-24.901 899 999 999 997 703 525 878 023 356 199 264 524 74 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -24.901 899 999 999 997 703 525 878 023 356 199 264 524 74(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
-24.901 899 999 999 997 703 525 878 023 356 199 264 524 74(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. Start with the positive version of the number:

|-24.901 899 999 999 997 703 525 878 023 356 199 264 524 74| = 24.901 899 999 999 997 703 525 878 023 356 199 264 524 74


2. First, convert to binary (in base 2) the integer part: 24.
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;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

3. 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.

24(10) =

1 1000(2)


4. Convert to binary (base 2) the fractional part: 0.901 899 999 999 997 703 525 878 023 356 199 264 524 74.

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) 0.901 899 999 999 997 703 525 878 023 356 199 264 524 74 × 2 = 1 + 0.803 799 999 999 995 407 051 756 046 712 398 529 049 48;
  • 2) 0.803 799 999 999 995 407 051 756 046 712 398 529 049 48 × 2 = 1 + 0.607 599 999 999 990 814 103 512 093 424 797 058 098 96;
  • 3) 0.607 599 999 999 990 814 103 512 093 424 797 058 098 96 × 2 = 1 + 0.215 199 999 999 981 628 207 024 186 849 594 116 197 92;
  • 4) 0.215 199 999 999 981 628 207 024 186 849 594 116 197 92 × 2 = 0 + 0.430 399 999 999 963 256 414 048 373 699 188 232 395 84;
  • 5) 0.430 399 999 999 963 256 414 048 373 699 188 232 395 84 × 2 = 0 + 0.860 799 999 999 926 512 828 096 747 398 376 464 791 68;
  • 6) 0.860 799 999 999 926 512 828 096 747 398 376 464 791 68 × 2 = 1 + 0.721 599 999 999 853 025 656 193 494 796 752 929 583 36;
  • 7) 0.721 599 999 999 853 025 656 193 494 796 752 929 583 36 × 2 = 1 + 0.443 199 999 999 706 051 312 386 989 593 505 859 166 72;
  • 8) 0.443 199 999 999 706 051 312 386 989 593 505 859 166 72 × 2 = 0 + 0.886 399 999 999 412 102 624 773 979 187 011 718 333 44;
  • 9) 0.886 399 999 999 412 102 624 773 979 187 011 718 333 44 × 2 = 1 + 0.772 799 999 998 824 205 249 547 958 374 023 436 666 88;
  • 10) 0.772 799 999 998 824 205 249 547 958 374 023 436 666 88 × 2 = 1 + 0.545 599 999 997 648 410 499 095 916 748 046 873 333 76;
  • 11) 0.545 599 999 997 648 410 499 095 916 748 046 873 333 76 × 2 = 1 + 0.091 199 999 995 296 820 998 191 833 496 093 746 667 52;
  • 12) 0.091 199 999 995 296 820 998 191 833 496 093 746 667 52 × 2 = 0 + 0.182 399 999 990 593 641 996 383 666 992 187 493 335 04;
  • 13) 0.182 399 999 990 593 641 996 383 666 992 187 493 335 04 × 2 = 0 + 0.364 799 999 981 187 283 992 767 333 984 374 986 670 08;
  • 14) 0.364 799 999 981 187 283 992 767 333 984 374 986 670 08 × 2 = 0 + 0.729 599 999 962 374 567 985 534 667 968 749 973 340 16;
  • 15) 0.729 599 999 962 374 567 985 534 667 968 749 973 340 16 × 2 = 1 + 0.459 199 999 924 749 135 971 069 335 937 499 946 680 32;
  • 16) 0.459 199 999 924 749 135 971 069 335 937 499 946 680 32 × 2 = 0 + 0.918 399 999 849 498 271 942 138 671 874 999 893 360 64;
  • 17) 0.918 399 999 849 498 271 942 138 671 874 999 893 360 64 × 2 = 1 + 0.836 799 999 698 996 543 884 277 343 749 999 786 721 28;
  • 18) 0.836 799 999 698 996 543 884 277 343 749 999 786 721 28 × 2 = 1 + 0.673 599 999 397 993 087 768 554 687 499 999 573 442 56;
  • 19) 0.673 599 999 397 993 087 768 554 687 499 999 573 442 56 × 2 = 1 + 0.347 199 998 795 986 175 537 109 374 999 999 146 885 12;
  • 20) 0.347 199 998 795 986 175 537 109 374 999 999 146 885 12 × 2 = 0 + 0.694 399 997 591 972 351 074 218 749 999 998 293 770 24;
  • 21) 0.694 399 997 591 972 351 074 218 749 999 998 293 770 24 × 2 = 1 + 0.388 799 995 183 944 702 148 437 499 999 996 587 540 48;
  • 22) 0.388 799 995 183 944 702 148 437 499 999 996 587 540 48 × 2 = 0 + 0.777 599 990 367 889 404 296 874 999 999 993 175 080 96;
  • 23) 0.777 599 990 367 889 404 296 874 999 999 993 175 080 96 × 2 = 1 + 0.555 199 980 735 778 808 593 749 999 999 986 350 161 92;
  • 24) 0.555 199 980 735 778 808 593 749 999 999 986 350 161 92 × 2 = 1 + 0.110 399 961 471 557 617 187 499 999 999 972 700 323 84;
  • 25) 0.110 399 961 471 557 617 187 499 999 999 972 700 323 84 × 2 = 0 + 0.220 799 922 943 115 234 374 999 999 999 945 400 647 68;
  • 26) 0.220 799 922 943 115 234 374 999 999 999 945 400 647 68 × 2 = 0 + 0.441 599 845 886 230 468 749 999 999 999 890 801 295 36;
  • 27) 0.441 599 845 886 230 468 749 999 999 999 890 801 295 36 × 2 = 0 + 0.883 199 691 772 460 937 499 999 999 999 781 602 590 72;
  • 28) 0.883 199 691 772 460 937 499 999 999 999 781 602 590 72 × 2 = 1 + 0.766 399 383 544 921 874 999 999 999 999 563 205 181 44;
  • 29) 0.766 399 383 544 921 874 999 999 999 999 563 205 181 44 × 2 = 1 + 0.532 798 767 089 843 749 999 999 999 999 126 410 362 88;
  • 30) 0.532 798 767 089 843 749 999 999 999 999 126 410 362 88 × 2 = 1 + 0.065 597 534 179 687 499 999 999 999 998 252 820 725 76;
  • 31) 0.065 597 534 179 687 499 999 999 999 998 252 820 725 76 × 2 = 0 + 0.131 195 068 359 374 999 999 999 999 996 505 641 451 52;
  • 32) 0.131 195 068 359 374 999 999 999 999 996 505 641 451 52 × 2 = 0 + 0.262 390 136 718 749 999 999 999 999 993 011 282 903 04;
  • 33) 0.262 390 136 718 749 999 999 999 999 993 011 282 903 04 × 2 = 0 + 0.524 780 273 437 499 999 999 999 999 986 022 565 806 08;
  • 34) 0.524 780 273 437 499 999 999 999 999 986 022 565 806 08 × 2 = 1 + 0.049 560 546 874 999 999 999 999 999 972 045 131 612 16;
  • 35) 0.049 560 546 874 999 999 999 999 999 972 045 131 612 16 × 2 = 0 + 0.099 121 093 749 999 999 999 999 999 944 090 263 224 32;
  • 36) 0.099 121 093 749 999 999 999 999 999 944 090 263 224 32 × 2 = 0 + 0.198 242 187 499 999 999 999 999 999 888 180 526 448 64;
  • 37) 0.198 242 187 499 999 999 999 999 999 888 180 526 448 64 × 2 = 0 + 0.396 484 374 999 999 999 999 999 999 776 361 052 897 28;
  • 38) 0.396 484 374 999 999 999 999 999 999 776 361 052 897 28 × 2 = 0 + 0.792 968 749 999 999 999 999 999 999 552 722 105 794 56;
  • 39) 0.792 968 749 999 999 999 999 999 999 552 722 105 794 56 × 2 = 1 + 0.585 937 499 999 999 999 999 999 999 105 444 211 589 12;
  • 40) 0.585 937 499 999 999 999 999 999 999 105 444 211 589 12 × 2 = 1 + 0.171 874 999 999 999 999 999 999 998 210 888 423 178 24;
  • 41) 0.171 874 999 999 999 999 999 999 998 210 888 423 178 24 × 2 = 0 + 0.343 749 999 999 999 999 999 999 996 421 776 846 356 48;
  • 42) 0.343 749 999 999 999 999 999 999 996 421 776 846 356 48 × 2 = 0 + 0.687 499 999 999 999 999 999 999 992 843 553 692 712 96;
  • 43) 0.687 499 999 999 999 999 999 999 992 843 553 692 712 96 × 2 = 1 + 0.374 999 999 999 999 999 999 999 985 687 107 385 425 92;
  • 44) 0.374 999 999 999 999 999 999 999 985 687 107 385 425 92 × 2 = 0 + 0.749 999 999 999 999 999 999 999 971 374 214 770 851 84;
  • 45) 0.749 999 999 999 999 999 999 999 971 374 214 770 851 84 × 2 = 1 + 0.499 999 999 999 999 999 999 999 942 748 429 541 703 68;
  • 46) 0.499 999 999 999 999 999 999 999 942 748 429 541 703 68 × 2 = 0 + 0.999 999 999 999 999 999 999 999 885 496 859 083 407 36;
  • 47) 0.999 999 999 999 999 999 999 999 885 496 859 083 407 36 × 2 = 1 + 0.999 999 999 999 999 999 999 999 770 993 718 166 814 72;
  • 48) 0.999 999 999 999 999 999 999 999 770 993 718 166 814 72 × 2 = 1 + 0.999 999 999 999 999 999 999 999 541 987 436 333 629 44;
  • 49) 0.999 999 999 999 999 999 999 999 541 987 436 333 629 44 × 2 = 1 + 0.999 999 999 999 999 999 999 999 083 974 872 667 258 88;
  • 50) 0.999 999 999 999 999 999 999 999 083 974 872 667 258 88 × 2 = 1 + 0.999 999 999 999 999 999 999 998 167 949 745 334 517 76;
  • 51) 0.999 999 999 999 999 999 999 998 167 949 745 334 517 76 × 2 = 1 + 0.999 999 999 999 999 999 999 996 335 899 490 669 035 52;
  • 52) 0.999 999 999 999 999 999 999 996 335 899 490 669 035 52 × 2 = 1 + 0.999 999 999 999 999 999 999 992 671 798 981 338 071 04;
  • 53) 0.999 999 999 999 999 999 999 992 671 798 981 338 071 04 × 2 = 1 + 0.999 999 999 999 999 999 999 985 343 597 962 676 142 08;

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).


5. 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:


0.901 899 999 999 997 703 525 878 023 356 199 264 524 74(10) =

0.1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1(2)

6. Positive number before normalization:

24.901 899 999 999 997 703 525 878 023 356 199 264 524 74(10) =

1 1000.1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1(2)

7. Normalize the binary representation of the number.

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


24.901 899 999 999 997 703 525 878 023 356 199 264 524 74(10) =

1 1000.1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1(2) =

1 1000.1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1(2) × 20 =

1.1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1(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):
1.1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1111 1


9. 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) =

1 027(10)


10. 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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 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;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1027(10) =

100 0000 0011(2)


12. 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. 1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011 1 1111 =

1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011


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

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
100 0000 0011

Mantissa (52 bits) =
1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011


Decimal number -24.901 899 999 999 997 703 525 878 023 356 199 264 524 74 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0011 - 1000 1110 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1011

» Convert decimal -24.901 899 999 999 997 703 525 878 023 356 199 264 525 17 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