0.029 999 999 999 999 998 889 776 975 374 843 459 576 462 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.029 999 999 999 999 998 889 776 975 374 843 459 576 462(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
0.029 999 999 999 999 998 889 776 975 374 843 459 576 462(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: 0.029 999 999 999 999 998 889 776 975 374 843 459 576 462.

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) 0.029 999 999 999 999 998 889 776 975 374 843 459 576 462 × 2 = 0 + 0.059 999 999 999 999 997 779 553 950 749 686 919 152 924;
  • 2) 0.059 999 999 999 999 997 779 553 950 749 686 919 152 924 × 2 = 0 + 0.119 999 999 999 999 995 559 107 901 499 373 838 305 848;
  • 3) 0.119 999 999 999 999 995 559 107 901 499 373 838 305 848 × 2 = 0 + 0.239 999 999 999 999 991 118 215 802 998 747 676 611 696;
  • 4) 0.239 999 999 999 999 991 118 215 802 998 747 676 611 696 × 2 = 0 + 0.479 999 999 999 999 982 236 431 605 997 495 353 223 392;
  • 5) 0.479 999 999 999 999 982 236 431 605 997 495 353 223 392 × 2 = 0 + 0.959 999 999 999 999 964 472 863 211 994 990 706 446 784;
  • 6) 0.959 999 999 999 999 964 472 863 211 994 990 706 446 784 × 2 = 1 + 0.919 999 999 999 999 928 945 726 423 989 981 412 893 568;
  • 7) 0.919 999 999 999 999 928 945 726 423 989 981 412 893 568 × 2 = 1 + 0.839 999 999 999 999 857 891 452 847 979 962 825 787 136;
  • 8) 0.839 999 999 999 999 857 891 452 847 979 962 825 787 136 × 2 = 1 + 0.679 999 999 999 999 715 782 905 695 959 925 651 574 272;
  • 9) 0.679 999 999 999 999 715 782 905 695 959 925 651 574 272 × 2 = 1 + 0.359 999 999 999 999 431 565 811 391 919 851 303 148 544;
  • 10) 0.359 999 999 999 999 431 565 811 391 919 851 303 148 544 × 2 = 0 + 0.719 999 999 999 998 863 131 622 783 839 702 606 297 088;
  • 11) 0.719 999 999 999 998 863 131 622 783 839 702 606 297 088 × 2 = 1 + 0.439 999 999 999 997 726 263 245 567 679 405 212 594 176;
  • 12) 0.439 999 999 999 997 726 263 245 567 679 405 212 594 176 × 2 = 0 + 0.879 999 999 999 995 452 526 491 135 358 810 425 188 352;
  • 13) 0.879 999 999 999 995 452 526 491 135 358 810 425 188 352 × 2 = 1 + 0.759 999 999 999 990 905 052 982 270 717 620 850 376 704;
  • 14) 0.759 999 999 999 990 905 052 982 270 717 620 850 376 704 × 2 = 1 + 0.519 999 999 999 981 810 105 964 541 435 241 700 753 408;
  • 15) 0.519 999 999 999 981 810 105 964 541 435 241 700 753 408 × 2 = 1 + 0.039 999 999 999 963 620 211 929 082 870 483 401 506 816;
  • 16) 0.039 999 999 999 963 620 211 929 082 870 483 401 506 816 × 2 = 0 + 0.079 999 999 999 927 240 423 858 165 740 966 803 013 632;
  • 17) 0.079 999 999 999 927 240 423 858 165 740 966 803 013 632 × 2 = 0 + 0.159 999 999 999 854 480 847 716 331 481 933 606 027 264;
  • 18) 0.159 999 999 999 854 480 847 716 331 481 933 606 027 264 × 2 = 0 + 0.319 999 999 999 708 961 695 432 662 963 867 212 054 528;
  • 19) 0.319 999 999 999 708 961 695 432 662 963 867 212 054 528 × 2 = 0 + 0.639 999 999 999 417 923 390 865 325 927 734 424 109 056;
  • 20) 0.639 999 999 999 417 923 390 865 325 927 734 424 109 056 × 2 = 1 + 0.279 999 999 998 835 846 781 730 651 855 468 848 218 112;
  • 21) 0.279 999 999 998 835 846 781 730 651 855 468 848 218 112 × 2 = 0 + 0.559 999 999 997 671 693 563 461 303 710 937 696 436 224;
  • 22) 0.559 999 999 997 671 693 563 461 303 710 937 696 436 224 × 2 = 1 + 0.119 999 999 995 343 387 126 922 607 421 875 392 872 448;
  • 23) 0.119 999 999 995 343 387 126 922 607 421 875 392 872 448 × 2 = 0 + 0.239 999 999 990 686 774 253 845 214 843 750 785 744 896;
  • 24) 0.239 999 999 990 686 774 253 845 214 843 750 785 744 896 × 2 = 0 + 0.479 999 999 981 373 548 507 690 429 687 501 571 489 792;
  • 25) 0.479 999 999 981 373 548 507 690 429 687 501 571 489 792 × 2 = 0 + 0.959 999 999 962 747 097 015 380 859 375 003 142 979 584;
  • 26) 0.959 999 999 962 747 097 015 380 859 375 003 142 979 584 × 2 = 1 + 0.919 999 999 925 494 194 030 761 718 750 006 285 959 168;
  • 27) 0.919 999 999 925 494 194 030 761 718 750 006 285 959 168 × 2 = 1 + 0.839 999 999 850 988 388 061 523 437 500 012 571 918 336;
  • 28) 0.839 999 999 850 988 388 061 523 437 500 012 571 918 336 × 2 = 1 + 0.679 999 999 701 976 776 123 046 875 000 025 143 836 672;
  • 29) 0.679 999 999 701 976 776 123 046 875 000 025 143 836 672 × 2 = 1 + 0.359 999 999 403 953 552 246 093 750 000 050 287 673 344;
  • 30) 0.359 999 999 403 953 552 246 093 750 000 050 287 673 344 × 2 = 0 + 0.719 999 998 807 907 104 492 187 500 000 100 575 346 688;
  • 31) 0.719 999 998 807 907 104 492 187 500 000 100 575 346 688 × 2 = 1 + 0.439 999 997 615 814 208 984 375 000 000 201 150 693 376;
  • 32) 0.439 999 997 615 814 208 984 375 000 000 201 150 693 376 × 2 = 0 + 0.879 999 995 231 628 417 968 750 000 000 402 301 386 752;
  • 33) 0.879 999 995 231 628 417 968 750 000 000 402 301 386 752 × 2 = 1 + 0.759 999 990 463 256 835 937 500 000 000 804 602 773 504;
  • 34) 0.759 999 990 463 256 835 937 500 000 000 804 602 773 504 × 2 = 1 + 0.519 999 980 926 513 671 875 000 000 001 609 205 547 008;
  • 35) 0.519 999 980 926 513 671 875 000 000 001 609 205 547 008 × 2 = 1 + 0.039 999 961 853 027 343 750 000 000 003 218 411 094 016;
  • 36) 0.039 999 961 853 027 343 750 000 000 003 218 411 094 016 × 2 = 0 + 0.079 999 923 706 054 687 500 000 000 006 436 822 188 032;
  • 37) 0.079 999 923 706 054 687 500 000 000 006 436 822 188 032 × 2 = 0 + 0.159 999 847 412 109 375 000 000 000 012 873 644 376 064;
  • 38) 0.159 999 847 412 109 375 000 000 000 012 873 644 376 064 × 2 = 0 + 0.319 999 694 824 218 750 000 000 000 025 747 288 752 128;
  • 39) 0.319 999 694 824 218 750 000 000 000 025 747 288 752 128 × 2 = 0 + 0.639 999 389 648 437 500 000 000 000 051 494 577 504 256;
  • 40) 0.639 999 389 648 437 500 000 000 000 051 494 577 504 256 × 2 = 1 + 0.279 998 779 296 875 000 000 000 000 102 989 155 008 512;
  • 41) 0.279 998 779 296 875 000 000 000 000 102 989 155 008 512 × 2 = 0 + 0.559 997 558 593 750 000 000 000 000 205 978 310 017 024;
  • 42) 0.559 997 558 593 750 000 000 000 000 205 978 310 017 024 × 2 = 1 + 0.119 995 117 187 500 000 000 000 000 411 956 620 034 048;
  • 43) 0.119 995 117 187 500 000 000 000 000 411 956 620 034 048 × 2 = 0 + 0.239 990 234 375 000 000 000 000 000 823 913 240 068 096;
  • 44) 0.239 990 234 375 000 000 000 000 000 823 913 240 068 096 × 2 = 0 + 0.479 980 468 750 000 000 000 000 001 647 826 480 136 192;
  • 45) 0.479 980 468 750 000 000 000 000 001 647 826 480 136 192 × 2 = 0 + 0.959 960 937 500 000 000 000 000 003 295 652 960 272 384;
  • 46) 0.959 960 937 500 000 000 000 000 003 295 652 960 272 384 × 2 = 1 + 0.919 921 875 000 000 000 000 000 006 591 305 920 544 768;
  • 47) 0.919 921 875 000 000 000 000 000 006 591 305 920 544 768 × 2 = 1 + 0.839 843 750 000 000 000 000 000 013 182 611 841 089 536;
  • 48) 0.839 843 750 000 000 000 000 000 013 182 611 841 089 536 × 2 = 1 + 0.679 687 500 000 000 000 000 000 026 365 223 682 179 072;
  • 49) 0.679 687 500 000 000 000 000 000 026 365 223 682 179 072 × 2 = 1 + 0.359 375 000 000 000 000 000 000 052 730 447 364 358 144;
  • 50) 0.359 375 000 000 000 000 000 000 052 730 447 364 358 144 × 2 = 0 + 0.718 750 000 000 000 000 000 000 105 460 894 728 716 288;
  • 51) 0.718 750 000 000 000 000 000 000 105 460 894 728 716 288 × 2 = 1 + 0.437 500 000 000 000 000 000 000 210 921 789 457 432 576;
  • 52) 0.437 500 000 000 000 000 000 000 210 921 789 457 432 576 × 2 = 0 + 0.875 000 000 000 000 000 000 000 421 843 578 914 865 152;
  • 53) 0.875 000 000 000 000 000 000 000 421 843 578 914 865 152 × 2 = 1 + 0.750 000 000 000 000 000 000 000 843 687 157 829 730 304;
  • 54) 0.750 000 000 000 000 000 000 000 843 687 157 829 730 304 × 2 = 1 + 0.500 000 000 000 000 000 000 001 687 374 315 659 460 608;
  • 55) 0.500 000 000 000 000 000 000 001 687 374 315 659 460 608 × 2 = 1 + 0.000 000 000 000 000 000 000 003 374 748 631 318 921 216;
  • 56) 0.000 000 000 000 000 000 000 003 374 748 631 318 921 216 × 2 = 0 + 0.000 000 000 000 000 000 000 006 749 497 262 637 842 432;
  • 57) 0.000 000 000 000 000 000 000 006 749 497 262 637 842 432 × 2 = 0 + 0.000 000 000 000 000 000 000 013 498 994 525 275 684 864;
  • 58) 0.000 000 000 000 000 000 000 013 498 994 525 275 684 864 × 2 = 0 + 0.000 000 000 000 000 000 000 026 997 989 050 551 369 728;

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:


0.029 999 999 999 999 998 889 776 975 374 843 459 576 462(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2)

5. Positive number before normalization:

0.029 999 999 999 999 998 889 776 975 374 843 459 576 462(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.029 999 999 999 999 998 889 776 975 374 843 459 576 462(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2) × 20 =

1.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000(2) × 2-6


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): -6

Mantissa (not normalized):
1.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 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) =

1017(10) =

011 1111 1001(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. 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 =

1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


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 1001

Mantissa (52 bits) =
1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


Decimal number 0.029 999 999 999 999 998 889 776 975 374 843 459 576 462 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1001 - 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000

» Convert decimal 0.029 999 999 999 999 998 889 776 975 374 843 459 576 465 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