32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4(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
32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4(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: 32.
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;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

32(10) =

10 0000(2)


3. Convert to binary (base 2) the fractional part: 0.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4.

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) 0.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4 × 2 = 0 + 0.007 813 453 790 745 938 931 650 016 456 842 422 485 351 426 8;
  • 2) 0.007 813 453 790 745 938 931 650 016 456 842 422 485 351 426 8 × 2 = 0 + 0.015 626 907 581 491 877 863 300 032 913 684 844 970 702 853 6;
  • 3) 0.015 626 907 581 491 877 863 300 032 913 684 844 970 702 853 6 × 2 = 0 + 0.031 253 815 162 983 755 726 600 065 827 369 689 941 405 707 2;
  • 4) 0.031 253 815 162 983 755 726 600 065 827 369 689 941 405 707 2 × 2 = 0 + 0.062 507 630 325 967 511 453 200 131 654 739 379 882 811 414 4;
  • 5) 0.062 507 630 325 967 511 453 200 131 654 739 379 882 811 414 4 × 2 = 0 + 0.125 015 260 651 935 022 906 400 263 309 478 759 765 622 828 8;
  • 6) 0.125 015 260 651 935 022 906 400 263 309 478 759 765 622 828 8 × 2 = 0 + 0.250 030 521 303 870 045 812 800 526 618 957 519 531 245 657 6;
  • 7) 0.250 030 521 303 870 045 812 800 526 618 957 519 531 245 657 6 × 2 = 0 + 0.500 061 042 607 740 091 625 601 053 237 915 039 062 491 315 2;
  • 8) 0.500 061 042 607 740 091 625 601 053 237 915 039 062 491 315 2 × 2 = 1 + 0.000 122 085 215 480 183 251 202 106 475 830 078 124 982 630 4;
  • 9) 0.000 122 085 215 480 183 251 202 106 475 830 078 124 982 630 4 × 2 = 0 + 0.000 244 170 430 960 366 502 404 212 951 660 156 249 965 260 8;
  • 10) 0.000 244 170 430 960 366 502 404 212 951 660 156 249 965 260 8 × 2 = 0 + 0.000 488 340 861 920 733 004 808 425 903 320 312 499 930 521 6;
  • 11) 0.000 488 340 861 920 733 004 808 425 903 320 312 499 930 521 6 × 2 = 0 + 0.000 976 681 723 841 466 009 616 851 806 640 624 999 861 043 2;
  • 12) 0.000 976 681 723 841 466 009 616 851 806 640 624 999 861 043 2 × 2 = 0 + 0.001 953 363 447 682 932 019 233 703 613 281 249 999 722 086 4;
  • 13) 0.001 953 363 447 682 932 019 233 703 613 281 249 999 722 086 4 × 2 = 0 + 0.003 906 726 895 365 864 038 467 407 226 562 499 999 444 172 8;
  • 14) 0.003 906 726 895 365 864 038 467 407 226 562 499 999 444 172 8 × 2 = 0 + 0.007 813 453 790 731 728 076 934 814 453 124 999 998 888 345 6;
  • 15) 0.007 813 453 790 731 728 076 934 814 453 124 999 998 888 345 6 × 2 = 0 + 0.015 626 907 581 463 456 153 869 628 906 249 999 997 776 691 2;
  • 16) 0.015 626 907 581 463 456 153 869 628 906 249 999 997 776 691 2 × 2 = 0 + 0.031 253 815 162 926 912 307 739 257 812 499 999 995 553 382 4;
  • 17) 0.031 253 815 162 926 912 307 739 257 812 499 999 995 553 382 4 × 2 = 0 + 0.062 507 630 325 853 824 615 478 515 624 999 999 991 106 764 8;
  • 18) 0.062 507 630 325 853 824 615 478 515 624 999 999 991 106 764 8 × 2 = 0 + 0.125 015 260 651 707 649 230 957 031 249 999 999 982 213 529 6;
  • 19) 0.125 015 260 651 707 649 230 957 031 249 999 999 982 213 529 6 × 2 = 0 + 0.250 030 521 303 415 298 461 914 062 499 999 999 964 427 059 2;
  • 20) 0.250 030 521 303 415 298 461 914 062 499 999 999 964 427 059 2 × 2 = 0 + 0.500 061 042 606 830 596 923 828 124 999 999 999 928 854 118 4;
  • 21) 0.500 061 042 606 830 596 923 828 124 999 999 999 928 854 118 4 × 2 = 1 + 0.000 122 085 213 661 193 847 656 249 999 999 999 857 708 236 8;
  • 22) 0.000 122 085 213 661 193 847 656 249 999 999 999 857 708 236 8 × 2 = 0 + 0.000 244 170 427 322 387 695 312 499 999 999 999 715 416 473 6;
  • 23) 0.000 244 170 427 322 387 695 312 499 999 999 999 715 416 473 6 × 2 = 0 + 0.000 488 340 854 644 775 390 624 999 999 999 999 430 832 947 2;
  • 24) 0.000 488 340 854 644 775 390 624 999 999 999 999 430 832 947 2 × 2 = 0 + 0.000 976 681 709 289 550 781 249 999 999 999 998 861 665 894 4;
  • 25) 0.000 976 681 709 289 550 781 249 999 999 999 998 861 665 894 4 × 2 = 0 + 0.001 953 363 418 579 101 562 499 999 999 999 997 723 331 788 8;
  • 26) 0.001 953 363 418 579 101 562 499 999 999 999 997 723 331 788 8 × 2 = 0 + 0.003 906 726 837 158 203 124 999 999 999 999 995 446 663 577 6;
  • 27) 0.003 906 726 837 158 203 124 999 999 999 999 995 446 663 577 6 × 2 = 0 + 0.007 813 453 674 316 406 249 999 999 999 999 990 893 327 155 2;
  • 28) 0.007 813 453 674 316 406 249 999 999 999 999 990 893 327 155 2 × 2 = 0 + 0.015 626 907 348 632 812 499 999 999 999 999 981 786 654 310 4;
  • 29) 0.015 626 907 348 632 812 499 999 999 999 999 981 786 654 310 4 × 2 = 0 + 0.031 253 814 697 265 624 999 999 999 999 999 963 573 308 620 8;
  • 30) 0.031 253 814 697 265 624 999 999 999 999 999 963 573 308 620 8 × 2 = 0 + 0.062 507 629 394 531 249 999 999 999 999 999 927 146 617 241 6;
  • 31) 0.062 507 629 394 531 249 999 999 999 999 999 927 146 617 241 6 × 2 = 0 + 0.125 015 258 789 062 499 999 999 999 999 999 854 293 234 483 2;
  • 32) 0.125 015 258 789 062 499 999 999 999 999 999 854 293 234 483 2 × 2 = 0 + 0.250 030 517 578 124 999 999 999 999 999 999 708 586 468 966 4;
  • 33) 0.250 030 517 578 124 999 999 999 999 999 999 708 586 468 966 4 × 2 = 0 + 0.500 061 035 156 249 999 999 999 999 999 999 417 172 937 932 8;
  • 34) 0.500 061 035 156 249 999 999 999 999 999 999 417 172 937 932 8 × 2 = 1 + 0.000 122 070 312 499 999 999 999 999 999 998 834 345 875 865 6;
  • 35) 0.000 122 070 312 499 999 999 999 999 999 998 834 345 875 865 6 × 2 = 0 + 0.000 244 140 624 999 999 999 999 999 999 997 668 691 751 731 2;
  • 36) 0.000 244 140 624 999 999 999 999 999 999 997 668 691 751 731 2 × 2 = 0 + 0.000 488 281 249 999 999 999 999 999 999 995 337 383 503 462 4;
  • 37) 0.000 488 281 249 999 999 999 999 999 999 995 337 383 503 462 4 × 2 = 0 + 0.000 976 562 499 999 999 999 999 999 999 990 674 767 006 924 8;
  • 38) 0.000 976 562 499 999 999 999 999 999 999 990 674 767 006 924 8 × 2 = 0 + 0.001 953 124 999 999 999 999 999 999 999 981 349 534 013 849 6;
  • 39) 0.001 953 124 999 999 999 999 999 999 999 981 349 534 013 849 6 × 2 = 0 + 0.003 906 249 999 999 999 999 999 999 999 962 699 068 027 699 2;
  • 40) 0.003 906 249 999 999 999 999 999 999 999 962 699 068 027 699 2 × 2 = 0 + 0.007 812 499 999 999 999 999 999 999 999 925 398 136 055 398 4;
  • 41) 0.007 812 499 999 999 999 999 999 999 999 925 398 136 055 398 4 × 2 = 0 + 0.015 624 999 999 999 999 999 999 999 999 850 796 272 110 796 8;
  • 42) 0.015 624 999 999 999 999 999 999 999 999 850 796 272 110 796 8 × 2 = 0 + 0.031 249 999 999 999 999 999 999 999 999 701 592 544 221 593 6;
  • 43) 0.031 249 999 999 999 999 999 999 999 999 701 592 544 221 593 6 × 2 = 0 + 0.062 499 999 999 999 999 999 999 999 999 403 185 088 443 187 2;
  • 44) 0.062 499 999 999 999 999 999 999 999 999 403 185 088 443 187 2 × 2 = 0 + 0.124 999 999 999 999 999 999 999 999 998 806 370 176 886 374 4;
  • 45) 0.124 999 999 999 999 999 999 999 999 998 806 370 176 886 374 4 × 2 = 0 + 0.249 999 999 999 999 999 999 999 999 997 612 740 353 772 748 8;
  • 46) 0.249 999 999 999 999 999 999 999 999 997 612 740 353 772 748 8 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 995 225 480 707 545 497 6;
  • 47) 0.499 999 999 999 999 999 999 999 999 995 225 480 707 545 497 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 990 450 961 415 090 995 2;
  • 48) 0.999 999 999 999 999 999 999 999 999 990 450 961 415 090 995 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 980 901 922 830 181 990 4;
  • 49) 0.999 999 999 999 999 999 999 999 999 980 901 922 830 181 990 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 961 803 845 660 363 980 8;
  • 50) 0.999 999 999 999 999 999 999 999 999 961 803 845 660 363 980 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 923 607 691 320 727 961 6;
  • 51) 0.999 999 999 999 999 999 999 999 999 923 607 691 320 727 961 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 847 215 382 641 455 923 2;
  • 52) 0.999 999 999 999 999 999 999 999 999 847 215 382 641 455 923 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 694 430 765 282 911 846 4;
  • 53) 0.999 999 999 999 999 999 999 999 999 694 430 765 282 911 846 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 388 861 530 565 823 692 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:


0.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4(10) =

0.0000 0001 0000 0000 0000 1000 0000 0000 0100 0000 0000 0001 1111 1(2)

5. Positive number before normalization:

32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4(10) =

10 0000.0000 0001 0000 0000 0000 1000 0000 0000 0100 0000 0000 0001 1111 1(2)

6. Normalize the binary representation of the number.

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


32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4(10) =

10 0000.0000 0001 0000 0000 0000 1000 0000 0000 0100 0000 0000 0001 1111 1(2) =

10 0000.0000 0001 0000 0000 0000 1000 0000 0000 0100 0000 0000 0001 1111 1(2) × 20 =

1.0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000 1111 11(2) × 25


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):
1.0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000 1111 11


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) =

1 028(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;
  • 1 028 ÷ 2 = 514 + 0;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 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) =

1028(10) =

100 0000 0100(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. 0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000 11 1111 =

0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000


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 0100

Mantissa (52 bits) =
0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000


Decimal number 32.003 906 726 895 372 969 465 825 008 228 421 211 242 675 713 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0100 - 0000 0000 0000 1000 0000 0000 0100 0000 0000 0010 0000 0000 0000

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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