0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079(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
0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079(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: 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079.

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) 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079 × 2 = 0 + 0.200 000 000 000 000 011 102 230 246 251 565 404 236 316 680 908 158;
  • 2) 0.200 000 000 000 000 011 102 230 246 251 565 404 236 316 680 908 158 × 2 = 0 + 0.400 000 000 000 000 022 204 460 492 503 130 808 472 633 361 816 316;
  • 3) 0.400 000 000 000 000 022 204 460 492 503 130 808 472 633 361 816 316 × 2 = 0 + 0.800 000 000 000 000 044 408 920 985 006 261 616 945 266 723 632 632;
  • 4) 0.800 000 000 000 000 044 408 920 985 006 261 616 945 266 723 632 632 × 2 = 1 + 0.600 000 000 000 000 088 817 841 970 012 523 233 890 533 447 265 264;
  • 5) 0.600 000 000 000 000 088 817 841 970 012 523 233 890 533 447 265 264 × 2 = 1 + 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 894 530 528;
  • 6) 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 894 530 528 × 2 = 0 + 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 789 061 056;
  • 7) 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 789 061 056 × 2 = 0 + 0.800 000 000 000 000 710 542 735 760 100 185 871 124 267 578 122 112;
  • 8) 0.800 000 000 000 000 710 542 735 760 100 185 871 124 267 578 122 112 × 2 = 1 + 0.600 000 000 000 001 421 085 471 520 200 371 742 248 535 156 244 224;
  • 9) 0.600 000 000 000 001 421 085 471 520 200 371 742 248 535 156 244 224 × 2 = 1 + 0.200 000 000 000 002 842 170 943 040 400 743 484 497 070 312 488 448;
  • 10) 0.200 000 000 000 002 842 170 943 040 400 743 484 497 070 312 488 448 × 2 = 0 + 0.400 000 000 000 005 684 341 886 080 801 486 968 994 140 624 976 896;
  • 11) 0.400 000 000 000 005 684 341 886 080 801 486 968 994 140 624 976 896 × 2 = 0 + 0.800 000 000 000 011 368 683 772 161 602 973 937 988 281 249 953 792;
  • 12) 0.800 000 000 000 011 368 683 772 161 602 973 937 988 281 249 953 792 × 2 = 1 + 0.600 000 000 000 022 737 367 544 323 205 947 875 976 562 499 907 584;
  • 13) 0.600 000 000 000 022 737 367 544 323 205 947 875 976 562 499 907 584 × 2 = 1 + 0.200 000 000 000 045 474 735 088 646 411 895 751 953 124 999 815 168;
  • 14) 0.200 000 000 000 045 474 735 088 646 411 895 751 953 124 999 815 168 × 2 = 0 + 0.400 000 000 000 090 949 470 177 292 823 791 503 906 249 999 630 336;
  • 15) 0.400 000 000 000 090 949 470 177 292 823 791 503 906 249 999 630 336 × 2 = 0 + 0.800 000 000 000 181 898 940 354 585 647 583 007 812 499 999 260 672;
  • 16) 0.800 000 000 000 181 898 940 354 585 647 583 007 812 499 999 260 672 × 2 = 1 + 0.600 000 000 000 363 797 880 709 171 295 166 015 624 999 998 521 344;
  • 17) 0.600 000 000 000 363 797 880 709 171 295 166 015 624 999 998 521 344 × 2 = 1 + 0.200 000 000 000 727 595 761 418 342 590 332 031 249 999 997 042 688;
  • 18) 0.200 000 000 000 727 595 761 418 342 590 332 031 249 999 997 042 688 × 2 = 0 + 0.400 000 000 001 455 191 522 836 685 180 664 062 499 999 994 085 376;
  • 19) 0.400 000 000 001 455 191 522 836 685 180 664 062 499 999 994 085 376 × 2 = 0 + 0.800 000 000 002 910 383 045 673 370 361 328 124 999 999 988 170 752;
  • 20) 0.800 000 000 002 910 383 045 673 370 361 328 124 999 999 988 170 752 × 2 = 1 + 0.600 000 000 005 820 766 091 346 740 722 656 249 999 999 976 341 504;
  • 21) 0.600 000 000 005 820 766 091 346 740 722 656 249 999 999 976 341 504 × 2 = 1 + 0.200 000 000 011 641 532 182 693 481 445 312 499 999 999 952 683 008;
  • 22) 0.200 000 000 011 641 532 182 693 481 445 312 499 999 999 952 683 008 × 2 = 0 + 0.400 000 000 023 283 064 365 386 962 890 624 999 999 999 905 366 016;
  • 23) 0.400 000 000 023 283 064 365 386 962 890 624 999 999 999 905 366 016 × 2 = 0 + 0.800 000 000 046 566 128 730 773 925 781 249 999 999 999 810 732 032;
  • 24) 0.800 000 000 046 566 128 730 773 925 781 249 999 999 999 810 732 032 × 2 = 1 + 0.600 000 000 093 132 257 461 547 851 562 499 999 999 999 621 464 064;
  • 25) 0.600 000 000 093 132 257 461 547 851 562 499 999 999 999 621 464 064 × 2 = 1 + 0.200 000 000 186 264 514 923 095 703 124 999 999 999 999 242 928 128;
  • 26) 0.200 000 000 186 264 514 923 095 703 124 999 999 999 999 242 928 128 × 2 = 0 + 0.400 000 000 372 529 029 846 191 406 249 999 999 999 998 485 856 256;
  • 27) 0.400 000 000 372 529 029 846 191 406 249 999 999 999 998 485 856 256 × 2 = 0 + 0.800 000 000 745 058 059 692 382 812 499 999 999 999 996 971 712 512;
  • 28) 0.800 000 000 745 058 059 692 382 812 499 999 999 999 996 971 712 512 × 2 = 1 + 0.600 000 001 490 116 119 384 765 624 999 999 999 999 993 943 425 024;
  • 29) 0.600 000 001 490 116 119 384 765 624 999 999 999 999 993 943 425 024 × 2 = 1 + 0.200 000 002 980 232 238 769 531 249 999 999 999 999 987 886 850 048;
  • 30) 0.200 000 002 980 232 238 769 531 249 999 999 999 999 987 886 850 048 × 2 = 0 + 0.400 000 005 960 464 477 539 062 499 999 999 999 999 975 773 700 096;
  • 31) 0.400 000 005 960 464 477 539 062 499 999 999 999 999 975 773 700 096 × 2 = 0 + 0.800 000 011 920 928 955 078 124 999 999 999 999 999 951 547 400 192;
  • 32) 0.800 000 011 920 928 955 078 124 999 999 999 999 999 951 547 400 192 × 2 = 1 + 0.600 000 023 841 857 910 156 249 999 999 999 999 999 903 094 800 384;
  • 33) 0.600 000 023 841 857 910 156 249 999 999 999 999 999 903 094 800 384 × 2 = 1 + 0.200 000 047 683 715 820 312 499 999 999 999 999 999 806 189 600 768;
  • 34) 0.200 000 047 683 715 820 312 499 999 999 999 999 999 806 189 600 768 × 2 = 0 + 0.400 000 095 367 431 640 624 999 999 999 999 999 999 612 379 201 536;
  • 35) 0.400 000 095 367 431 640 624 999 999 999 999 999 999 612 379 201 536 × 2 = 0 + 0.800 000 190 734 863 281 249 999 999 999 999 999 999 224 758 403 072;
  • 36) 0.800 000 190 734 863 281 249 999 999 999 999 999 999 224 758 403 072 × 2 = 1 + 0.600 000 381 469 726 562 499 999 999 999 999 999 998 449 516 806 144;
  • 37) 0.600 000 381 469 726 562 499 999 999 999 999 999 998 449 516 806 144 × 2 = 1 + 0.200 000 762 939 453 124 999 999 999 999 999 999 996 899 033 612 288;
  • 38) 0.200 000 762 939 453 124 999 999 999 999 999 999 996 899 033 612 288 × 2 = 0 + 0.400 001 525 878 906 249 999 999 999 999 999 999 993 798 067 224 576;
  • 39) 0.400 001 525 878 906 249 999 999 999 999 999 999 993 798 067 224 576 × 2 = 0 + 0.800 003 051 757 812 499 999 999 999 999 999 999 987 596 134 449 152;
  • 40) 0.800 003 051 757 812 499 999 999 999 999 999 999 987 596 134 449 152 × 2 = 1 + 0.600 006 103 515 624 999 999 999 999 999 999 999 975 192 268 898 304;
  • 41) 0.600 006 103 515 624 999 999 999 999 999 999 999 975 192 268 898 304 × 2 = 1 + 0.200 012 207 031 249 999 999 999 999 999 999 999 950 384 537 796 608;
  • 42) 0.200 012 207 031 249 999 999 999 999 999 999 999 950 384 537 796 608 × 2 = 0 + 0.400 024 414 062 499 999 999 999 999 999 999 999 900 769 075 593 216;
  • 43) 0.400 024 414 062 499 999 999 999 999 999 999 999 900 769 075 593 216 × 2 = 0 + 0.800 048 828 124 999 999 999 999 999 999 999 999 801 538 151 186 432;
  • 44) 0.800 048 828 124 999 999 999 999 999 999 999 999 801 538 151 186 432 × 2 = 1 + 0.600 097 656 249 999 999 999 999 999 999 999 999 603 076 302 372 864;
  • 45) 0.600 097 656 249 999 999 999 999 999 999 999 999 603 076 302 372 864 × 2 = 1 + 0.200 195 312 499 999 999 999 999 999 999 999 999 206 152 604 745 728;
  • 46) 0.200 195 312 499 999 999 999 999 999 999 999 999 206 152 604 745 728 × 2 = 0 + 0.400 390 624 999 999 999 999 999 999 999 999 998 412 305 209 491 456;
  • 47) 0.400 390 624 999 999 999 999 999 999 999 999 998 412 305 209 491 456 × 2 = 0 + 0.800 781 249 999 999 999 999 999 999 999 999 996 824 610 418 982 912;
  • 48) 0.800 781 249 999 999 999 999 999 999 999 999 996 824 610 418 982 912 × 2 = 1 + 0.601 562 499 999 999 999 999 999 999 999 999 993 649 220 837 965 824;
  • 49) 0.601 562 499 999 999 999 999 999 999 999 999 993 649 220 837 965 824 × 2 = 1 + 0.203 124 999 999 999 999 999 999 999 999 999 987 298 441 675 931 648;
  • 50) 0.203 124 999 999 999 999 999 999 999 999 999 987 298 441 675 931 648 × 2 = 0 + 0.406 249 999 999 999 999 999 999 999 999 999 974 596 883 351 863 296;
  • 51) 0.406 249 999 999 999 999 999 999 999 999 999 974 596 883 351 863 296 × 2 = 0 + 0.812 499 999 999 999 999 999 999 999 999 999 949 193 766 703 726 592;
  • 52) 0.812 499 999 999 999 999 999 999 999 999 999 949 193 766 703 726 592 × 2 = 1 + 0.624 999 999 999 999 999 999 999 999 999 999 898 387 533 407 453 184;
  • 53) 0.624 999 999 999 999 999 999 999 999 999 999 898 387 533 407 453 184 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 796 775 066 814 906 368;
  • 54) 0.249 999 999 999 999 999 999 999 999 999 999 796 775 066 814 906 368 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 593 550 133 629 812 736;
  • 55) 0.499 999 999 999 999 999 999 999 999 999 999 593 550 133 629 812 736 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 187 100 267 259 625 472;
  • 56) 0.999 999 999 999 999 999 999 999 999 999 999 187 100 267 259 625 472 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 998 374 200 534 519 250 944;

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:


0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079(10) =

0.0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001(2)

5. Positive number before normalization:

0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079(10) =

0.0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the right, so that only one non zero digit remains to the left of it:


0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079(10) =

0.0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001(2) =

0.0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001(2) × 20 =

1.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001(2) × 2-4


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): -4

Mantissa (not normalized):
1.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-4 + 2(11-1) - 1 =

(-4 + 1 023)(10) =

1 019(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;
  • 1 019 ÷ 2 = 509 + 1;
  • 509 ÷ 2 = 254 + 1;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 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) =

1019(10) =

011 1111 1011(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. 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 =

1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001


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

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

Exponent (11 bits) =
011 1111 1011

Mantissa (52 bits) =
1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001


Decimal number 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 079 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1011 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001

» Convert decimal 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 137 to 64 bit double precision IEEE 754 binary floating point representation standard

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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