0.000 000 000 000 000 000 008 514 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 514 4(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
0.000 000 000 000 000 000 008 514 4(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: 0.000 000 000 000 000 000 008 514 4.

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) 0.000 000 000 000 000 000 008 514 4 × 2 = 0 + 0.000 000 000 000 000 000 017 028 8;
  • 2) 0.000 000 000 000 000 000 017 028 8 × 2 = 0 + 0.000 000 000 000 000 000 034 057 6;
  • 3) 0.000 000 000 000 000 000 034 057 6 × 2 = 0 + 0.000 000 000 000 000 000 068 115 2;
  • 4) 0.000 000 000 000 000 000 068 115 2 × 2 = 0 + 0.000 000 000 000 000 000 136 230 4;
  • 5) 0.000 000 000 000 000 000 136 230 4 × 2 = 0 + 0.000 000 000 000 000 000 272 460 8;
  • 6) 0.000 000 000 000 000 000 272 460 8 × 2 = 0 + 0.000 000 000 000 000 000 544 921 6;
  • 7) 0.000 000 000 000 000 000 544 921 6 × 2 = 0 + 0.000 000 000 000 000 001 089 843 2;
  • 8) 0.000 000 000 000 000 001 089 843 2 × 2 = 0 + 0.000 000 000 000 000 002 179 686 4;
  • 9) 0.000 000 000 000 000 002 179 686 4 × 2 = 0 + 0.000 000 000 000 000 004 359 372 8;
  • 10) 0.000 000 000 000 000 004 359 372 8 × 2 = 0 + 0.000 000 000 000 000 008 718 745 6;
  • 11) 0.000 000 000 000 000 008 718 745 6 × 2 = 0 + 0.000 000 000 000 000 017 437 491 2;
  • 12) 0.000 000 000 000 000 017 437 491 2 × 2 = 0 + 0.000 000 000 000 000 034 874 982 4;
  • 13) 0.000 000 000 000 000 034 874 982 4 × 2 = 0 + 0.000 000 000 000 000 069 749 964 8;
  • 14) 0.000 000 000 000 000 069 749 964 8 × 2 = 0 + 0.000 000 000 000 000 139 499 929 6;
  • 15) 0.000 000 000 000 000 139 499 929 6 × 2 = 0 + 0.000 000 000 000 000 278 999 859 2;
  • 16) 0.000 000 000 000 000 278 999 859 2 × 2 = 0 + 0.000 000 000 000 000 557 999 718 4;
  • 17) 0.000 000 000 000 000 557 999 718 4 × 2 = 0 + 0.000 000 000 000 001 115 999 436 8;
  • 18) 0.000 000 000 000 001 115 999 436 8 × 2 = 0 + 0.000 000 000 000 002 231 998 873 6;
  • 19) 0.000 000 000 000 002 231 998 873 6 × 2 = 0 + 0.000 000 000 000 004 463 997 747 2;
  • 20) 0.000 000 000 000 004 463 997 747 2 × 2 = 0 + 0.000 000 000 000 008 927 995 494 4;
  • 21) 0.000 000 000 000 008 927 995 494 4 × 2 = 0 + 0.000 000 000 000 017 855 990 988 8;
  • 22) 0.000 000 000 000 017 855 990 988 8 × 2 = 0 + 0.000 000 000 000 035 711 981 977 6;
  • 23) 0.000 000 000 000 035 711 981 977 6 × 2 = 0 + 0.000 000 000 000 071 423 963 955 2;
  • 24) 0.000 000 000 000 071 423 963 955 2 × 2 = 0 + 0.000 000 000 000 142 847 927 910 4;
  • 25) 0.000 000 000 000 142 847 927 910 4 × 2 = 0 + 0.000 000 000 000 285 695 855 820 8;
  • 26) 0.000 000 000 000 285 695 855 820 8 × 2 = 0 + 0.000 000 000 000 571 391 711 641 6;
  • 27) 0.000 000 000 000 571 391 711 641 6 × 2 = 0 + 0.000 000 000 001 142 783 423 283 2;
  • 28) 0.000 000 000 001 142 783 423 283 2 × 2 = 0 + 0.000 000 000 002 285 566 846 566 4;
  • 29) 0.000 000 000 002 285 566 846 566 4 × 2 = 0 + 0.000 000 000 004 571 133 693 132 8;
  • 30) 0.000 000 000 004 571 133 693 132 8 × 2 = 0 + 0.000 000 000 009 142 267 386 265 6;
  • 31) 0.000 000 000 009 142 267 386 265 6 × 2 = 0 + 0.000 000 000 018 284 534 772 531 2;
  • 32) 0.000 000 000 018 284 534 772 531 2 × 2 = 0 + 0.000 000 000 036 569 069 545 062 4;
  • 33) 0.000 000 000 036 569 069 545 062 4 × 2 = 0 + 0.000 000 000 073 138 139 090 124 8;
  • 34) 0.000 000 000 073 138 139 090 124 8 × 2 = 0 + 0.000 000 000 146 276 278 180 249 6;
  • 35) 0.000 000 000 146 276 278 180 249 6 × 2 = 0 + 0.000 000 000 292 552 556 360 499 2;
  • 36) 0.000 000 000 292 552 556 360 499 2 × 2 = 0 + 0.000 000 000 585 105 112 720 998 4;
  • 37) 0.000 000 000 585 105 112 720 998 4 × 2 = 0 + 0.000 000 001 170 210 225 441 996 8;
  • 38) 0.000 000 001 170 210 225 441 996 8 × 2 = 0 + 0.000 000 002 340 420 450 883 993 6;
  • 39) 0.000 000 002 340 420 450 883 993 6 × 2 = 0 + 0.000 000 004 680 840 901 767 987 2;
  • 40) 0.000 000 004 680 840 901 767 987 2 × 2 = 0 + 0.000 000 009 361 681 803 535 974 4;
  • 41) 0.000 000 009 361 681 803 535 974 4 × 2 = 0 + 0.000 000 018 723 363 607 071 948 8;
  • 42) 0.000 000 018 723 363 607 071 948 8 × 2 = 0 + 0.000 000 037 446 727 214 143 897 6;
  • 43) 0.000 000 037 446 727 214 143 897 6 × 2 = 0 + 0.000 000 074 893 454 428 287 795 2;
  • 44) 0.000 000 074 893 454 428 287 795 2 × 2 = 0 + 0.000 000 149 786 908 856 575 590 4;
  • 45) 0.000 000 149 786 908 856 575 590 4 × 2 = 0 + 0.000 000 299 573 817 713 151 180 8;
  • 46) 0.000 000 299 573 817 713 151 180 8 × 2 = 0 + 0.000 000 599 147 635 426 302 361 6;
  • 47) 0.000 000 599 147 635 426 302 361 6 × 2 = 0 + 0.000 001 198 295 270 852 604 723 2;
  • 48) 0.000 001 198 295 270 852 604 723 2 × 2 = 0 + 0.000 002 396 590 541 705 209 446 4;
  • 49) 0.000 002 396 590 541 705 209 446 4 × 2 = 0 + 0.000 004 793 181 083 410 418 892 8;
  • 50) 0.000 004 793 181 083 410 418 892 8 × 2 = 0 + 0.000 009 586 362 166 820 837 785 6;
  • 51) 0.000 009 586 362 166 820 837 785 6 × 2 = 0 + 0.000 019 172 724 333 641 675 571 2;
  • 52) 0.000 019 172 724 333 641 675 571 2 × 2 = 0 + 0.000 038 345 448 667 283 351 142 4;
  • 53) 0.000 038 345 448 667 283 351 142 4 × 2 = 0 + 0.000 076 690 897 334 566 702 284 8;
  • 54) 0.000 076 690 897 334 566 702 284 8 × 2 = 0 + 0.000 153 381 794 669 133 404 569 6;
  • 55) 0.000 153 381 794 669 133 404 569 6 × 2 = 0 + 0.000 306 763 589 338 266 809 139 2;
  • 56) 0.000 306 763 589 338 266 809 139 2 × 2 = 0 + 0.000 613 527 178 676 533 618 278 4;
  • 57) 0.000 613 527 178 676 533 618 278 4 × 2 = 0 + 0.001 227 054 357 353 067 236 556 8;
  • 58) 0.001 227 054 357 353 067 236 556 8 × 2 = 0 + 0.002 454 108 714 706 134 473 113 6;
  • 59) 0.002 454 108 714 706 134 473 113 6 × 2 = 0 + 0.004 908 217 429 412 268 946 227 2;
  • 60) 0.004 908 217 429 412 268 946 227 2 × 2 = 0 + 0.009 816 434 858 824 537 892 454 4;
  • 61) 0.009 816 434 858 824 537 892 454 4 × 2 = 0 + 0.019 632 869 717 649 075 784 908 8;
  • 62) 0.019 632 869 717 649 075 784 908 8 × 2 = 0 + 0.039 265 739 435 298 151 569 817 6;
  • 63) 0.039 265 739 435 298 151 569 817 6 × 2 = 0 + 0.078 531 478 870 596 303 139 635 2;
  • 64) 0.078 531 478 870 596 303 139 635 2 × 2 = 0 + 0.157 062 957 741 192 606 279 270 4;
  • 65) 0.157 062 957 741 192 606 279 270 4 × 2 = 0 + 0.314 125 915 482 385 212 558 540 8;
  • 66) 0.314 125 915 482 385 212 558 540 8 × 2 = 0 + 0.628 251 830 964 770 425 117 081 6;
  • 67) 0.628 251 830 964 770 425 117 081 6 × 2 = 1 + 0.256 503 661 929 540 850 234 163 2;
  • 68) 0.256 503 661 929 540 850 234 163 2 × 2 = 0 + 0.513 007 323 859 081 700 468 326 4;
  • 69) 0.513 007 323 859 081 700 468 326 4 × 2 = 1 + 0.026 014 647 718 163 400 936 652 8;
  • 70) 0.026 014 647 718 163 400 936 652 8 × 2 = 0 + 0.052 029 295 436 326 801 873 305 6;
  • 71) 0.052 029 295 436 326 801 873 305 6 × 2 = 0 + 0.104 058 590 872 653 603 746 611 2;
  • 72) 0.104 058 590 872 653 603 746 611 2 × 2 = 0 + 0.208 117 181 745 307 207 493 222 4;
  • 73) 0.208 117 181 745 307 207 493 222 4 × 2 = 0 + 0.416 234 363 490 614 414 986 444 8;
  • 74) 0.416 234 363 490 614 414 986 444 8 × 2 = 0 + 0.832 468 726 981 228 829 972 889 6;
  • 75) 0.832 468 726 981 228 829 972 889 6 × 2 = 1 + 0.664 937 453 962 457 659 945 779 2;
  • 76) 0.664 937 453 962 457 659 945 779 2 × 2 = 1 + 0.329 874 907 924 915 319 891 558 4;
  • 77) 0.329 874 907 924 915 319 891 558 4 × 2 = 0 + 0.659 749 815 849 830 639 783 116 8;
  • 78) 0.659 749 815 849 830 639 783 116 8 × 2 = 1 + 0.319 499 631 699 661 279 566 233 6;
  • 79) 0.319 499 631 699 661 279 566 233 6 × 2 = 0 + 0.638 999 263 399 322 559 132 467 2;
  • 80) 0.638 999 263 399 322 559 132 467 2 × 2 = 1 + 0.277 998 526 798 645 118 264 934 4;
  • 81) 0.277 998 526 798 645 118 264 934 4 × 2 = 0 + 0.555 997 053 597 290 236 529 868 8;
  • 82) 0.555 997 053 597 290 236 529 868 8 × 2 = 1 + 0.111 994 107 194 580 473 059 737 6;
  • 83) 0.111 994 107 194 580 473 059 737 6 × 2 = 0 + 0.223 988 214 389 160 946 119 475 2;
  • 84) 0.223 988 214 389 160 946 119 475 2 × 2 = 0 + 0.447 976 428 778 321 892 238 950 4;
  • 85) 0.447 976 428 778 321 892 238 950 4 × 2 = 0 + 0.895 952 857 556 643 784 477 900 8;
  • 86) 0.895 952 857 556 643 784 477 900 8 × 2 = 1 + 0.791 905 715 113 287 568 955 801 6;
  • 87) 0.791 905 715 113 287 568 955 801 6 × 2 = 1 + 0.583 811 430 226 575 137 911 603 2;
  • 88) 0.583 811 430 226 575 137 911 603 2 × 2 = 1 + 0.167 622 860 453 150 275 823 206 4;
  • 89) 0.167 622 860 453 150 275 823 206 4 × 2 = 0 + 0.335 245 720 906 300 551 646 412 8;
  • 90) 0.335 245 720 906 300 551 646 412 8 × 2 = 0 + 0.670 491 441 812 601 103 292 825 6;
  • 91) 0.670 491 441 812 601 103 292 825 6 × 2 = 1 + 0.340 982 883 625 202 206 585 651 2;
  • 92) 0.340 982 883 625 202 206 585 651 2 × 2 = 0 + 0.681 965 767 250 404 413 171 302 4;
  • 93) 0.681 965 767 250 404 413 171 302 4 × 2 = 1 + 0.363 931 534 500 808 826 342 604 8;
  • 94) 0.363 931 534 500 808 826 342 604 8 × 2 = 0 + 0.727 863 069 001 617 652 685 209 6;
  • 95) 0.727 863 069 001 617 652 685 209 6 × 2 = 1 + 0.455 726 138 003 235 305 370 419 2;
  • 96) 0.455 726 138 003 235 305 370 419 2 × 2 = 0 + 0.911 452 276 006 470 610 740 838 4;
  • 97) 0.911 452 276 006 470 610 740 838 4 × 2 = 1 + 0.822 904 552 012 941 221 481 676 8;
  • 98) 0.822 904 552 012 941 221 481 676 8 × 2 = 1 + 0.645 809 104 025 882 442 963 353 6;
  • 99) 0.645 809 104 025 882 442 963 353 6 × 2 = 1 + 0.291 618 208 051 764 885 926 707 2;
  • 100) 0.291 618 208 051 764 885 926 707 2 × 2 = 0 + 0.583 236 416 103 529 771 853 414 4;
  • 101) 0.583 236 416 103 529 771 853 414 4 × 2 = 1 + 0.166 472 832 207 059 543 706 828 8;
  • 102) 0.166 472 832 207 059 543 706 828 8 × 2 = 0 + 0.332 945 664 414 119 087 413 657 6;
  • 103) 0.332 945 664 414 119 087 413 657 6 × 2 = 0 + 0.665 891 328 828 238 174 827 315 2;
  • 104) 0.665 891 328 828 238 174 827 315 2 × 2 = 1 + 0.331 782 657 656 476 349 654 630 4;
  • 105) 0.331 782 657 656 476 349 654 630 4 × 2 = 0 + 0.663 565 315 312 952 699 309 260 8;
  • 106) 0.663 565 315 312 952 699 309 260 8 × 2 = 1 + 0.327 130 630 625 905 398 618 521 6;
  • 107) 0.327 130 630 625 905 398 618 521 6 × 2 = 0 + 0.654 261 261 251 810 797 237 043 2;
  • 108) 0.654 261 261 251 810 797 237 043 2 × 2 = 1 + 0.308 522 522 503 621 594 474 086 4;
  • 109) 0.308 522 522 503 621 594 474 086 4 × 2 = 0 + 0.617 045 045 007 243 188 948 172 8;
  • 110) 0.617 045 045 007 243 188 948 172 8 × 2 = 1 + 0.234 090 090 014 486 377 896 345 6;
  • 111) 0.234 090 090 014 486 377 896 345 6 × 2 = 0 + 0.468 180 180 028 972 755 792 691 2;
  • 112) 0.468 180 180 028 972 755 792 691 2 × 2 = 0 + 0.936 360 360 057 945 511 585 382 4;
  • 113) 0.936 360 360 057 945 511 585 382 4 × 2 = 1 + 0.872 720 720 115 891 023 170 764 8;
  • 114) 0.872 720 720 115 891 023 170 764 8 × 2 = 1 + 0.745 441 440 231 782 046 341 529 6;
  • 115) 0.745 441 440 231 782 046 341 529 6 × 2 = 1 + 0.490 882 880 463 564 092 683 059 2;
  • 116) 0.490 882 880 463 564 092 683 059 2 × 2 = 0 + 0.981 765 760 927 128 185 366 118 4;
  • 117) 0.981 765 760 927 128 185 366 118 4 × 2 = 1 + 0.963 531 521 854 256 370 732 236 8;
  • 118) 0.963 531 521 854 256 370 732 236 8 × 2 = 1 + 0.927 063 043 708 512 741 464 473 6;
  • 119) 0.927 063 043 708 512 741 464 473 6 × 2 = 1 + 0.854 126 087 417 025 482 928 947 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:


0.000 000 000 000 000 000 008 514 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 0101 0100 0111 0010 1010 1110 1001 0101 0100 1110 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 514 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 0101 0100 0111 0010 1010 1110 1001 0101 0100 1110 111(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:


0.000 000 000 000 000 000 008 514 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 0101 0100 0111 0010 1010 1110 1001 0101 0100 1110 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0011 0101 0100 0111 0010 1010 1110 1001 0101 0100 1110 111(2) × 20 =

1.0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111(2) × 2-67


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

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-67 + 1 023)(10) =

956(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

1. 0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111 =

0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111


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

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

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111


Decimal number 0.000 000 000 000 000 000 008 514 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0001 1010 1010 0011 1001 0101 0111 0100 1010 1010 0111 0111

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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