0.000 000 000 000 000 000 008 529 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 529 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
0.000 000 000 000 000 000 008 529 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: 0.000 000 000 000 000 000 008 529 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) 0.000 000 000 000 000 000 008 529 9 × 2 = 0 + 0.000 000 000 000 000 000 017 059 8;
  • 2) 0.000 000 000 000 000 000 017 059 8 × 2 = 0 + 0.000 000 000 000 000 000 034 119 6;
  • 3) 0.000 000 000 000 000 000 034 119 6 × 2 = 0 + 0.000 000 000 000 000 000 068 239 2;
  • 4) 0.000 000 000 000 000 000 068 239 2 × 2 = 0 + 0.000 000 000 000 000 000 136 478 4;
  • 5) 0.000 000 000 000 000 000 136 478 4 × 2 = 0 + 0.000 000 000 000 000 000 272 956 8;
  • 6) 0.000 000 000 000 000 000 272 956 8 × 2 = 0 + 0.000 000 000 000 000 000 545 913 6;
  • 7) 0.000 000 000 000 000 000 545 913 6 × 2 = 0 + 0.000 000 000 000 000 001 091 827 2;
  • 8) 0.000 000 000 000 000 001 091 827 2 × 2 = 0 + 0.000 000 000 000 000 002 183 654 4;
  • 9) 0.000 000 000 000 000 002 183 654 4 × 2 = 0 + 0.000 000 000 000 000 004 367 308 8;
  • 10) 0.000 000 000 000 000 004 367 308 8 × 2 = 0 + 0.000 000 000 000 000 008 734 617 6;
  • 11) 0.000 000 000 000 000 008 734 617 6 × 2 = 0 + 0.000 000 000 000 000 017 469 235 2;
  • 12) 0.000 000 000 000 000 017 469 235 2 × 2 = 0 + 0.000 000 000 000 000 034 938 470 4;
  • 13) 0.000 000 000 000 000 034 938 470 4 × 2 = 0 + 0.000 000 000 000 000 069 876 940 8;
  • 14) 0.000 000 000 000 000 069 876 940 8 × 2 = 0 + 0.000 000 000 000 000 139 753 881 6;
  • 15) 0.000 000 000 000 000 139 753 881 6 × 2 = 0 + 0.000 000 000 000 000 279 507 763 2;
  • 16) 0.000 000 000 000 000 279 507 763 2 × 2 = 0 + 0.000 000 000 000 000 559 015 526 4;
  • 17) 0.000 000 000 000 000 559 015 526 4 × 2 = 0 + 0.000 000 000 000 001 118 031 052 8;
  • 18) 0.000 000 000 000 001 118 031 052 8 × 2 = 0 + 0.000 000 000 000 002 236 062 105 6;
  • 19) 0.000 000 000 000 002 236 062 105 6 × 2 = 0 + 0.000 000 000 000 004 472 124 211 2;
  • 20) 0.000 000 000 000 004 472 124 211 2 × 2 = 0 + 0.000 000 000 000 008 944 248 422 4;
  • 21) 0.000 000 000 000 008 944 248 422 4 × 2 = 0 + 0.000 000 000 000 017 888 496 844 8;
  • 22) 0.000 000 000 000 017 888 496 844 8 × 2 = 0 + 0.000 000 000 000 035 776 993 689 6;
  • 23) 0.000 000 000 000 035 776 993 689 6 × 2 = 0 + 0.000 000 000 000 071 553 987 379 2;
  • 24) 0.000 000 000 000 071 553 987 379 2 × 2 = 0 + 0.000 000 000 000 143 107 974 758 4;
  • 25) 0.000 000 000 000 143 107 974 758 4 × 2 = 0 + 0.000 000 000 000 286 215 949 516 8;
  • 26) 0.000 000 000 000 286 215 949 516 8 × 2 = 0 + 0.000 000 000 000 572 431 899 033 6;
  • 27) 0.000 000 000 000 572 431 899 033 6 × 2 = 0 + 0.000 000 000 001 144 863 798 067 2;
  • 28) 0.000 000 000 001 144 863 798 067 2 × 2 = 0 + 0.000 000 000 002 289 727 596 134 4;
  • 29) 0.000 000 000 002 289 727 596 134 4 × 2 = 0 + 0.000 000 000 004 579 455 192 268 8;
  • 30) 0.000 000 000 004 579 455 192 268 8 × 2 = 0 + 0.000 000 000 009 158 910 384 537 6;
  • 31) 0.000 000 000 009 158 910 384 537 6 × 2 = 0 + 0.000 000 000 018 317 820 769 075 2;
  • 32) 0.000 000 000 018 317 820 769 075 2 × 2 = 0 + 0.000 000 000 036 635 641 538 150 4;
  • 33) 0.000 000 000 036 635 641 538 150 4 × 2 = 0 + 0.000 000 000 073 271 283 076 300 8;
  • 34) 0.000 000 000 073 271 283 076 300 8 × 2 = 0 + 0.000 000 000 146 542 566 152 601 6;
  • 35) 0.000 000 000 146 542 566 152 601 6 × 2 = 0 + 0.000 000 000 293 085 132 305 203 2;
  • 36) 0.000 000 000 293 085 132 305 203 2 × 2 = 0 + 0.000 000 000 586 170 264 610 406 4;
  • 37) 0.000 000 000 586 170 264 610 406 4 × 2 = 0 + 0.000 000 001 172 340 529 220 812 8;
  • 38) 0.000 000 001 172 340 529 220 812 8 × 2 = 0 + 0.000 000 002 344 681 058 441 625 6;
  • 39) 0.000 000 002 344 681 058 441 625 6 × 2 = 0 + 0.000 000 004 689 362 116 883 251 2;
  • 40) 0.000 000 004 689 362 116 883 251 2 × 2 = 0 + 0.000 000 009 378 724 233 766 502 4;
  • 41) 0.000 000 009 378 724 233 766 502 4 × 2 = 0 + 0.000 000 018 757 448 467 533 004 8;
  • 42) 0.000 000 018 757 448 467 533 004 8 × 2 = 0 + 0.000 000 037 514 896 935 066 009 6;
  • 43) 0.000 000 037 514 896 935 066 009 6 × 2 = 0 + 0.000 000 075 029 793 870 132 019 2;
  • 44) 0.000 000 075 029 793 870 132 019 2 × 2 = 0 + 0.000 000 150 059 587 740 264 038 4;
  • 45) 0.000 000 150 059 587 740 264 038 4 × 2 = 0 + 0.000 000 300 119 175 480 528 076 8;
  • 46) 0.000 000 300 119 175 480 528 076 8 × 2 = 0 + 0.000 000 600 238 350 961 056 153 6;
  • 47) 0.000 000 600 238 350 961 056 153 6 × 2 = 0 + 0.000 001 200 476 701 922 112 307 2;
  • 48) 0.000 001 200 476 701 922 112 307 2 × 2 = 0 + 0.000 002 400 953 403 844 224 614 4;
  • 49) 0.000 002 400 953 403 844 224 614 4 × 2 = 0 + 0.000 004 801 906 807 688 449 228 8;
  • 50) 0.000 004 801 906 807 688 449 228 8 × 2 = 0 + 0.000 009 603 813 615 376 898 457 6;
  • 51) 0.000 009 603 813 615 376 898 457 6 × 2 = 0 + 0.000 019 207 627 230 753 796 915 2;
  • 52) 0.000 019 207 627 230 753 796 915 2 × 2 = 0 + 0.000 038 415 254 461 507 593 830 4;
  • 53) 0.000 038 415 254 461 507 593 830 4 × 2 = 0 + 0.000 076 830 508 923 015 187 660 8;
  • 54) 0.000 076 830 508 923 015 187 660 8 × 2 = 0 + 0.000 153 661 017 846 030 375 321 6;
  • 55) 0.000 153 661 017 846 030 375 321 6 × 2 = 0 + 0.000 307 322 035 692 060 750 643 2;
  • 56) 0.000 307 322 035 692 060 750 643 2 × 2 = 0 + 0.000 614 644 071 384 121 501 286 4;
  • 57) 0.000 614 644 071 384 121 501 286 4 × 2 = 0 + 0.001 229 288 142 768 243 002 572 8;
  • 58) 0.001 229 288 142 768 243 002 572 8 × 2 = 0 + 0.002 458 576 285 536 486 005 145 6;
  • 59) 0.002 458 576 285 536 486 005 145 6 × 2 = 0 + 0.004 917 152 571 072 972 010 291 2;
  • 60) 0.004 917 152 571 072 972 010 291 2 × 2 = 0 + 0.009 834 305 142 145 944 020 582 4;
  • 61) 0.009 834 305 142 145 944 020 582 4 × 2 = 0 + 0.019 668 610 284 291 888 041 164 8;
  • 62) 0.019 668 610 284 291 888 041 164 8 × 2 = 0 + 0.039 337 220 568 583 776 082 329 6;
  • 63) 0.039 337 220 568 583 776 082 329 6 × 2 = 0 + 0.078 674 441 137 167 552 164 659 2;
  • 64) 0.078 674 441 137 167 552 164 659 2 × 2 = 0 + 0.157 348 882 274 335 104 329 318 4;
  • 65) 0.157 348 882 274 335 104 329 318 4 × 2 = 0 + 0.314 697 764 548 670 208 658 636 8;
  • 66) 0.314 697 764 548 670 208 658 636 8 × 2 = 0 + 0.629 395 529 097 340 417 317 273 6;
  • 67) 0.629 395 529 097 340 417 317 273 6 × 2 = 1 + 0.258 791 058 194 680 834 634 547 2;
  • 68) 0.258 791 058 194 680 834 634 547 2 × 2 = 0 + 0.517 582 116 389 361 669 269 094 4;
  • 69) 0.517 582 116 389 361 669 269 094 4 × 2 = 1 + 0.035 164 232 778 723 338 538 188 8;
  • 70) 0.035 164 232 778 723 338 538 188 8 × 2 = 0 + 0.070 328 465 557 446 677 076 377 6;
  • 71) 0.070 328 465 557 446 677 076 377 6 × 2 = 0 + 0.140 656 931 114 893 354 152 755 2;
  • 72) 0.140 656 931 114 893 354 152 755 2 × 2 = 0 + 0.281 313 862 229 786 708 305 510 4;
  • 73) 0.281 313 862 229 786 708 305 510 4 × 2 = 0 + 0.562 627 724 459 573 416 611 020 8;
  • 74) 0.562 627 724 459 573 416 611 020 8 × 2 = 1 + 0.125 255 448 919 146 833 222 041 6;
  • 75) 0.125 255 448 919 146 833 222 041 6 × 2 = 0 + 0.250 510 897 838 293 666 444 083 2;
  • 76) 0.250 510 897 838 293 666 444 083 2 × 2 = 0 + 0.501 021 795 676 587 332 888 166 4;
  • 77) 0.501 021 795 676 587 332 888 166 4 × 2 = 1 + 0.002 043 591 353 174 665 776 332 8;
  • 78) 0.002 043 591 353 174 665 776 332 8 × 2 = 0 + 0.004 087 182 706 349 331 552 665 6;
  • 79) 0.004 087 182 706 349 331 552 665 6 × 2 = 0 + 0.008 174 365 412 698 663 105 331 2;
  • 80) 0.008 174 365 412 698 663 105 331 2 × 2 = 0 + 0.016 348 730 825 397 326 210 662 4;
  • 81) 0.016 348 730 825 397 326 210 662 4 × 2 = 0 + 0.032 697 461 650 794 652 421 324 8;
  • 82) 0.032 697 461 650 794 652 421 324 8 × 2 = 0 + 0.065 394 923 301 589 304 842 649 6;
  • 83) 0.065 394 923 301 589 304 842 649 6 × 2 = 0 + 0.130 789 846 603 178 609 685 299 2;
  • 84) 0.130 789 846 603 178 609 685 299 2 × 2 = 0 + 0.261 579 693 206 357 219 370 598 4;
  • 85) 0.261 579 693 206 357 219 370 598 4 × 2 = 0 + 0.523 159 386 412 714 438 741 196 8;
  • 86) 0.523 159 386 412 714 438 741 196 8 × 2 = 1 + 0.046 318 772 825 428 877 482 393 6;
  • 87) 0.046 318 772 825 428 877 482 393 6 × 2 = 0 + 0.092 637 545 650 857 754 964 787 2;
  • 88) 0.092 637 545 650 857 754 964 787 2 × 2 = 0 + 0.185 275 091 301 715 509 929 574 4;
  • 89) 0.185 275 091 301 715 509 929 574 4 × 2 = 0 + 0.370 550 182 603 431 019 859 148 8;
  • 90) 0.370 550 182 603 431 019 859 148 8 × 2 = 0 + 0.741 100 365 206 862 039 718 297 6;
  • 91) 0.741 100 365 206 862 039 718 297 6 × 2 = 1 + 0.482 200 730 413 724 079 436 595 2;
  • 92) 0.482 200 730 413 724 079 436 595 2 × 2 = 0 + 0.964 401 460 827 448 158 873 190 4;
  • 93) 0.964 401 460 827 448 158 873 190 4 × 2 = 1 + 0.928 802 921 654 896 317 746 380 8;
  • 94) 0.928 802 921 654 896 317 746 380 8 × 2 = 1 + 0.857 605 843 309 792 635 492 761 6;
  • 95) 0.857 605 843 309 792 635 492 761 6 × 2 = 1 + 0.715 211 686 619 585 270 985 523 2;
  • 96) 0.715 211 686 619 585 270 985 523 2 × 2 = 1 + 0.430 423 373 239 170 541 971 046 4;
  • 97) 0.430 423 373 239 170 541 971 046 4 × 2 = 0 + 0.860 846 746 478 341 083 942 092 8;
  • 98) 0.860 846 746 478 341 083 942 092 8 × 2 = 1 + 0.721 693 492 956 682 167 884 185 6;
  • 99) 0.721 693 492 956 682 167 884 185 6 × 2 = 1 + 0.443 386 985 913 364 335 768 371 2;
  • 100) 0.443 386 985 913 364 335 768 371 2 × 2 = 0 + 0.886 773 971 826 728 671 536 742 4;
  • 101) 0.886 773 971 826 728 671 536 742 4 × 2 = 1 + 0.773 547 943 653 457 343 073 484 8;
  • 102) 0.773 547 943 653 457 343 073 484 8 × 2 = 1 + 0.547 095 887 306 914 686 146 969 6;
  • 103) 0.547 095 887 306 914 686 146 969 6 × 2 = 1 + 0.094 191 774 613 829 372 293 939 2;
  • 104) 0.094 191 774 613 829 372 293 939 2 × 2 = 0 + 0.188 383 549 227 658 744 587 878 4;
  • 105) 0.188 383 549 227 658 744 587 878 4 × 2 = 0 + 0.376 767 098 455 317 489 175 756 8;
  • 106) 0.376 767 098 455 317 489 175 756 8 × 2 = 0 + 0.753 534 196 910 634 978 351 513 6;
  • 107) 0.753 534 196 910 634 978 351 513 6 × 2 = 1 + 0.507 068 393 821 269 956 703 027 2;
  • 108) 0.507 068 393 821 269 956 703 027 2 × 2 = 1 + 0.014 136 787 642 539 913 406 054 4;
  • 109) 0.014 136 787 642 539 913 406 054 4 × 2 = 0 + 0.028 273 575 285 079 826 812 108 8;
  • 110) 0.028 273 575 285 079 826 812 108 8 × 2 = 0 + 0.056 547 150 570 159 653 624 217 6;
  • 111) 0.056 547 150 570 159 653 624 217 6 × 2 = 0 + 0.113 094 301 140 319 307 248 435 2;
  • 112) 0.113 094 301 140 319 307 248 435 2 × 2 = 0 + 0.226 188 602 280 638 614 496 870 4;
  • 113) 0.226 188 602 280 638 614 496 870 4 × 2 = 0 + 0.452 377 204 561 277 228 993 740 8;
  • 114) 0.452 377 204 561 277 228 993 740 8 × 2 = 0 + 0.904 754 409 122 554 457 987 481 6;
  • 115) 0.904 754 409 122 554 457 987 481 6 × 2 = 1 + 0.809 508 818 245 108 915 974 963 2;
  • 116) 0.809 508 818 245 108 915 974 963 2 × 2 = 1 + 0.619 017 636 490 217 831 949 926 4;
  • 117) 0.619 017 636 490 217 831 949 926 4 × 2 = 1 + 0.238 035 272 980 435 663 899 852 8;
  • 118) 0.238 035 272 980 435 663 899 852 8 × 2 = 0 + 0.476 070 545 960 871 327 799 705 6;
  • 119) 0.476 070 545 960 871 327 799 705 6 × 2 = 0 + 0.952 141 091 921 742 655 599 411 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:


0.000 000 000 000 000 000 008 529 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1000 0000 0100 0010 1111 0110 1110 0011 0000 0011 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 529 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1000 0000 0100 0010 1111 0110 1110 0011 0000 0011 100(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 529 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1000 0000 0100 0010 1111 0110 1110 0011 0000 0011 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1000 0000 0100 0010 1111 0110 1110 0011 0000 0011 100(2) × 20 =

1.0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100(2) × 2-67


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

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-67 + 1 023)(10) =

956(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

1. 0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100 =

0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100


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

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

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100


Decimal number 0.000 000 000 000 000 000 008 529 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0100 0000 0010 0001 0111 1011 0111 0001 1000 0001 1100

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

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

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100