0.000 000 000 000 000 000 008 539 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 539 9(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
0.000 000 000 000 000 000 008 539 9(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: 0.000 000 000 000 000 000 008 539 9.

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) 0.000 000 000 000 000 000 008 539 9 × 2 = 0 + 0.000 000 000 000 000 000 017 079 8;
  • 2) 0.000 000 000 000 000 000 017 079 8 × 2 = 0 + 0.000 000 000 000 000 000 034 159 6;
  • 3) 0.000 000 000 000 000 000 034 159 6 × 2 = 0 + 0.000 000 000 000 000 000 068 319 2;
  • 4) 0.000 000 000 000 000 000 068 319 2 × 2 = 0 + 0.000 000 000 000 000 000 136 638 4;
  • 5) 0.000 000 000 000 000 000 136 638 4 × 2 = 0 + 0.000 000 000 000 000 000 273 276 8;
  • 6) 0.000 000 000 000 000 000 273 276 8 × 2 = 0 + 0.000 000 000 000 000 000 546 553 6;
  • 7) 0.000 000 000 000 000 000 546 553 6 × 2 = 0 + 0.000 000 000 000 000 001 093 107 2;
  • 8) 0.000 000 000 000 000 001 093 107 2 × 2 = 0 + 0.000 000 000 000 000 002 186 214 4;
  • 9) 0.000 000 000 000 000 002 186 214 4 × 2 = 0 + 0.000 000 000 000 000 004 372 428 8;
  • 10) 0.000 000 000 000 000 004 372 428 8 × 2 = 0 + 0.000 000 000 000 000 008 744 857 6;
  • 11) 0.000 000 000 000 000 008 744 857 6 × 2 = 0 + 0.000 000 000 000 000 017 489 715 2;
  • 12) 0.000 000 000 000 000 017 489 715 2 × 2 = 0 + 0.000 000 000 000 000 034 979 430 4;
  • 13) 0.000 000 000 000 000 034 979 430 4 × 2 = 0 + 0.000 000 000 000 000 069 958 860 8;
  • 14) 0.000 000 000 000 000 069 958 860 8 × 2 = 0 + 0.000 000 000 000 000 139 917 721 6;
  • 15) 0.000 000 000 000 000 139 917 721 6 × 2 = 0 + 0.000 000 000 000 000 279 835 443 2;
  • 16) 0.000 000 000 000 000 279 835 443 2 × 2 = 0 + 0.000 000 000 000 000 559 670 886 4;
  • 17) 0.000 000 000 000 000 559 670 886 4 × 2 = 0 + 0.000 000 000 000 001 119 341 772 8;
  • 18) 0.000 000 000 000 001 119 341 772 8 × 2 = 0 + 0.000 000 000 000 002 238 683 545 6;
  • 19) 0.000 000 000 000 002 238 683 545 6 × 2 = 0 + 0.000 000 000 000 004 477 367 091 2;
  • 20) 0.000 000 000 000 004 477 367 091 2 × 2 = 0 + 0.000 000 000 000 008 954 734 182 4;
  • 21) 0.000 000 000 000 008 954 734 182 4 × 2 = 0 + 0.000 000 000 000 017 909 468 364 8;
  • 22) 0.000 000 000 000 017 909 468 364 8 × 2 = 0 + 0.000 000 000 000 035 818 936 729 6;
  • 23) 0.000 000 000 000 035 818 936 729 6 × 2 = 0 + 0.000 000 000 000 071 637 873 459 2;
  • 24) 0.000 000 000 000 071 637 873 459 2 × 2 = 0 + 0.000 000 000 000 143 275 746 918 4;
  • 25) 0.000 000 000 000 143 275 746 918 4 × 2 = 0 + 0.000 000 000 000 286 551 493 836 8;
  • 26) 0.000 000 000 000 286 551 493 836 8 × 2 = 0 + 0.000 000 000 000 573 102 987 673 6;
  • 27) 0.000 000 000 000 573 102 987 673 6 × 2 = 0 + 0.000 000 000 001 146 205 975 347 2;
  • 28) 0.000 000 000 001 146 205 975 347 2 × 2 = 0 + 0.000 000 000 002 292 411 950 694 4;
  • 29) 0.000 000 000 002 292 411 950 694 4 × 2 = 0 + 0.000 000 000 004 584 823 901 388 8;
  • 30) 0.000 000 000 004 584 823 901 388 8 × 2 = 0 + 0.000 000 000 009 169 647 802 777 6;
  • 31) 0.000 000 000 009 169 647 802 777 6 × 2 = 0 + 0.000 000 000 018 339 295 605 555 2;
  • 32) 0.000 000 000 018 339 295 605 555 2 × 2 = 0 + 0.000 000 000 036 678 591 211 110 4;
  • 33) 0.000 000 000 036 678 591 211 110 4 × 2 = 0 + 0.000 000 000 073 357 182 422 220 8;
  • 34) 0.000 000 000 073 357 182 422 220 8 × 2 = 0 + 0.000 000 000 146 714 364 844 441 6;
  • 35) 0.000 000 000 146 714 364 844 441 6 × 2 = 0 + 0.000 000 000 293 428 729 688 883 2;
  • 36) 0.000 000 000 293 428 729 688 883 2 × 2 = 0 + 0.000 000 000 586 857 459 377 766 4;
  • 37) 0.000 000 000 586 857 459 377 766 4 × 2 = 0 + 0.000 000 001 173 714 918 755 532 8;
  • 38) 0.000 000 001 173 714 918 755 532 8 × 2 = 0 + 0.000 000 002 347 429 837 511 065 6;
  • 39) 0.000 000 002 347 429 837 511 065 6 × 2 = 0 + 0.000 000 004 694 859 675 022 131 2;
  • 40) 0.000 000 004 694 859 675 022 131 2 × 2 = 0 + 0.000 000 009 389 719 350 044 262 4;
  • 41) 0.000 000 009 389 719 350 044 262 4 × 2 = 0 + 0.000 000 018 779 438 700 088 524 8;
  • 42) 0.000 000 018 779 438 700 088 524 8 × 2 = 0 + 0.000 000 037 558 877 400 177 049 6;
  • 43) 0.000 000 037 558 877 400 177 049 6 × 2 = 0 + 0.000 000 075 117 754 800 354 099 2;
  • 44) 0.000 000 075 117 754 800 354 099 2 × 2 = 0 + 0.000 000 150 235 509 600 708 198 4;
  • 45) 0.000 000 150 235 509 600 708 198 4 × 2 = 0 + 0.000 000 300 471 019 201 416 396 8;
  • 46) 0.000 000 300 471 019 201 416 396 8 × 2 = 0 + 0.000 000 600 942 038 402 832 793 6;
  • 47) 0.000 000 600 942 038 402 832 793 6 × 2 = 0 + 0.000 001 201 884 076 805 665 587 2;
  • 48) 0.000 001 201 884 076 805 665 587 2 × 2 = 0 + 0.000 002 403 768 153 611 331 174 4;
  • 49) 0.000 002 403 768 153 611 331 174 4 × 2 = 0 + 0.000 004 807 536 307 222 662 348 8;
  • 50) 0.000 004 807 536 307 222 662 348 8 × 2 = 0 + 0.000 009 615 072 614 445 324 697 6;
  • 51) 0.000 009 615 072 614 445 324 697 6 × 2 = 0 + 0.000 019 230 145 228 890 649 395 2;
  • 52) 0.000 019 230 145 228 890 649 395 2 × 2 = 0 + 0.000 038 460 290 457 781 298 790 4;
  • 53) 0.000 038 460 290 457 781 298 790 4 × 2 = 0 + 0.000 076 920 580 915 562 597 580 8;
  • 54) 0.000 076 920 580 915 562 597 580 8 × 2 = 0 + 0.000 153 841 161 831 125 195 161 6;
  • 55) 0.000 153 841 161 831 125 195 161 6 × 2 = 0 + 0.000 307 682 323 662 250 390 323 2;
  • 56) 0.000 307 682 323 662 250 390 323 2 × 2 = 0 + 0.000 615 364 647 324 500 780 646 4;
  • 57) 0.000 615 364 647 324 500 780 646 4 × 2 = 0 + 0.001 230 729 294 649 001 561 292 8;
  • 58) 0.001 230 729 294 649 001 561 292 8 × 2 = 0 + 0.002 461 458 589 298 003 122 585 6;
  • 59) 0.002 461 458 589 298 003 122 585 6 × 2 = 0 + 0.004 922 917 178 596 006 245 171 2;
  • 60) 0.004 922 917 178 596 006 245 171 2 × 2 = 0 + 0.009 845 834 357 192 012 490 342 4;
  • 61) 0.009 845 834 357 192 012 490 342 4 × 2 = 0 + 0.019 691 668 714 384 024 980 684 8;
  • 62) 0.019 691 668 714 384 024 980 684 8 × 2 = 0 + 0.039 383 337 428 768 049 961 369 6;
  • 63) 0.039 383 337 428 768 049 961 369 6 × 2 = 0 + 0.078 766 674 857 536 099 922 739 2;
  • 64) 0.078 766 674 857 536 099 922 739 2 × 2 = 0 + 0.157 533 349 715 072 199 845 478 4;
  • 65) 0.157 533 349 715 072 199 845 478 4 × 2 = 0 + 0.315 066 699 430 144 399 690 956 8;
  • 66) 0.315 066 699 430 144 399 690 956 8 × 2 = 0 + 0.630 133 398 860 288 799 381 913 6;
  • 67) 0.630 133 398 860 288 799 381 913 6 × 2 = 1 + 0.260 266 797 720 577 598 763 827 2;
  • 68) 0.260 266 797 720 577 598 763 827 2 × 2 = 0 + 0.520 533 595 441 155 197 527 654 4;
  • 69) 0.520 533 595 441 155 197 527 654 4 × 2 = 1 + 0.041 067 190 882 310 395 055 308 8;
  • 70) 0.041 067 190 882 310 395 055 308 8 × 2 = 0 + 0.082 134 381 764 620 790 110 617 6;
  • 71) 0.082 134 381 764 620 790 110 617 6 × 2 = 0 + 0.164 268 763 529 241 580 221 235 2;
  • 72) 0.164 268 763 529 241 580 221 235 2 × 2 = 0 + 0.328 537 527 058 483 160 442 470 4;
  • 73) 0.328 537 527 058 483 160 442 470 4 × 2 = 0 + 0.657 075 054 116 966 320 884 940 8;
  • 74) 0.657 075 054 116 966 320 884 940 8 × 2 = 1 + 0.314 150 108 233 932 641 769 881 6;
  • 75) 0.314 150 108 233 932 641 769 881 6 × 2 = 0 + 0.628 300 216 467 865 283 539 763 2;
  • 76) 0.628 300 216 467 865 283 539 763 2 × 2 = 1 + 0.256 600 432 935 730 567 079 526 4;
  • 77) 0.256 600 432 935 730 567 079 526 4 × 2 = 0 + 0.513 200 865 871 461 134 159 052 8;
  • 78) 0.513 200 865 871 461 134 159 052 8 × 2 = 1 + 0.026 401 731 742 922 268 318 105 6;
  • 79) 0.026 401 731 742 922 268 318 105 6 × 2 = 0 + 0.052 803 463 485 844 536 636 211 2;
  • 80) 0.052 803 463 485 844 536 636 211 2 × 2 = 0 + 0.105 606 926 971 689 073 272 422 4;
  • 81) 0.105 606 926 971 689 073 272 422 4 × 2 = 0 + 0.211 213 853 943 378 146 544 844 8;
  • 82) 0.211 213 853 943 378 146 544 844 8 × 2 = 0 + 0.422 427 707 886 756 293 089 689 6;
  • 83) 0.422 427 707 886 756 293 089 689 6 × 2 = 0 + 0.844 855 415 773 512 586 179 379 2;
  • 84) 0.844 855 415 773 512 586 179 379 2 × 2 = 1 + 0.689 710 831 547 025 172 358 758 4;
  • 85) 0.689 710 831 547 025 172 358 758 4 × 2 = 1 + 0.379 421 663 094 050 344 717 516 8;
  • 86) 0.379 421 663 094 050 344 717 516 8 × 2 = 0 + 0.758 843 326 188 100 689 435 033 6;
  • 87) 0.758 843 326 188 100 689 435 033 6 × 2 = 1 + 0.517 686 652 376 201 378 870 067 2;
  • 88) 0.517 686 652 376 201 378 870 067 2 × 2 = 1 + 0.035 373 304 752 402 757 740 134 4;
  • 89) 0.035 373 304 752 402 757 740 134 4 × 2 = 0 + 0.070 746 609 504 805 515 480 268 8;
  • 90) 0.070 746 609 504 805 515 480 268 8 × 2 = 0 + 0.141 493 219 009 611 030 960 537 6;
  • 91) 0.141 493 219 009 611 030 960 537 6 × 2 = 0 + 0.282 986 438 019 222 061 921 075 2;
  • 92) 0.282 986 438 019 222 061 921 075 2 × 2 = 0 + 0.565 972 876 038 444 123 842 150 4;
  • 93) 0.565 972 876 038 444 123 842 150 4 × 2 = 1 + 0.131 945 752 076 888 247 684 300 8;
  • 94) 0.131 945 752 076 888 247 684 300 8 × 2 = 0 + 0.263 891 504 153 776 495 368 601 6;
  • 95) 0.263 891 504 153 776 495 368 601 6 × 2 = 0 + 0.527 783 008 307 552 990 737 203 2;
  • 96) 0.527 783 008 307 552 990 737 203 2 × 2 = 1 + 0.055 566 016 615 105 981 474 406 4;
  • 97) 0.055 566 016 615 105 981 474 406 4 × 2 = 0 + 0.111 132 033 230 211 962 948 812 8;
  • 98) 0.111 132 033 230 211 962 948 812 8 × 2 = 0 + 0.222 264 066 460 423 925 897 625 6;
  • 99) 0.222 264 066 460 423 925 897 625 6 × 2 = 0 + 0.444 528 132 920 847 851 795 251 2;
  • 100) 0.444 528 132 920 847 851 795 251 2 × 2 = 0 + 0.889 056 265 841 695 703 590 502 4;
  • 101) 0.889 056 265 841 695 703 590 502 4 × 2 = 1 + 0.778 112 531 683 391 407 181 004 8;
  • 102) 0.778 112 531 683 391 407 181 004 8 × 2 = 1 + 0.556 225 063 366 782 814 362 009 6;
  • 103) 0.556 225 063 366 782 814 362 009 6 × 2 = 1 + 0.112 450 126 733 565 628 724 019 2;
  • 104) 0.112 450 126 733 565 628 724 019 2 × 2 = 0 + 0.224 900 253 467 131 257 448 038 4;
  • 105) 0.224 900 253 467 131 257 448 038 4 × 2 = 0 + 0.449 800 506 934 262 514 896 076 8;
  • 106) 0.449 800 506 934 262 514 896 076 8 × 2 = 0 + 0.899 601 013 868 525 029 792 153 6;
  • 107) 0.899 601 013 868 525 029 792 153 6 × 2 = 1 + 0.799 202 027 737 050 059 584 307 2;
  • 108) 0.799 202 027 737 050 059 584 307 2 × 2 = 1 + 0.598 404 055 474 100 119 168 614 4;
  • 109) 0.598 404 055 474 100 119 168 614 4 × 2 = 1 + 0.196 808 110 948 200 238 337 228 8;
  • 110) 0.196 808 110 948 200 238 337 228 8 × 2 = 0 + 0.393 616 221 896 400 476 674 457 6;
  • 111) 0.393 616 221 896 400 476 674 457 6 × 2 = 0 + 0.787 232 443 792 800 953 348 915 2;
  • 112) 0.787 232 443 792 800 953 348 915 2 × 2 = 1 + 0.574 464 887 585 601 906 697 830 4;
  • 113) 0.574 464 887 585 601 906 697 830 4 × 2 = 1 + 0.148 929 775 171 203 813 395 660 8;
  • 114) 0.148 929 775 171 203 813 395 660 8 × 2 = 0 + 0.297 859 550 342 407 626 791 321 6;
  • 115) 0.297 859 550 342 407 626 791 321 6 × 2 = 0 + 0.595 719 100 684 815 253 582 643 2;
  • 116) 0.595 719 100 684 815 253 582 643 2 × 2 = 1 + 0.191 438 201 369 630 507 165 286 4;
  • 117) 0.191 438 201 369 630 507 165 286 4 × 2 = 0 + 0.382 876 402 739 261 014 330 572 8;
  • 118) 0.382 876 402 739 261 014 330 572 8 × 2 = 0 + 0.765 752 805 478 522 028 661 145 6;
  • 119) 0.765 752 805 478 522 028 661 145 6 × 2 = 1 + 0.531 505 610 957 044 057 322 291 2;

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:


0.000 000 000 000 000 000 008 539 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0100 0001 1011 0000 1001 0000 1110 0011 1001 1001 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 539 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0100 0001 1011 0000 1001 0000 1110 0011 1001 1001 001(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:


0.000 000 000 000 000 000 008 539 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0100 0001 1011 0000 1001 0000 1110 0011 1001 1001 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0100 0001 1011 0000 1001 0000 1110 0011 1001 1001 001(2) × 20 =

1.0100 0010 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001(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 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001


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 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001 =

0100 0010 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001


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 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001


Decimal number 0.000 000 000 000 000 000 008 539 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1010 0000 1101 1000 0100 1000 0111 0001 1100 1100 1001

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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