307 257 183 857 426 615 500 8 × 2 = 0 + 0.000 614 514 367 714 853 231 001 6;
  • 57) 0.000 614 514 367 714 853 231 001 6 × 2 = 0 + 0.001 229 028 735 429 706 462 003 2;
  • 58) 0.001 229 028 735 429 706 462 003 2 × 2 = 0 + 0.002 458 057 470 859 412 924 006 4;
  • 59) 0.002 458 057 470 859 412 924 006 4 × 2 = 0 + 0.004 916 114 941 718 825 848 012 8;
  • 60) 0.004 916 114 941 718 825 848 012 8 × 2 = 0 + 0.009 832 229 883 437 651 696 025 6;
  • 61) 0.009 832 229 883 437 651 696 025 6 × 2 = 0 + 0.019 664 459 766 875 303 392 051 2;
  • 62) 0.019 664 459 766 875 303 392 051 2 × 2 = 0 + 0.039 328 919 533 750 606 784 102 4;
  • 63) 0.039 328 919 533 750 606 784 102 4 × 2 = 0 + 0.078 657 839 067 501 213 568 204 8;
  • 64) 0.078 657 839 067 501 213 568 204 8 × 2 = 0 + 0.157 315 678 135 002 427 136 409 6;
  • 65) 0.157 315 678 135 002 427 136 409 6 × 2 = 0 + 0.314 631 356 270 004 854 272 819 2;
  • 66) 0.314 631 356 270 004 854 272 819 2 × 2 = 0 + 0.629 262 712 540 009 708 545 638 4;
  • 67) 0.629 262 712 540 009 708 545 638 4 × 2 = 1 + 0.258 525 425 080 019 417 091 276 8;
  • 68) 0.258 525 425 080 019 417 091 276 8 × 2 = 0 + 0.517 050 850 160 038 834 182 553 6;
  • 69) 0.517 050 850 160 038 834 182 553 6 × 2 = 1 + 0.034 101 700 320 077 668 365 107 2;
  • 70) 0.034 101 700 320 077 668 365 107 2 × 2 = 0 + 0.068 203 400 640 155 336 730 214 4;
  • 71) 0.068 203 400 640 155 336 730 214 4 × 2 = 0 + 0.136 406 801 280 310 673 460 428 8;
  • 72) 0.136 406 801 280 310 673 460 428 8 × 2 = 0 + 0.272 813 602 560 621 346 920 857 6;
  • 73) 0.272 813 602 560 621 346 920 857 6 × 2 = 0 + 0.545 627 205 121 242 693 841 715 2;
  • 74) 0.545 627 205 121 242 693 841 715 2 × 2 = 1 + 0.091 254 410 242 485 387 683 430 4;
  • 75) 0.091 254 410 242 485 387 683 430 4 × 2 = 0 + 0.182 508 820 484 970 775 366 860 8;
  • 76) 0.182 508 820 484 970 775 366 860 8 × 2 = 0 + 0.365 017 640 969 941 550 733 721 6;
  • 77) 0.365 017 640 969 941 550 733 721 6 × 2 = 0 + 0.730 035 281 939 883 101 467 443 2;
  • 78) 0.730 035 281 939 883 101 467 443 2 × 2 = 1 + 0.460 070 563 879 766 202 934 886 4;
  • 79) 0.460 070 563 879 766 202 934 886 4 × 2 = 0 + 0.920 141 127 759 532 405 869 772 8;
  • 80) 0.920 141 127 759 532 405 869 772 8 × 2 = 1 + 0.840 282 255 519 064 811 739 545 6;
  • 81) 0.840 282 255 519 064 811 739 545 6 × 2 = 1 + 0.680 564 511 038 129 623 479 091 2;
  • 82) 0.680 564 511 038 129 623 479 091 2 × 2 = 1 + 0.361 129 022 076 259 246 958 182 4;
  • 83) 0.361 129 022 076 259 246 958 182 4 × 2 = 0 + 0.722 258 044 152 518 493 916 364 8;
  • 84) 0.722 258 044 152 518 493 916 364 8 × 2 = 1 + 0.444 516 088 305 036 987 832 729 6;
  • 85) 0.444 516 088 305 036 987 832 729 6 × 2 = 0 + 0.889 032 176 610 073 975 665 459 2;
  • 86) 0.889 032 176 610 073 975 665 459 2 × 2 = 1 + 0.778 064 353 220 147 951 330 918 4;
  • 87) 0.778 064 353 220 147 951 330 918 4 × 2 = 1 + 0.556 128 706 440 295 902 661 836 8;
  • 88) 0.556 128 706 440 295 902 661 836 8 × 2 = 1 + 0.112 257 412 880 591 805 323 673 6;
  • 89) 0.112 257 412 880 591 805 323 673 6 × 2 = 0 + 0.224 514 825 761 183 610 647 347 2;
  • 90) 0.224 514 825 761 183 610 647 347 2 × 2 = 0 + 0.449 029 651 522 367 221 294 694 4;
  • 91) 0.449 029 651 522 367 221 294 694 4 × 2 = 0 + 0.898 059 303 044 734 442 589 388 8;
  • 92) 0.898 059 303 044 734 442 589 388 8 × 2 = 1 + 0.796 118 606 089 468 885 178 777 6;
  • 93) 0.796 118 606 089 468 885 178 777 6 × 2 = 1 + 0.592 237 212 178 937 770 357 555 2;
  • 94) 0.592 237 212 178 937 770 357 555 2 × 2 = 1 + 0.184 474 424 357 875 540 715 110 4;
  • 95) 0.184 474 424 357 875 540 715 110 4 × 2 = 0 + 0.368 948 848 715 751 081 430 220 8;
  • 96) 0.368 948 848 715 751 081 430 220 8 × 2 = 0 + 0.737 897 697 431 502 162 860 441 6;
  • 97) 0.737 897 697 431 502 162 860 441 6 × 2 = 1 + 0.475 795 394 863 004 325 720 883 2;
  • 98) 0.475 795 394 863 004 325 720 883 2 × 2 = 0 + 0.951 590 789 726 008 651 441 766 4;
  • 99) 0.951 590 789 726 008 651 441 766 4 × 2 = 1 + 0.903 181 579 452 017 302 883 532 8;
  • 100) 0.903 181 579 452 017 302 883 532 8 × 2 = 1 + 0.806 363 158 904 034 605 767 065 6;
  • 101) 0.806 363 158 904 034 605 767 065 6 × 2 = 1 + 0.612 726 317 808 069 211 534 131 2;
  • 102) 0.612 726 317 808 069 211 534 131 2 × 2 = 1 + 0.225 452 635 616 138 423 068 262 4;
  • 103) 0.225 452 635 616 138 423 068 262 4 × 2 = 0 + 0.450 905 271 232 276 846 136 524 8;
  • 104) 0.450 905 271 232 276 846 136 524 8 × 2 = 0 + 0.901 810 542 464 553 692 273 049 6;
  • 105) 0.901 810 542 464 553 692 273 049 6 × 2 = 1 + 0.803 621 084 929 107 384 546 099 2;
  • 106) 0.803 621 084 929 107 384 546 099 2 × 2 = 1 + 0.607 242 169 858 214 769 092 198 4;
  • 107) 0.607 242 169 858 214 769 092 198 4 × 2 = 1 + 0.214 484 339 716 429 538 184 396 8;
  • 108) 0.214 484 339 716 429 538 184 396 8 × 2 = 0 + 0.428 968 679 432 859 076 368 793 6;
  • 109) 0.428 968 679 432 859 076 368 793 6 × 2 = 0 + 0.857 937 358 865 718 152 737 587 2;
  • 110) 0.857 937 358 865 718 152 737 587 2 × 2 = 1 + 0.715 874 717 731 436 305 475 174 4;
  • 111) 0.715 874 717 731 436 305 475 174 4 × 2 = 1 + 0.431 749 435 462 872 610 950 348 8;
  • 112) 0.431 749 435 462 872 610 950 348 8 × 2 = 0 + 0.863 498 870 925 745 221 900 697 6;
  • 113) 0.863 498 870 925 745 221 900 697 6 × 2 = 1 + 0.726 997 741 851 490 443 801 395 2;
  • 114) 0.726 997 741 851 490 443 801 395 2 × 2 = 1 + 0.453 995 483 702 980 887 602 790 4;
  • 115) 0.453 995 483 702 980 887 602 790 4 × 2 = 0 + 0.907 990 967 405 961 775 205 580 8;
  • 116) 0.907 990 967 405 961 775 205 580 8 × 2 = 1 + 0.815 981 934 811 923 550 411 161 6;
  • 117) 0.815 981 934 811 923 550 411 161 6 × 2 = 1 + 0.631 963 869 623 847 100 822 323 2;
  • 118) 0.631 963 869 623 847 100 822 323 2 × 2 = 1 + 0.263 927 739 247 694 201 644 646 4;
  • 119) 0.263 927 739 247 694 201 644 646 4 × 2 = 0 + 0.527 855 478 495 388 403 289 292 8;

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:


0.000 000 000 000 000 000 008 528 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0101 1101 0111 0001 1100 1011 1100 1110 0110 1101 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 528 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0101 1101 0111 0001 1100 1011 1100 1110 0110 1101 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:


0.000 000 000 000 000 000 008 528 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0101 1101 0111 0001 1100 1011 1100 1110 0110 1101 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0101 1101 0111 0001 1100 1011 1100 1110 0110 1101 110(2) × 20 =

1.0100 0010 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110(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 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110


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 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110 =

0100 0010 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110


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 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110


Decimal number 0.000 000 000 000 000 000 008 528 1 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0010 1110 1011 1000 1110 0101 1110 0111 0011 0110 1110

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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