0.000 000 000 000 000 000 008 552 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 552 3(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
0.000 000 000 000 000 000 008 552 3(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: 0.000 000 000 000 000 000 008 552 3.

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) 0.000 000 000 000 000 000 008 552 3 × 2 = 0 + 0.000 000 000 000 000 000 017 104 6;
  • 2) 0.000 000 000 000 000 000 017 104 6 × 2 = 0 + 0.000 000 000 000 000 000 034 209 2;
  • 3) 0.000 000 000 000 000 000 034 209 2 × 2 = 0 + 0.000 000 000 000 000 000 068 418 4;
  • 4) 0.000 000 000 000 000 000 068 418 4 × 2 = 0 + 0.000 000 000 000 000 000 136 836 8;
  • 5) 0.000 000 000 000 000 000 136 836 8 × 2 = 0 + 0.000 000 000 000 000 000 273 673 6;
  • 6) 0.000 000 000 000 000 000 273 673 6 × 2 = 0 + 0.000 000 000 000 000 000 547 347 2;
  • 7) 0.000 000 000 000 000 000 547 347 2 × 2 = 0 + 0.000 000 000 000 000 001 094 694 4;
  • 8) 0.000 000 000 000 000 001 094 694 4 × 2 = 0 + 0.000 000 000 000 000 002 189 388 8;
  • 9) 0.000 000 000 000 000 002 189 388 8 × 2 = 0 + 0.000 000 000 000 000 004 378 777 6;
  • 10) 0.000 000 000 000 000 004 378 777 6 × 2 = 0 + 0.000 000 000 000 000 008 757 555 2;
  • 11) 0.000 000 000 000 000 008 757 555 2 × 2 = 0 + 0.000 000 000 000 000 017 515 110 4;
  • 12) 0.000 000 000 000 000 017 515 110 4 × 2 = 0 + 0.000 000 000 000 000 035 030 220 8;
  • 13) 0.000 000 000 000 000 035 030 220 8 × 2 = 0 + 0.000 000 000 000 000 070 060 441 6;
  • 14) 0.000 000 000 000 000 070 060 441 6 × 2 = 0 + 0.000 000 000 000 000 140 120 883 2;
  • 15) 0.000 000 000 000 000 140 120 883 2 × 2 = 0 + 0.000 000 000 000 000 280 241 766 4;
  • 16) 0.000 000 000 000 000 280 241 766 4 × 2 = 0 + 0.000 000 000 000 000 560 483 532 8;
  • 17) 0.000 000 000 000 000 560 483 532 8 × 2 = 0 + 0.000 000 000 000 001 120 967 065 6;
  • 18) 0.000 000 000 000 001 120 967 065 6 × 2 = 0 + 0.000 000 000 000 002 241 934 131 2;
  • 19) 0.000 000 000 000 002 241 934 131 2 × 2 = 0 + 0.000 000 000 000 004 483 868 262 4;
  • 20) 0.000 000 000 000 004 483 868 262 4 × 2 = 0 + 0.000 000 000 000 008 967 736 524 8;
  • 21) 0.000 000 000 000 008 967 736 524 8 × 2 = 0 + 0.000 000 000 000 017 935 473 049 6;
  • 22) 0.000 000 000 000 017 935 473 049 6 × 2 = 0 + 0.000 000 000 000 035 870 946 099 2;
  • 23) 0.000 000 000 000 035 870 946 099 2 × 2 = 0 + 0.000 000 000 000 071 741 892 198 4;
  • 24) 0.000 000 000 000 071 741 892 198 4 × 2 = 0 + 0.000 000 000 000 143 483 784 396 8;
  • 25) 0.000 000 000 000 143 483 784 396 8 × 2 = 0 + 0.000 000 000 000 286 967 568 793 6;
  • 26) 0.000 000 000 000 286 967 568 793 6 × 2 = 0 + 0.000 000 000 000 573 935 137 587 2;
  • 27) 0.000 000 000 000 573 935 137 587 2 × 2 = 0 + 0.000 000 000 001 147 870 275 174 4;
  • 28) 0.000 000 000 001 147 870 275 174 4 × 2 = 0 + 0.000 000 000 002 295 740 550 348 8;
  • 29) 0.000 000 000 002 295 740 550 348 8 × 2 = 0 + 0.000 000 000 004 591 481 100 697 6;
  • 30) 0.000 000 000 004 591 481 100 697 6 × 2 = 0 + 0.000 000 000 009 182 962 201 395 2;
  • 31) 0.000 000 000 009 182 962 201 395 2 × 2 = 0 + 0.000 000 000 018 365 924 402 790 4;
  • 32) 0.000 000 000 018 365 924 402 790 4 × 2 = 0 + 0.000 000 000 036 731 848 805 580 8;
  • 33) 0.000 000 000 036 731 848 805 580 8 × 2 = 0 + 0.000 000 000 073 463 697 611 161 6;
  • 34) 0.000 000 000 073 463 697 611 161 6 × 2 = 0 + 0.000 000 000 146 927 395 222 323 2;
  • 35) 0.000 000 000 146 927 395 222 323 2 × 2 = 0 + 0.000 000 000 293 854 790 444 646 4;
  • 36) 0.000 000 000 293 854 790 444 646 4 × 2 = 0 + 0.000 000 000 587 709 580 889 292 8;
  • 37) 0.000 000 000 587 709 580 889 292 8 × 2 = 0 + 0.000 000 001 175 419 161 778 585 6;
  • 38) 0.000 000 001 175 419 161 778 585 6 × 2 = 0 + 0.000 000 002 350 838 323 557 171 2;
  • 39) 0.000 000 002 350 838 323 557 171 2 × 2 = 0 + 0.000 000 004 701 676 647 114 342 4;
  • 40) 0.000 000 004 701 676 647 114 342 4 × 2 = 0 + 0.000 000 009 403 353 294 228 684 8;
  • 41) 0.000 000 009 403 353 294 228 684 8 × 2 = 0 + 0.000 000 018 806 706 588 457 369 6;
  • 42) 0.000 000 018 806 706 588 457 369 6 × 2 = 0 + 0.000 000 037 613 413 176 914 739 2;
  • 43) 0.000 000 037 613 413 176 914 739 2 × 2 = 0 + 0.000 000 075 226 826 353 829 478 4;
  • 44) 0.000 000 075 226 826 353 829 478 4 × 2 = 0 + 0.000 000 150 453 652 707 658 956 8;
  • 45) 0.000 000 150 453 652 707 658 956 8 × 2 = 0 + 0.000 000 300 907 305 415 317 913 6;
  • 46) 0.000 000 300 907 305 415 317 913 6 × 2 = 0 + 0.000 000 601 814 610 830 635 827 2;
  • 47) 0.000 000 601 814 610 830 635 827 2 × 2 = 0 + 0.000 001 203 629 221 661 271 654 4;
  • 48) 0.000 001 203 629 221 661 271 654 4 × 2 = 0 + 0.000 002 407 258 443 322 543 308 8;
  • 49) 0.000 002 407 258 443 322 543 308 8 × 2 = 0 + 0.000 004 814 516 886 645 086 617 6;
  • 50) 0.000 004 814 516 886 645 086 617 6 × 2 = 0 + 0.000 009 629 033 773 290 173 235 2;
  • 51) 0.000 009 629 033 773 290 173 235 2 × 2 = 0 + 0.000 019 258 067 546 580 346 470 4;
  • 52) 0.000 019 258 067 546 580 346 470 4 × 2 = 0 + 0.000 038 516 135 093 160 692 940 8;
  • 53) 0.000 038 516 135 093 160 692 940 8 × 2 = 0 + 0.000 077 032 270 186 321 385 881 6;
  • 54) 0.000 077 032 270 186 321 385 881 6 × 2 = 0 + 0.000 154 064 540 372 642 771 763 2;
  • 55) 0.000 154 064 540 372 642 771 763 2 × 2 = 0 + 0.000 308 129 080 745 285 543 526 4;
  • 56) 0.000 308 129 080 745 285 543 526 4 × 2 = 0 + 0.000 616 258 161 490 571 087 052 8;
  • 57) 0.000 616 258 161 490 571 087 052 8 × 2 = 0 + 0.001 232 516 322 981 142 174 105 6;
  • 58) 0.001 232 516 322 981 142 174 105 6 × 2 = 0 + 0.002 465 032 645 962 284 348 211 2;
  • 59) 0.002 465 032 645 962 284 348 211 2 × 2 = 0 + 0.004 930 065 291 924 568 696 422 4;
  • 60) 0.004 930 065 291 924 568 696 422 4 × 2 = 0 + 0.009 860 130 583 849 137 392 844 8;
  • 61) 0.009 860 130 583 849 137 392 844 8 × 2 = 0 + 0.019 720 261 167 698 274 785 689 6;
  • 62) 0.019 720 261 167 698 274 785 689 6 × 2 = 0 + 0.039 440 522 335 396 549 571 379 2;
  • 63) 0.039 440 522 335 396 549 571 379 2 × 2 = 0 + 0.078 881 044 670 793 099 142 758 4;
  • 64) 0.078 881 044 670 793 099 142 758 4 × 2 = 0 + 0.157 762 089 341 586 198 285 516 8;
  • 65) 0.157 762 089 341 586 198 285 516 8 × 2 = 0 + 0.315 524 178 683 172 396 571 033 6;
  • 66) 0.315 524 178 683 172 396 571 033 6 × 2 = 0 + 0.631 048 357 366 344 793 142 067 2;
  • 67) 0.631 048 357 366 344 793 142 067 2 × 2 = 1 + 0.262 096 714 732 689 586 284 134 4;
  • 68) 0.262 096 714 732 689 586 284 134 4 × 2 = 0 + 0.524 193 429 465 379 172 568 268 8;
  • 69) 0.524 193 429 465 379 172 568 268 8 × 2 = 1 + 0.048 386 858 930 758 345 136 537 6;
  • 70) 0.048 386 858 930 758 345 136 537 6 × 2 = 0 + 0.096 773 717 861 516 690 273 075 2;
  • 71) 0.096 773 717 861 516 690 273 075 2 × 2 = 0 + 0.193 547 435 723 033 380 546 150 4;
  • 72) 0.193 547 435 723 033 380 546 150 4 × 2 = 0 + 0.387 094 871 446 066 761 092 300 8;
  • 73) 0.387 094 871 446 066 761 092 300 8 × 2 = 0 + 0.774 189 742 892 133 522 184 601 6;
  • 74) 0.774 189 742 892 133 522 184 601 6 × 2 = 1 + 0.548 379 485 784 267 044 369 203 2;
  • 75) 0.548 379 485 784 267 044 369 203 2 × 2 = 1 + 0.096 758 971 568 534 088 738 406 4;
  • 76) 0.096 758 971 568 534 088 738 406 4 × 2 = 0 + 0.193 517 943 137 068 177 476 812 8;
  • 77) 0.193 517 943 137 068 177 476 812 8 × 2 = 0 + 0.387 035 886 274 136 354 953 625 6;
  • 78) 0.387 035 886 274 136 354 953 625 6 × 2 = 0 + 0.774 071 772 548 272 709 907 251 2;
  • 79) 0.774 071 772 548 272 709 907 251 2 × 2 = 1 + 0.548 143 545 096 545 419 814 502 4;
  • 80) 0.548 143 545 096 545 419 814 502 4 × 2 = 1 + 0.096 287 090 193 090 839 629 004 8;
  • 81) 0.096 287 090 193 090 839 629 004 8 × 2 = 0 + 0.192 574 180 386 181 679 258 009 6;
  • 82) 0.192 574 180 386 181 679 258 009 6 × 2 = 0 + 0.385 148 360 772 363 358 516 019 2;
  • 83) 0.385 148 360 772 363 358 516 019 2 × 2 = 0 + 0.770 296 721 544 726 717 032 038 4;
  • 84) 0.770 296 721 544 726 717 032 038 4 × 2 = 1 + 0.540 593 443 089 453 434 064 076 8;
  • 85) 0.540 593 443 089 453 434 064 076 8 × 2 = 1 + 0.081 186 886 178 906 868 128 153 6;
  • 86) 0.081 186 886 178 906 868 128 153 6 × 2 = 0 + 0.162 373 772 357 813 736 256 307 2;
  • 87) 0.162 373 772 357 813 736 256 307 2 × 2 = 0 + 0.324 747 544 715 627 472 512 614 4;
  • 88) 0.324 747 544 715 627 472 512 614 4 × 2 = 0 + 0.649 495 089 431 254 945 025 228 8;
  • 89) 0.649 495 089 431 254 945 025 228 8 × 2 = 1 + 0.298 990 178 862 509 890 050 457 6;
  • 90) 0.298 990 178 862 509 890 050 457 6 × 2 = 0 + 0.597 980 357 725 019 780 100 915 2;
  • 91) 0.597 980 357 725 019 780 100 915 2 × 2 = 1 + 0.195 960 715 450 039 560 201 830 4;
  • 92) 0.195 960 715 450 039 560 201 830 4 × 2 = 0 + 0.391 921 430 900 079 120 403 660 8;
  • 93) 0.391 921 430 900 079 120 403 660 8 × 2 = 0 + 0.783 842 861 800 158 240 807 321 6;
  • 94) 0.783 842 861 800 158 240 807 321 6 × 2 = 1 + 0.567 685 723 600 316 481 614 643 2;
  • 95) 0.567 685 723 600 316 481 614 643 2 × 2 = 1 + 0.135 371 447 200 632 963 229 286 4;
  • 96) 0.135 371 447 200 632 963 229 286 4 × 2 = 0 + 0.270 742 894 401 265 926 458 572 8;
  • 97) 0.270 742 894 401 265 926 458 572 8 × 2 = 0 + 0.541 485 788 802 531 852 917 145 6;
  • 98) 0.541 485 788 802 531 852 917 145 6 × 2 = 1 + 0.082 971 577 605 063 705 834 291 2;
  • 99) 0.082 971 577 605 063 705 834 291 2 × 2 = 0 + 0.165 943 155 210 127 411 668 582 4;
  • 100) 0.165 943 155 210 127 411 668 582 4 × 2 = 0 + 0.331 886 310 420 254 823 337 164 8;
  • 101) 0.331 886 310 420 254 823 337 164 8 × 2 = 0 + 0.663 772 620 840 509 646 674 329 6;
  • 102) 0.663 772 620 840 509 646 674 329 6 × 2 = 1 + 0.327 545 241 681 019 293 348 659 2;
  • 103) 0.327 545 241 681 019 293 348 659 2 × 2 = 0 + 0.655 090 483 362 038 586 697 318 4;
  • 104) 0.655 090 483 362 038 586 697 318 4 × 2 = 1 + 0.310 180 966 724 077 173 394 636 8;
  • 105) 0.310 180 966 724 077 173 394 636 8 × 2 = 0 + 0.620 361 933 448 154 346 789 273 6;
  • 106) 0.620 361 933 448 154 346 789 273 6 × 2 = 1 + 0.240 723 866 896 308 693 578 547 2;
  • 107) 0.240 723 866 896 308 693 578 547 2 × 2 = 0 + 0.481 447 733 792 617 387 157 094 4;
  • 108) 0.481 447 733 792 617 387 157 094 4 × 2 = 0 + 0.962 895 467 585 234 774 314 188 8;
  • 109) 0.962 895 467 585 234 774 314 188 8 × 2 = 1 + 0.925 790 935 170 469 548 628 377 6;
  • 110) 0.925 790 935 170 469 548 628 377 6 × 2 = 1 + 0.851 581 870 340 939 097 256 755 2;
  • 111) 0.851 581 870 340 939 097 256 755 2 × 2 = 1 + 0.703 163 740 681 878 194 513 510 4;
  • 112) 0.703 163 740 681 878 194 513 510 4 × 2 = 1 + 0.406 327 481 363 756 389 027 020 8;
  • 113) 0.406 327 481 363 756 389 027 020 8 × 2 = 0 + 0.812 654 962 727 512 778 054 041 6;
  • 114) 0.812 654 962 727 512 778 054 041 6 × 2 = 1 + 0.625 309 925 455 025 556 108 083 2;
  • 115) 0.625 309 925 455 025 556 108 083 2 × 2 = 1 + 0.250 619 850 910 051 112 216 166 4;
  • 116) 0.250 619 850 910 051 112 216 166 4 × 2 = 0 + 0.501 239 701 820 102 224 432 332 8;
  • 117) 0.501 239 701 820 102 224 432 332 8 × 2 = 1 + 0.002 479 403 640 204 448 864 665 6;
  • 118) 0.002 479 403 640 204 448 864 665 6 × 2 = 0 + 0.004 958 807 280 408 897 729 331 2;
  • 119) 0.004 958 807 280 408 897 729 331 2 × 2 = 0 + 0.009 917 614 560 817 795 458 662 4;

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:


0.000 000 000 000 000 000 008 552 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0011 0001 1000 1010 0110 0100 0101 0100 1111 0110 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 552 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0011 0001 1000 1010 0110 0100 0101 0100 1111 0110 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:


0.000 000 000 000 000 000 008 552 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0011 0001 1000 1010 0110 0100 0101 0100 1111 0110 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0011 0001 1000 1010 0110 0100 0101 0100 1111 0110 100(2) × 20 =

1.0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100(2) × 2-67


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100 =

0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100


Decimal number 0.000 000 000 000 000 000 008 552 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0001 1000 1100 0101 0011 0010 0010 1010 0111 1011 0100

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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