0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816(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
0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816(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: 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816.

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) 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816 × 2 = 0 + 0.080 000 000 000 000 001 665 334 536 937 734 810 635 447 501 632;
  • 2) 0.080 000 000 000 000 001 665 334 536 937 734 810 635 447 501 632 × 2 = 0 + 0.160 000 000 000 000 003 330 669 073 875 469 621 270 895 003 264;
  • 3) 0.160 000 000 000 000 003 330 669 073 875 469 621 270 895 003 264 × 2 = 0 + 0.320 000 000 000 000 006 661 338 147 750 939 242 541 790 006 528;
  • 4) 0.320 000 000 000 000 006 661 338 147 750 939 242 541 790 006 528 × 2 = 0 + 0.640 000 000 000 000 013 322 676 295 501 878 485 083 580 013 056;
  • 5) 0.640 000 000 000 000 013 322 676 295 501 878 485 083 580 013 056 × 2 = 1 + 0.280 000 000 000 000 026 645 352 591 003 756 970 167 160 026 112;
  • 6) 0.280 000 000 000 000 026 645 352 591 003 756 970 167 160 026 112 × 2 = 0 + 0.560 000 000 000 000 053 290 705 182 007 513 940 334 320 052 224;
  • 7) 0.560 000 000 000 000 053 290 705 182 007 513 940 334 320 052 224 × 2 = 1 + 0.120 000 000 000 000 106 581 410 364 015 027 880 668 640 104 448;
  • 8) 0.120 000 000 000 000 106 581 410 364 015 027 880 668 640 104 448 × 2 = 0 + 0.240 000 000 000 000 213 162 820 728 030 055 761 337 280 208 896;
  • 9) 0.240 000 000 000 000 213 162 820 728 030 055 761 337 280 208 896 × 2 = 0 + 0.480 000 000 000 000 426 325 641 456 060 111 522 674 560 417 792;
  • 10) 0.480 000 000 000 000 426 325 641 456 060 111 522 674 560 417 792 × 2 = 0 + 0.960 000 000 000 000 852 651 282 912 120 223 045 349 120 835 584;
  • 11) 0.960 000 000 000 000 852 651 282 912 120 223 045 349 120 835 584 × 2 = 1 + 0.920 000 000 000 001 705 302 565 824 240 446 090 698 241 671 168;
  • 12) 0.920 000 000 000 001 705 302 565 824 240 446 090 698 241 671 168 × 2 = 1 + 0.840 000 000 000 003 410 605 131 648 480 892 181 396 483 342 336;
  • 13) 0.840 000 000 000 003 410 605 131 648 480 892 181 396 483 342 336 × 2 = 1 + 0.680 000 000 000 006 821 210 263 296 961 784 362 792 966 684 672;
  • 14) 0.680 000 000 000 006 821 210 263 296 961 784 362 792 966 684 672 × 2 = 1 + 0.360 000 000 000 013 642 420 526 593 923 568 725 585 933 369 344;
  • 15) 0.360 000 000 000 013 642 420 526 593 923 568 725 585 933 369 344 × 2 = 0 + 0.720 000 000 000 027 284 841 053 187 847 137 451 171 866 738 688;
  • 16) 0.720 000 000 000 027 284 841 053 187 847 137 451 171 866 738 688 × 2 = 1 + 0.440 000 000 000 054 569 682 106 375 694 274 902 343 733 477 376;
  • 17) 0.440 000 000 000 054 569 682 106 375 694 274 902 343 733 477 376 × 2 = 0 + 0.880 000 000 000 109 139 364 212 751 388 549 804 687 466 954 752;
  • 18) 0.880 000 000 000 109 139 364 212 751 388 549 804 687 466 954 752 × 2 = 1 + 0.760 000 000 000 218 278 728 425 502 777 099 609 374 933 909 504;
  • 19) 0.760 000 000 000 218 278 728 425 502 777 099 609 374 933 909 504 × 2 = 1 + 0.520 000 000 000 436 557 456 851 005 554 199 218 749 867 819 008;
  • 20) 0.520 000 000 000 436 557 456 851 005 554 199 218 749 867 819 008 × 2 = 1 + 0.040 000 000 000 873 114 913 702 011 108 398 437 499 735 638 016;
  • 21) 0.040 000 000 000 873 114 913 702 011 108 398 437 499 735 638 016 × 2 = 0 + 0.080 000 000 001 746 229 827 404 022 216 796 874 999 471 276 032;
  • 22) 0.080 000 000 001 746 229 827 404 022 216 796 874 999 471 276 032 × 2 = 0 + 0.160 000 000 003 492 459 654 808 044 433 593 749 998 942 552 064;
  • 23) 0.160 000 000 003 492 459 654 808 044 433 593 749 998 942 552 064 × 2 = 0 + 0.320 000 000 006 984 919 309 616 088 867 187 499 997 885 104 128;
  • 24) 0.320 000 000 006 984 919 309 616 088 867 187 499 997 885 104 128 × 2 = 0 + 0.640 000 000 013 969 838 619 232 177 734 374 999 995 770 208 256;
  • 25) 0.640 000 000 013 969 838 619 232 177 734 374 999 995 770 208 256 × 2 = 1 + 0.280 000 000 027 939 677 238 464 355 468 749 999 991 540 416 512;
  • 26) 0.280 000 000 027 939 677 238 464 355 468 749 999 991 540 416 512 × 2 = 0 + 0.560 000 000 055 879 354 476 928 710 937 499 999 983 080 833 024;
  • 27) 0.560 000 000 055 879 354 476 928 710 937 499 999 983 080 833 024 × 2 = 1 + 0.120 000 000 111 758 708 953 857 421 874 999 999 966 161 666 048;
  • 28) 0.120 000 000 111 758 708 953 857 421 874 999 999 966 161 666 048 × 2 = 0 + 0.240 000 000 223 517 417 907 714 843 749 999 999 932 323 332 096;
  • 29) 0.240 000 000 223 517 417 907 714 843 749 999 999 932 323 332 096 × 2 = 0 + 0.480 000 000 447 034 835 815 429 687 499 999 999 864 646 664 192;
  • 30) 0.480 000 000 447 034 835 815 429 687 499 999 999 864 646 664 192 × 2 = 0 + 0.960 000 000 894 069 671 630 859 374 999 999 999 729 293 328 384;
  • 31) 0.960 000 000 894 069 671 630 859 374 999 999 999 729 293 328 384 × 2 = 1 + 0.920 000 001 788 139 343 261 718 749 999 999 999 458 586 656 768;
  • 32) 0.920 000 001 788 139 343 261 718 749 999 999 999 458 586 656 768 × 2 = 1 + 0.840 000 003 576 278 686 523 437 499 999 999 998 917 173 313 536;
  • 33) 0.840 000 003 576 278 686 523 437 499 999 999 998 917 173 313 536 × 2 = 1 + 0.680 000 007 152 557 373 046 874 999 999 999 997 834 346 627 072;
  • 34) 0.680 000 007 152 557 373 046 874 999 999 999 997 834 346 627 072 × 2 = 1 + 0.360 000 014 305 114 746 093 749 999 999 999 995 668 693 254 144;
  • 35) 0.360 000 014 305 114 746 093 749 999 999 999 995 668 693 254 144 × 2 = 0 + 0.720 000 028 610 229 492 187 499 999 999 999 991 337 386 508 288;
  • 36) 0.720 000 028 610 229 492 187 499 999 999 999 991 337 386 508 288 × 2 = 1 + 0.440 000 057 220 458 984 374 999 999 999 999 982 674 773 016 576;
  • 37) 0.440 000 057 220 458 984 374 999 999 999 999 982 674 773 016 576 × 2 = 0 + 0.880 000 114 440 917 968 749 999 999 999 999 965 349 546 033 152;
  • 38) 0.880 000 114 440 917 968 749 999 999 999 999 965 349 546 033 152 × 2 = 1 + 0.760 000 228 881 835 937 499 999 999 999 999 930 699 092 066 304;
  • 39) 0.760 000 228 881 835 937 499 999 999 999 999 930 699 092 066 304 × 2 = 1 + 0.520 000 457 763 671 874 999 999 999 999 999 861 398 184 132 608;
  • 40) 0.520 000 457 763 671 874 999 999 999 999 999 861 398 184 132 608 × 2 = 1 + 0.040 000 915 527 343 749 999 999 999 999 999 722 796 368 265 216;
  • 41) 0.040 000 915 527 343 749 999 999 999 999 999 722 796 368 265 216 × 2 = 0 + 0.080 001 831 054 687 499 999 999 999 999 999 445 592 736 530 432;
  • 42) 0.080 001 831 054 687 499 999 999 999 999 999 445 592 736 530 432 × 2 = 0 + 0.160 003 662 109 374 999 999 999 999 999 998 891 185 473 060 864;
  • 43) 0.160 003 662 109 374 999 999 999 999 999 998 891 185 473 060 864 × 2 = 0 + 0.320 007 324 218 749 999 999 999 999 999 997 782 370 946 121 728;
  • 44) 0.320 007 324 218 749 999 999 999 999 999 997 782 370 946 121 728 × 2 = 0 + 0.640 014 648 437 499 999 999 999 999 999 995 564 741 892 243 456;
  • 45) 0.640 014 648 437 499 999 999 999 999 999 995 564 741 892 243 456 × 2 = 1 + 0.280 029 296 874 999 999 999 999 999 999 991 129 483 784 486 912;
  • 46) 0.280 029 296 874 999 999 999 999 999 999 991 129 483 784 486 912 × 2 = 0 + 0.560 058 593 749 999 999 999 999 999 999 982 258 967 568 973 824;
  • 47) 0.560 058 593 749 999 999 999 999 999 999 982 258 967 568 973 824 × 2 = 1 + 0.120 117 187 499 999 999 999 999 999 999 964 517 935 137 947 648;
  • 48) 0.120 117 187 499 999 999 999 999 999 999 964 517 935 137 947 648 × 2 = 0 + 0.240 234 374 999 999 999 999 999 999 999 929 035 870 275 895 296;
  • 49) 0.240 234 374 999 999 999 999 999 999 999 929 035 870 275 895 296 × 2 = 0 + 0.480 468 749 999 999 999 999 999 999 999 858 071 740 551 790 592;
  • 50) 0.480 468 749 999 999 999 999 999 999 999 858 071 740 551 790 592 × 2 = 0 + 0.960 937 499 999 999 999 999 999 999 999 716 143 481 103 581 184;
  • 51) 0.960 937 499 999 999 999 999 999 999 999 716 143 481 103 581 184 × 2 = 1 + 0.921 874 999 999 999 999 999 999 999 999 432 286 962 207 162 368;
  • 52) 0.921 874 999 999 999 999 999 999 999 999 432 286 962 207 162 368 × 2 = 1 + 0.843 749 999 999 999 999 999 999 999 998 864 573 924 414 324 736;
  • 53) 0.843 749 999 999 999 999 999 999 999 998 864 573 924 414 324 736 × 2 = 1 + 0.687 499 999 999 999 999 999 999 999 997 729 147 848 828 649 472;
  • 54) 0.687 499 999 999 999 999 999 999 999 997 729 147 848 828 649 472 × 2 = 1 + 0.374 999 999 999 999 999 999 999 999 995 458 295 697 657 298 944;
  • 55) 0.374 999 999 999 999 999 999 999 999 995 458 295 697 657 298 944 × 2 = 0 + 0.749 999 999 999 999 999 999 999 999 990 916 591 395 314 597 888;
  • 56) 0.749 999 999 999 999 999 999 999 999 990 916 591 395 314 597 888 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 981 833 182 790 629 195 776;
  • 57) 0.499 999 999 999 999 999 999 999 999 981 833 182 790 629 195 776 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 963 666 365 581 258 391 552;

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:


0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2)

5. Positive number before normalization:

0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2) × 20 =

1.0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010(2) × 2-5


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): -5

Mantissa (not normalized):
1.0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 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) =

1018(10) =

011 1111 1010(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 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 =

0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


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 1010

Mantissa (52 bits) =
0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


Decimal number 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 750 816 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1010 - 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010

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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