1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2(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
1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2.

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) 0.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2 × 2 = 0 + 0.666 666 666 666 666 518 636 930 049 979 127 943 515 777 566 4;
  • 2) 0.666 666 666 666 666 518 636 930 049 979 127 943 515 777 566 4 × 2 = 1 + 0.333 333 333 333 333 037 273 860 099 958 255 887 031 555 132 8;
  • 3) 0.333 333 333 333 333 037 273 860 099 958 255 887 031 555 132 8 × 2 = 0 + 0.666 666 666 666 666 074 547 720 199 916 511 774 063 110 265 6;
  • 4) 0.666 666 666 666 666 074 547 720 199 916 511 774 063 110 265 6 × 2 = 1 + 0.333 333 333 333 332 149 095 440 399 833 023 548 126 220 531 2;
  • 5) 0.333 333 333 333 332 149 095 440 399 833 023 548 126 220 531 2 × 2 = 0 + 0.666 666 666 666 664 298 190 880 799 666 047 096 252 441 062 4;
  • 6) 0.666 666 666 666 664 298 190 880 799 666 047 096 252 441 062 4 × 2 = 1 + 0.333 333 333 333 328 596 381 761 599 332 094 192 504 882 124 8;
  • 7) 0.333 333 333 333 328 596 381 761 599 332 094 192 504 882 124 8 × 2 = 0 + 0.666 666 666 666 657 192 763 523 198 664 188 385 009 764 249 6;
  • 8) 0.666 666 666 666 657 192 763 523 198 664 188 385 009 764 249 6 × 2 = 1 + 0.333 333 333 333 314 385 527 046 397 328 376 770 019 528 499 2;
  • 9) 0.333 333 333 333 314 385 527 046 397 328 376 770 019 528 499 2 × 2 = 0 + 0.666 666 666 666 628 771 054 092 794 656 753 540 039 056 998 4;
  • 10) 0.666 666 666 666 628 771 054 092 794 656 753 540 039 056 998 4 × 2 = 1 + 0.333 333 333 333 257 542 108 185 589 313 507 080 078 113 996 8;
  • 11) 0.333 333 333 333 257 542 108 185 589 313 507 080 078 113 996 8 × 2 = 0 + 0.666 666 666 666 515 084 216 371 178 627 014 160 156 227 993 6;
  • 12) 0.666 666 666 666 515 084 216 371 178 627 014 160 156 227 993 6 × 2 = 1 + 0.333 333 333 333 030 168 432 742 357 254 028 320 312 455 987 2;
  • 13) 0.333 333 333 333 030 168 432 742 357 254 028 320 312 455 987 2 × 2 = 0 + 0.666 666 666 666 060 336 865 484 714 508 056 640 624 911 974 4;
  • 14) 0.666 666 666 666 060 336 865 484 714 508 056 640 624 911 974 4 × 2 = 1 + 0.333 333 333 332 120 673 730 969 429 016 113 281 249 823 948 8;
  • 15) 0.333 333 333 332 120 673 730 969 429 016 113 281 249 823 948 8 × 2 = 0 + 0.666 666 666 664 241 347 461 938 858 032 226 562 499 647 897 6;
  • 16) 0.666 666 666 664 241 347 461 938 858 032 226 562 499 647 897 6 × 2 = 1 + 0.333 333 333 328 482 694 923 877 716 064 453 124 999 295 795 2;
  • 17) 0.333 333 333 328 482 694 923 877 716 064 453 124 999 295 795 2 × 2 = 0 + 0.666 666 666 656 965 389 847 755 432 128 906 249 998 591 590 4;
  • 18) 0.666 666 666 656 965 389 847 755 432 128 906 249 998 591 590 4 × 2 = 1 + 0.333 333 333 313 930 779 695 510 864 257 812 499 997 183 180 8;
  • 19) 0.333 333 333 313 930 779 695 510 864 257 812 499 997 183 180 8 × 2 = 0 + 0.666 666 666 627 861 559 391 021 728 515 624 999 994 366 361 6;
  • 20) 0.666 666 666 627 861 559 391 021 728 515 624 999 994 366 361 6 × 2 = 1 + 0.333 333 333 255 723 118 782 043 457 031 249 999 988 732 723 2;
  • 21) 0.333 333 333 255 723 118 782 043 457 031 249 999 988 732 723 2 × 2 = 0 + 0.666 666 666 511 446 237 564 086 914 062 499 999 977 465 446 4;
  • 22) 0.666 666 666 511 446 237 564 086 914 062 499 999 977 465 446 4 × 2 = 1 + 0.333 333 333 022 892 475 128 173 828 124 999 999 954 930 892 8;
  • 23) 0.333 333 333 022 892 475 128 173 828 124 999 999 954 930 892 8 × 2 = 0 + 0.666 666 666 045 784 950 256 347 656 249 999 999 909 861 785 6;
  • 24) 0.666 666 666 045 784 950 256 347 656 249 999 999 909 861 785 6 × 2 = 1 + 0.333 333 332 091 569 900 512 695 312 499 999 999 819 723 571 2;
  • 25) 0.333 333 332 091 569 900 512 695 312 499 999 999 819 723 571 2 × 2 = 0 + 0.666 666 664 183 139 801 025 390 624 999 999 999 639 447 142 4;
  • 26) 0.666 666 664 183 139 801 025 390 624 999 999 999 639 447 142 4 × 2 = 1 + 0.333 333 328 366 279 602 050 781 249 999 999 999 278 894 284 8;
  • 27) 0.333 333 328 366 279 602 050 781 249 999 999 999 278 894 284 8 × 2 = 0 + 0.666 666 656 732 559 204 101 562 499 999 999 998 557 788 569 6;
  • 28) 0.666 666 656 732 559 204 101 562 499 999 999 998 557 788 569 6 × 2 = 1 + 0.333 333 313 465 118 408 203 124 999 999 999 997 115 577 139 2;
  • 29) 0.333 333 313 465 118 408 203 124 999 999 999 997 115 577 139 2 × 2 = 0 + 0.666 666 626 930 236 816 406 249 999 999 999 994 231 154 278 4;
  • 30) 0.666 666 626 930 236 816 406 249 999 999 999 994 231 154 278 4 × 2 = 1 + 0.333 333 253 860 473 632 812 499 999 999 999 988 462 308 556 8;
  • 31) 0.333 333 253 860 473 632 812 499 999 999 999 988 462 308 556 8 × 2 = 0 + 0.666 666 507 720 947 265 624 999 999 999 999 976 924 617 113 6;
  • 32) 0.666 666 507 720 947 265 624 999 999 999 999 976 924 617 113 6 × 2 = 1 + 0.333 333 015 441 894 531 249 999 999 999 999 953 849 234 227 2;
  • 33) 0.333 333 015 441 894 531 249 999 999 999 999 953 849 234 227 2 × 2 = 0 + 0.666 666 030 883 789 062 499 999 999 999 999 907 698 468 454 4;
  • 34) 0.666 666 030 883 789 062 499 999 999 999 999 907 698 468 454 4 × 2 = 1 + 0.333 332 061 767 578 124 999 999 999 999 999 815 396 936 908 8;
  • 35) 0.333 332 061 767 578 124 999 999 999 999 999 815 396 936 908 8 × 2 = 0 + 0.666 664 123 535 156 249 999 999 999 999 999 630 793 873 817 6;
  • 36) 0.666 664 123 535 156 249 999 999 999 999 999 630 793 873 817 6 × 2 = 1 + 0.333 328 247 070 312 499 999 999 999 999 999 261 587 747 635 2;
  • 37) 0.333 328 247 070 312 499 999 999 999 999 999 261 587 747 635 2 × 2 = 0 + 0.666 656 494 140 624 999 999 999 999 999 998 523 175 495 270 4;
  • 38) 0.666 656 494 140 624 999 999 999 999 999 998 523 175 495 270 4 × 2 = 1 + 0.333 312 988 281 249 999 999 999 999 999 997 046 350 990 540 8;
  • 39) 0.333 312 988 281 249 999 999 999 999 999 997 046 350 990 540 8 × 2 = 0 + 0.666 625 976 562 499 999 999 999 999 999 994 092 701 981 081 6;
  • 40) 0.666 625 976 562 499 999 999 999 999 999 994 092 701 981 081 6 × 2 = 1 + 0.333 251 953 124 999 999 999 999 999 999 988 185 403 962 163 2;
  • 41) 0.333 251 953 124 999 999 999 999 999 999 988 185 403 962 163 2 × 2 = 0 + 0.666 503 906 249 999 999 999 999 999 999 976 370 807 924 326 4;
  • 42) 0.666 503 906 249 999 999 999 999 999 999 976 370 807 924 326 4 × 2 = 1 + 0.333 007 812 499 999 999 999 999 999 999 952 741 615 848 652 8;
  • 43) 0.333 007 812 499 999 999 999 999 999 999 952 741 615 848 652 8 × 2 = 0 + 0.666 015 624 999 999 999 999 999 999 999 905 483 231 697 305 6;
  • 44) 0.666 015 624 999 999 999 999 999 999 999 905 483 231 697 305 6 × 2 = 1 + 0.332 031 249 999 999 999 999 999 999 999 810 966 463 394 611 2;
  • 45) 0.332 031 249 999 999 999 999 999 999 999 810 966 463 394 611 2 × 2 = 0 + 0.664 062 499 999 999 999 999 999 999 999 621 932 926 789 222 4;
  • 46) 0.664 062 499 999 999 999 999 999 999 999 621 932 926 789 222 4 × 2 = 1 + 0.328 124 999 999 999 999 999 999 999 999 243 865 853 578 444 8;
  • 47) 0.328 124 999 999 999 999 999 999 999 999 243 865 853 578 444 8 × 2 = 0 + 0.656 249 999 999 999 999 999 999 999 998 487 731 707 156 889 6;
  • 48) 0.656 249 999 999 999 999 999 999 999 998 487 731 707 156 889 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 996 975 463 414 313 779 2;
  • 49) 0.312 499 999 999 999 999 999 999 999 996 975 463 414 313 779 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 993 950 926 828 627 558 4;
  • 50) 0.624 999 999 999 999 999 999 999 999 993 950 926 828 627 558 4 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 987 901 853 657 255 116 8;
  • 51) 0.249 999 999 999 999 999 999 999 999 987 901 853 657 255 116 8 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 975 803 707 314 510 233 6;
  • 52) 0.499 999 999 999 999 999 999 999 999 975 803 707 314 510 233 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 951 607 414 629 020 467 2;
  • 53) 0.999 999 999 999 999 999 999 999 999 951 607 414 629 020 467 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 903 214 829 258 040 934 4;

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:


0.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1(2)

5. Positive number before normalization:

1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2(10) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2(10) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1(2) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1(2) × 20


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): 0

Mantissa (not normalized):
1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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) =

1023(10) =

011 1111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 1 =

0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100


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 1111

Mantissa (52 bits) =
0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100


Decimal number 1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 783 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100

» Convert decimal 1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 793 1 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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