0.000 000 000 000 100 000 000 000 029 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 100 000 000 000 029 7(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
0.000 000 000 000 100 000 000 000 029 7(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: 0.000 000 000 000 100 000 000 000 029 7.

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) 0.000 000 000 000 100 000 000 000 029 7 × 2 = 0 + 0.000 000 000 000 200 000 000 000 059 4;
  • 2) 0.000 000 000 000 200 000 000 000 059 4 × 2 = 0 + 0.000 000 000 000 400 000 000 000 118 8;
  • 3) 0.000 000 000 000 400 000 000 000 118 8 × 2 = 0 + 0.000 000 000 000 800 000 000 000 237 6;
  • 4) 0.000 000 000 000 800 000 000 000 237 6 × 2 = 0 + 0.000 000 000 001 600 000 000 000 475 2;
  • 5) 0.000 000 000 001 600 000 000 000 475 2 × 2 = 0 + 0.000 000 000 003 200 000 000 000 950 4;
  • 6) 0.000 000 000 003 200 000 000 000 950 4 × 2 = 0 + 0.000 000 000 006 400 000 000 001 900 8;
  • 7) 0.000 000 000 006 400 000 000 001 900 8 × 2 = 0 + 0.000 000 000 012 800 000 000 003 801 6;
  • 8) 0.000 000 000 012 800 000 000 003 801 6 × 2 = 0 + 0.000 000 000 025 600 000 000 007 603 2;
  • 9) 0.000 000 000 025 600 000 000 007 603 2 × 2 = 0 + 0.000 000 000 051 200 000 000 015 206 4;
  • 10) 0.000 000 000 051 200 000 000 015 206 4 × 2 = 0 + 0.000 000 000 102 400 000 000 030 412 8;
  • 11) 0.000 000 000 102 400 000 000 030 412 8 × 2 = 0 + 0.000 000 000 204 800 000 000 060 825 6;
  • 12) 0.000 000 000 204 800 000 000 060 825 6 × 2 = 0 + 0.000 000 000 409 600 000 000 121 651 2;
  • 13) 0.000 000 000 409 600 000 000 121 651 2 × 2 = 0 + 0.000 000 000 819 200 000 000 243 302 4;
  • 14) 0.000 000 000 819 200 000 000 243 302 4 × 2 = 0 + 0.000 000 001 638 400 000 000 486 604 8;
  • 15) 0.000 000 001 638 400 000 000 486 604 8 × 2 = 0 + 0.000 000 003 276 800 000 000 973 209 6;
  • 16) 0.000 000 003 276 800 000 000 973 209 6 × 2 = 0 + 0.000 000 006 553 600 000 001 946 419 2;
  • 17) 0.000 000 006 553 600 000 001 946 419 2 × 2 = 0 + 0.000 000 013 107 200 000 003 892 838 4;
  • 18) 0.000 000 013 107 200 000 003 892 838 4 × 2 = 0 + 0.000 000 026 214 400 000 007 785 676 8;
  • 19) 0.000 000 026 214 400 000 007 785 676 8 × 2 = 0 + 0.000 000 052 428 800 000 015 571 353 6;
  • 20) 0.000 000 052 428 800 000 015 571 353 6 × 2 = 0 + 0.000 000 104 857 600 000 031 142 707 2;
  • 21) 0.000 000 104 857 600 000 031 142 707 2 × 2 = 0 + 0.000 000 209 715 200 000 062 285 414 4;
  • 22) 0.000 000 209 715 200 000 062 285 414 4 × 2 = 0 + 0.000 000 419 430 400 000 124 570 828 8;
  • 23) 0.000 000 419 430 400 000 124 570 828 8 × 2 = 0 + 0.000 000 838 860 800 000 249 141 657 6;
  • 24) 0.000 000 838 860 800 000 249 141 657 6 × 2 = 0 + 0.000 001 677 721 600 000 498 283 315 2;
  • 25) 0.000 001 677 721 600 000 498 283 315 2 × 2 = 0 + 0.000 003 355 443 200 000 996 566 630 4;
  • 26) 0.000 003 355 443 200 000 996 566 630 4 × 2 = 0 + 0.000 006 710 886 400 001 993 133 260 8;
  • 27) 0.000 006 710 886 400 001 993 133 260 8 × 2 = 0 + 0.000 013 421 772 800 003 986 266 521 6;
  • 28) 0.000 013 421 772 800 003 986 266 521 6 × 2 = 0 + 0.000 026 843 545 600 007 972 533 043 2;
  • 29) 0.000 026 843 545 600 007 972 533 043 2 × 2 = 0 + 0.000 053 687 091 200 015 945 066 086 4;
  • 30) 0.000 053 687 091 200 015 945 066 086 4 × 2 = 0 + 0.000 107 374 182 400 031 890 132 172 8;
  • 31) 0.000 107 374 182 400 031 890 132 172 8 × 2 = 0 + 0.000 214 748 364 800 063 780 264 345 6;
  • 32) 0.000 214 748 364 800 063 780 264 345 6 × 2 = 0 + 0.000 429 496 729 600 127 560 528 691 2;
  • 33) 0.000 429 496 729 600 127 560 528 691 2 × 2 = 0 + 0.000 858 993 459 200 255 121 057 382 4;
  • 34) 0.000 858 993 459 200 255 121 057 382 4 × 2 = 0 + 0.001 717 986 918 400 510 242 114 764 8;
  • 35) 0.001 717 986 918 400 510 242 114 764 8 × 2 = 0 + 0.003 435 973 836 801 020 484 229 529 6;
  • 36) 0.003 435 973 836 801 020 484 229 529 6 × 2 = 0 + 0.006 871 947 673 602 040 968 459 059 2;
  • 37) 0.006 871 947 673 602 040 968 459 059 2 × 2 = 0 + 0.013 743 895 347 204 081 936 918 118 4;
  • 38) 0.013 743 895 347 204 081 936 918 118 4 × 2 = 0 + 0.027 487 790 694 408 163 873 836 236 8;
  • 39) 0.027 487 790 694 408 163 873 836 236 8 × 2 = 0 + 0.054 975 581 388 816 327 747 672 473 6;
  • 40) 0.054 975 581 388 816 327 747 672 473 6 × 2 = 0 + 0.109 951 162 777 632 655 495 344 947 2;
  • 41) 0.109 951 162 777 632 655 495 344 947 2 × 2 = 0 + 0.219 902 325 555 265 310 990 689 894 4;
  • 42) 0.219 902 325 555 265 310 990 689 894 4 × 2 = 0 + 0.439 804 651 110 530 621 981 379 788 8;
  • 43) 0.439 804 651 110 530 621 981 379 788 8 × 2 = 0 + 0.879 609 302 221 061 243 962 759 577 6;
  • 44) 0.879 609 302 221 061 243 962 759 577 6 × 2 = 1 + 0.759 218 604 442 122 487 925 519 155 2;
  • 45) 0.759 218 604 442 122 487 925 519 155 2 × 2 = 1 + 0.518 437 208 884 244 975 851 038 310 4;
  • 46) 0.518 437 208 884 244 975 851 038 310 4 × 2 = 1 + 0.036 874 417 768 489 951 702 076 620 8;
  • 47) 0.036 874 417 768 489 951 702 076 620 8 × 2 = 0 + 0.073 748 835 536 979 903 404 153 241 6;
  • 48) 0.073 748 835 536 979 903 404 153 241 6 × 2 = 0 + 0.147 497 671 073 959 806 808 306 483 2;
  • 49) 0.147 497 671 073 959 806 808 306 483 2 × 2 = 0 + 0.294 995 342 147 919 613 616 612 966 4;
  • 50) 0.294 995 342 147 919 613 616 612 966 4 × 2 = 0 + 0.589 990 684 295 839 227 233 225 932 8;
  • 51) 0.589 990 684 295 839 227 233 225 932 8 × 2 = 1 + 0.179 981 368 591 678 454 466 451 865 6;
  • 52) 0.179 981 368 591 678 454 466 451 865 6 × 2 = 0 + 0.359 962 737 183 356 908 932 903 731 2;
  • 53) 0.359 962 737 183 356 908 932 903 731 2 × 2 = 0 + 0.719 925 474 366 713 817 865 807 462 4;
  • 54) 0.719 925 474 366 713 817 865 807 462 4 × 2 = 1 + 0.439 850 948 733 427 635 731 614 924 8;
  • 55) 0.439 850 948 733 427 635 731 614 924 8 × 2 = 0 + 0.879 701 897 466 855 271 463 229 849 6;
  • 56) 0.879 701 897 466 855 271 463 229 849 6 × 2 = 1 + 0.759 403 794 933 710 542 926 459 699 2;
  • 57) 0.759 403 794 933 710 542 926 459 699 2 × 2 = 1 + 0.518 807 589 867 421 085 852 919 398 4;
  • 58) 0.518 807 589 867 421 085 852 919 398 4 × 2 = 1 + 0.037 615 179 734 842 171 705 838 796 8;
  • 59) 0.037 615 179 734 842 171 705 838 796 8 × 2 = 0 + 0.075 230 359 469 684 343 411 677 593 6;
  • 60) 0.075 230 359 469 684 343 411 677 593 6 × 2 = 0 + 0.150 460 718 939 368 686 823 355 187 2;
  • 61) 0.150 460 718 939 368 686 823 355 187 2 × 2 = 0 + 0.300 921 437 878 737 373 646 710 374 4;
  • 62) 0.300 921 437 878 737 373 646 710 374 4 × 2 = 0 + 0.601 842 875 757 474 747 293 420 748 8;
  • 63) 0.601 842 875 757 474 747 293 420 748 8 × 2 = 1 + 0.203 685 751 514 949 494 586 841 497 6;
  • 64) 0.203 685 751 514 949 494 586 841 497 6 × 2 = 0 + 0.407 371 503 029 898 989 173 682 995 2;
  • 65) 0.407 371 503 029 898 989 173 682 995 2 × 2 = 0 + 0.814 743 006 059 797 978 347 365 990 4;
  • 66) 0.814 743 006 059 797 978 347 365 990 4 × 2 = 1 + 0.629 486 012 119 595 956 694 731 980 8;
  • 67) 0.629 486 012 119 595 956 694 731 980 8 × 2 = 1 + 0.258 972 024 239 191 913 389 463 961 6;
  • 68) 0.258 972 024 239 191 913 389 463 961 6 × 2 = 0 + 0.517 944 048 478 383 826 778 927 923 2;
  • 69) 0.517 944 048 478 383 826 778 927 923 2 × 2 = 1 + 0.035 888 096 956 767 653 557 855 846 4;
  • 70) 0.035 888 096 956 767 653 557 855 846 4 × 2 = 0 + 0.071 776 193 913 535 307 115 711 692 8;
  • 71) 0.071 776 193 913 535 307 115 711 692 8 × 2 = 0 + 0.143 552 387 827 070 614 231 423 385 6;
  • 72) 0.143 552 387 827 070 614 231 423 385 6 × 2 = 0 + 0.287 104 775 654 141 228 462 846 771 2;
  • 73) 0.287 104 775 654 141 228 462 846 771 2 × 2 = 0 + 0.574 209 551 308 282 456 925 693 542 4;
  • 74) 0.574 209 551 308 282 456 925 693 542 4 × 2 = 1 + 0.148 419 102 616 564 913 851 387 084 8;
  • 75) 0.148 419 102 616 564 913 851 387 084 8 × 2 = 0 + 0.296 838 205 233 129 827 702 774 169 6;
  • 76) 0.296 838 205 233 129 827 702 774 169 6 × 2 = 0 + 0.593 676 410 466 259 655 405 548 339 2;
  • 77) 0.593 676 410 466 259 655 405 548 339 2 × 2 = 1 + 0.187 352 820 932 519 310 811 096 678 4;
  • 78) 0.187 352 820 932 519 310 811 096 678 4 × 2 = 0 + 0.374 705 641 865 038 621 622 193 356 8;
  • 79) 0.374 705 641 865 038 621 622 193 356 8 × 2 = 0 + 0.749 411 283 730 077 243 244 386 713 6;
  • 80) 0.749 411 283 730 077 243 244 386 713 6 × 2 = 1 + 0.498 822 567 460 154 486 488 773 427 2;
  • 81) 0.498 822 567 460 154 486 488 773 427 2 × 2 = 0 + 0.997 645 134 920 308 972 977 546 854 4;
  • 82) 0.997 645 134 920 308 972 977 546 854 4 × 2 = 1 + 0.995 290 269 840 617 945 955 093 708 8;
  • 83) 0.995 290 269 840 617 945 955 093 708 8 × 2 = 1 + 0.990 580 539 681 235 891 910 187 417 6;
  • 84) 0.990 580 539 681 235 891 910 187 417 6 × 2 = 1 + 0.981 161 079 362 471 783 820 374 835 2;
  • 85) 0.981 161 079 362 471 783 820 374 835 2 × 2 = 1 + 0.962 322 158 724 943 567 640 749 670 4;
  • 86) 0.962 322 158 724 943 567 640 749 670 4 × 2 = 1 + 0.924 644 317 449 887 135 281 499 340 8;
  • 87) 0.924 644 317 449 887 135 281 499 340 8 × 2 = 1 + 0.849 288 634 899 774 270 562 998 681 6;
  • 88) 0.849 288 634 899 774 270 562 998 681 6 × 2 = 1 + 0.698 577 269 799 548 541 125 997 363 2;
  • 89) 0.698 577 269 799 548 541 125 997 363 2 × 2 = 1 + 0.397 154 539 599 097 082 251 994 726 4;
  • 90) 0.397 154 539 599 097 082 251 994 726 4 × 2 = 0 + 0.794 309 079 198 194 164 503 989 452 8;
  • 91) 0.794 309 079 198 194 164 503 989 452 8 × 2 = 1 + 0.588 618 158 396 388 329 007 978 905 6;
  • 92) 0.588 618 158 396 388 329 007 978 905 6 × 2 = 1 + 0.177 236 316 792 776 658 015 957 811 2;
  • 93) 0.177 236 316 792 776 658 015 957 811 2 × 2 = 0 + 0.354 472 633 585 553 316 031 915 622 4;
  • 94) 0.354 472 633 585 553 316 031 915 622 4 × 2 = 0 + 0.708 945 267 171 106 632 063 831 244 8;
  • 95) 0.708 945 267 171 106 632 063 831 244 8 × 2 = 1 + 0.417 890 534 342 213 264 127 662 489 6;
  • 96) 0.417 890 534 342 213 264 127 662 489 6 × 2 = 0 + 0.835 781 068 684 426 528 255 324 979 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:


0.000 000 000 000 100 000 000 000 029 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010(2)

5. Positive number before normalization:

0.000 000 000 000 100 000 000 000 029 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 44 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 100 000 000 000 029 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010(2) × 20 =

1.1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010(2) × 2-44


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): -44

Mantissa (not normalized):
1.1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-44 + 2(11-1) - 1 =

(-44 + 1 023)(10) =

979(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;
  • 979 ÷ 2 = 489 + 1;
  • 489 ÷ 2 = 244 + 1;
  • 244 ÷ 2 = 122 + 0;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

979(10) =

011 1101 0011(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. 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010 =

1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010


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 1101 0011

Mantissa (52 bits) =
1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010


Decimal number 0.000 000 000 000 100 000 000 000 029 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 0011 - 1100 0010 0101 1100 0010 0110 1000 0100 1001 0111 1111 1011 0010

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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