7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775(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
7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775(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: 7.
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;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

7(10) =

111(2)


3. Convert to binary (base 2) the fractional part: 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775.

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) 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775 × 2 = 0 + 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 55;
  • 2) 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 55 × 2 = 0 + 0.800 000 000 000 000 710 542 735 760 100 185 871 124 267 1;
  • 3) 0.800 000 000 000 000 710 542 735 760 100 185 871 124 267 1 × 2 = 1 + 0.600 000 000 000 001 421 085 471 520 200 371 742 248 534 2;
  • 4) 0.600 000 000 000 001 421 085 471 520 200 371 742 248 534 2 × 2 = 1 + 0.200 000 000 000 002 842 170 943 040 400 743 484 497 068 4;
  • 5) 0.200 000 000 000 002 842 170 943 040 400 743 484 497 068 4 × 2 = 0 + 0.400 000 000 000 005 684 341 886 080 801 486 968 994 136 8;
  • 6) 0.400 000 000 000 005 684 341 886 080 801 486 968 994 136 8 × 2 = 0 + 0.800 000 000 000 011 368 683 772 161 602 973 937 988 273 6;
  • 7) 0.800 000 000 000 011 368 683 772 161 602 973 937 988 273 6 × 2 = 1 + 0.600 000 000 000 022 737 367 544 323 205 947 875 976 547 2;
  • 8) 0.600 000 000 000 022 737 367 544 323 205 947 875 976 547 2 × 2 = 1 + 0.200 000 000 000 045 474 735 088 646 411 895 751 953 094 4;
  • 9) 0.200 000 000 000 045 474 735 088 646 411 895 751 953 094 4 × 2 = 0 + 0.400 000 000 000 090 949 470 177 292 823 791 503 906 188 8;
  • 10) 0.400 000 000 000 090 949 470 177 292 823 791 503 906 188 8 × 2 = 0 + 0.800 000 000 000 181 898 940 354 585 647 583 007 812 377 6;
  • 11) 0.800 000 000 000 181 898 940 354 585 647 583 007 812 377 6 × 2 = 1 + 0.600 000 000 000 363 797 880 709 171 295 166 015 624 755 2;
  • 12) 0.600 000 000 000 363 797 880 709 171 295 166 015 624 755 2 × 2 = 1 + 0.200 000 000 000 727 595 761 418 342 590 332 031 249 510 4;
  • 13) 0.200 000 000 000 727 595 761 418 342 590 332 031 249 510 4 × 2 = 0 + 0.400 000 000 001 455 191 522 836 685 180 664 062 499 020 8;
  • 14) 0.400 000 000 001 455 191 522 836 685 180 664 062 499 020 8 × 2 = 0 + 0.800 000 000 002 910 383 045 673 370 361 328 124 998 041 6;
  • 15) 0.800 000 000 002 910 383 045 673 370 361 328 124 998 041 6 × 2 = 1 + 0.600 000 000 005 820 766 091 346 740 722 656 249 996 083 2;
  • 16) 0.600 000 000 005 820 766 091 346 740 722 656 249 996 083 2 × 2 = 1 + 0.200 000 000 011 641 532 182 693 481 445 312 499 992 166 4;
  • 17) 0.200 000 000 011 641 532 182 693 481 445 312 499 992 166 4 × 2 = 0 + 0.400 000 000 023 283 064 365 386 962 890 624 999 984 332 8;
  • 18) 0.400 000 000 023 283 064 365 386 962 890 624 999 984 332 8 × 2 = 0 + 0.800 000 000 046 566 128 730 773 925 781 249 999 968 665 6;
  • 19) 0.800 000 000 046 566 128 730 773 925 781 249 999 968 665 6 × 2 = 1 + 0.600 000 000 093 132 257 461 547 851 562 499 999 937 331 2;
  • 20) 0.600 000 000 093 132 257 461 547 851 562 499 999 937 331 2 × 2 = 1 + 0.200 000 000 186 264 514 923 095 703 124 999 999 874 662 4;
  • 21) 0.200 000 000 186 264 514 923 095 703 124 999 999 874 662 4 × 2 = 0 + 0.400 000 000 372 529 029 846 191 406 249 999 999 749 324 8;
  • 22) 0.400 000 000 372 529 029 846 191 406 249 999 999 749 324 8 × 2 = 0 + 0.800 000 000 745 058 059 692 382 812 499 999 999 498 649 6;
  • 23) 0.800 000 000 745 058 059 692 382 812 499 999 999 498 649 6 × 2 = 1 + 0.600 000 001 490 116 119 384 765 624 999 999 998 997 299 2;
  • 24) 0.600 000 001 490 116 119 384 765 624 999 999 998 997 299 2 × 2 = 1 + 0.200 000 002 980 232 238 769 531 249 999 999 997 994 598 4;
  • 25) 0.200 000 002 980 232 238 769 531 249 999 999 997 994 598 4 × 2 = 0 + 0.400 000 005 960 464 477 539 062 499 999 999 995 989 196 8;
  • 26) 0.400 000 005 960 464 477 539 062 499 999 999 995 989 196 8 × 2 = 0 + 0.800 000 011 920 928 955 078 124 999 999 999 991 978 393 6;
  • 27) 0.800 000 011 920 928 955 078 124 999 999 999 991 978 393 6 × 2 = 1 + 0.600 000 023 841 857 910 156 249 999 999 999 983 956 787 2;
  • 28) 0.600 000 023 841 857 910 156 249 999 999 999 983 956 787 2 × 2 = 1 + 0.200 000 047 683 715 820 312 499 999 999 999 967 913 574 4;
  • 29) 0.200 000 047 683 715 820 312 499 999 999 999 967 913 574 4 × 2 = 0 + 0.400 000 095 367 431 640 624 999 999 999 999 935 827 148 8;
  • 30) 0.400 000 095 367 431 640 624 999 999 999 999 935 827 148 8 × 2 = 0 + 0.800 000 190 734 863 281 249 999 999 999 999 871 654 297 6;
  • 31) 0.800 000 190 734 863 281 249 999 999 999 999 871 654 297 6 × 2 = 1 + 0.600 000 381 469 726 562 499 999 999 999 999 743 308 595 2;
  • 32) 0.600 000 381 469 726 562 499 999 999 999 999 743 308 595 2 × 2 = 1 + 0.200 000 762 939 453 124 999 999 999 999 999 486 617 190 4;
  • 33) 0.200 000 762 939 453 124 999 999 999 999 999 486 617 190 4 × 2 = 0 + 0.400 001 525 878 906 249 999 999 999 999 998 973 234 380 8;
  • 34) 0.400 001 525 878 906 249 999 999 999 999 998 973 234 380 8 × 2 = 0 + 0.800 003 051 757 812 499 999 999 999 999 997 946 468 761 6;
  • 35) 0.800 003 051 757 812 499 999 999 999 999 997 946 468 761 6 × 2 = 1 + 0.600 006 103 515 624 999 999 999 999 999 995 892 937 523 2;
  • 36) 0.600 006 103 515 624 999 999 999 999 999 995 892 937 523 2 × 2 = 1 + 0.200 012 207 031 249 999 999 999 999 999 991 785 875 046 4;
  • 37) 0.200 012 207 031 249 999 999 999 999 999 991 785 875 046 4 × 2 = 0 + 0.400 024 414 062 499 999 999 999 999 999 983 571 750 092 8;
  • 38) 0.400 024 414 062 499 999 999 999 999 999 983 571 750 092 8 × 2 = 0 + 0.800 048 828 124 999 999 999 999 999 999 967 143 500 185 6;
  • 39) 0.800 048 828 124 999 999 999 999 999 999 967 143 500 185 6 × 2 = 1 + 0.600 097 656 249 999 999 999 999 999 999 934 287 000 371 2;
  • 40) 0.600 097 656 249 999 999 999 999 999 999 934 287 000 371 2 × 2 = 1 + 0.200 195 312 499 999 999 999 999 999 999 868 574 000 742 4;
  • 41) 0.200 195 312 499 999 999 999 999 999 999 868 574 000 742 4 × 2 = 0 + 0.400 390 624 999 999 999 999 999 999 999 737 148 001 484 8;
  • 42) 0.400 390 624 999 999 999 999 999 999 999 737 148 001 484 8 × 2 = 0 + 0.800 781 249 999 999 999 999 999 999 999 474 296 002 969 6;
  • 43) 0.800 781 249 999 999 999 999 999 999 999 474 296 002 969 6 × 2 = 1 + 0.601 562 499 999 999 999 999 999 999 998 948 592 005 939 2;
  • 44) 0.601 562 499 999 999 999 999 999 999 998 948 592 005 939 2 × 2 = 1 + 0.203 124 999 999 999 999 999 999 999 997 897 184 011 878 4;
  • 45) 0.203 124 999 999 999 999 999 999 999 997 897 184 011 878 4 × 2 = 0 + 0.406 249 999 999 999 999 999 999 999 995 794 368 023 756 8;
  • 46) 0.406 249 999 999 999 999 999 999 999 995 794 368 023 756 8 × 2 = 0 + 0.812 499 999 999 999 999 999 999 999 991 588 736 047 513 6;
  • 47) 0.812 499 999 999 999 999 999 999 999 991 588 736 047 513 6 × 2 = 1 + 0.624 999 999 999 999 999 999 999 999 983 177 472 095 027 2;
  • 48) 0.624 999 999 999 999 999 999 999 999 983 177 472 095 027 2 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 966 354 944 190 054 4;
  • 49) 0.249 999 999 999 999 999 999 999 999 966 354 944 190 054 4 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 932 709 888 380 108 8;
  • 50) 0.499 999 999 999 999 999 999 999 999 932 709 888 380 108 8 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 865 419 776 760 217 6;
  • 51) 0.999 999 999 999 999 999 999 999 999 865 419 776 760 217 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 730 839 553 520 435 2;
  • 52) 0.999 999 999 999 999 999 999 999 999 730 839 553 520 435 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 461 679 107 040 870 4;
  • 53) 0.999 999 999 999 999 999 999 999 999 461 679 107 040 870 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 998 923 358 214 081 740 8;

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:


0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775(10) =

0.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2)

5. Positive number before normalization:

7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775(10) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the left, so that only one non zero digit remains to the left of it:


7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775(10) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2) × 20 =

1.1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111(2) × 22


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): 2

Mantissa (not normalized):
1.1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

2 + 2(11-1) - 1 =

(2 + 1 023)(10) =

1 025(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;
  • 1 025 ÷ 2 = 512 + 1;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1025(10) =

100 0000 0001(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. 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111 =

1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100


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) =
100 0000 0001

Mantissa (52 bits) =
1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100


Decimal number 7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 775 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0001 - 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100

Share

» Convert decimal 7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 753 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 04, 2026 [April]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: April - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 minutes

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