1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37(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
1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37(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: 0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37.

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) 0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37 × 2 = 0 + 0.828 427 124 746 190 097 603 377 448 419 396 157 139 343 750 764 74;
  • 2) 0.828 427 124 746 190 097 603 377 448 419 396 157 139 343 750 764 74 × 2 = 1 + 0.656 854 249 492 380 195 206 754 896 838 792 314 278 687 501 529 48;
  • 3) 0.656 854 249 492 380 195 206 754 896 838 792 314 278 687 501 529 48 × 2 = 1 + 0.313 708 498 984 760 390 413 509 793 677 584 628 557 375 003 058 96;
  • 4) 0.313 708 498 984 760 390 413 509 793 677 584 628 557 375 003 058 96 × 2 = 0 + 0.627 416 997 969 520 780 827 019 587 355 169 257 114 750 006 117 92;
  • 5) 0.627 416 997 969 520 780 827 019 587 355 169 257 114 750 006 117 92 × 2 = 1 + 0.254 833 995 939 041 561 654 039 174 710 338 514 229 500 012 235 84;
  • 6) 0.254 833 995 939 041 561 654 039 174 710 338 514 229 500 012 235 84 × 2 = 0 + 0.509 667 991 878 083 123 308 078 349 420 677 028 459 000 024 471 68;
  • 7) 0.509 667 991 878 083 123 308 078 349 420 677 028 459 000 024 471 68 × 2 = 1 + 0.019 335 983 756 166 246 616 156 698 841 354 056 918 000 048 943 36;
  • 8) 0.019 335 983 756 166 246 616 156 698 841 354 056 918 000 048 943 36 × 2 = 0 + 0.038 671 967 512 332 493 232 313 397 682 708 113 836 000 097 886 72;
  • 9) 0.038 671 967 512 332 493 232 313 397 682 708 113 836 000 097 886 72 × 2 = 0 + 0.077 343 935 024 664 986 464 626 795 365 416 227 672 000 195 773 44;
  • 10) 0.077 343 935 024 664 986 464 626 795 365 416 227 672 000 195 773 44 × 2 = 0 + 0.154 687 870 049 329 972 929 253 590 730 832 455 344 000 391 546 88;
  • 11) 0.154 687 870 049 329 972 929 253 590 730 832 455 344 000 391 546 88 × 2 = 0 + 0.309 375 740 098 659 945 858 507 181 461 664 910 688 000 783 093 76;
  • 12) 0.309 375 740 098 659 945 858 507 181 461 664 910 688 000 783 093 76 × 2 = 0 + 0.618 751 480 197 319 891 717 014 362 923 329 821 376 001 566 187 52;
  • 13) 0.618 751 480 197 319 891 717 014 362 923 329 821 376 001 566 187 52 × 2 = 1 + 0.237 502 960 394 639 783 434 028 725 846 659 642 752 003 132 375 04;
  • 14) 0.237 502 960 394 639 783 434 028 725 846 659 642 752 003 132 375 04 × 2 = 0 + 0.475 005 920 789 279 566 868 057 451 693 319 285 504 006 264 750 08;
  • 15) 0.475 005 920 789 279 566 868 057 451 693 319 285 504 006 264 750 08 × 2 = 0 + 0.950 011 841 578 559 133 736 114 903 386 638 571 008 012 529 500 16;
  • 16) 0.950 011 841 578 559 133 736 114 903 386 638 571 008 012 529 500 16 × 2 = 1 + 0.900 023 683 157 118 267 472 229 806 773 277 142 016 025 059 000 32;
  • 17) 0.900 023 683 157 118 267 472 229 806 773 277 142 016 025 059 000 32 × 2 = 1 + 0.800 047 366 314 236 534 944 459 613 546 554 284 032 050 118 000 64;
  • 18) 0.800 047 366 314 236 534 944 459 613 546 554 284 032 050 118 000 64 × 2 = 1 + 0.600 094 732 628 473 069 888 919 227 093 108 568 064 100 236 001 28;
  • 19) 0.600 094 732 628 473 069 888 919 227 093 108 568 064 100 236 001 28 × 2 = 1 + 0.200 189 465 256 946 139 777 838 454 186 217 136 128 200 472 002 56;
  • 20) 0.200 189 465 256 946 139 777 838 454 186 217 136 128 200 472 002 56 × 2 = 0 + 0.400 378 930 513 892 279 555 676 908 372 434 272 256 400 944 005 12;
  • 21) 0.400 378 930 513 892 279 555 676 908 372 434 272 256 400 944 005 12 × 2 = 0 + 0.800 757 861 027 784 559 111 353 816 744 868 544 512 801 888 010 24;
  • 22) 0.800 757 861 027 784 559 111 353 816 744 868 544 512 801 888 010 24 × 2 = 1 + 0.601 515 722 055 569 118 222 707 633 489 737 089 025 603 776 020 48;
  • 23) 0.601 515 722 055 569 118 222 707 633 489 737 089 025 603 776 020 48 × 2 = 1 + 0.203 031 444 111 138 236 445 415 266 979 474 178 051 207 552 040 96;
  • 24) 0.203 031 444 111 138 236 445 415 266 979 474 178 051 207 552 040 96 × 2 = 0 + 0.406 062 888 222 276 472 890 830 533 958 948 356 102 415 104 081 92;
  • 25) 0.406 062 888 222 276 472 890 830 533 958 948 356 102 415 104 081 92 × 2 = 0 + 0.812 125 776 444 552 945 781 661 067 917 896 712 204 830 208 163 84;
  • 26) 0.812 125 776 444 552 945 781 661 067 917 896 712 204 830 208 163 84 × 2 = 1 + 0.624 251 552 889 105 891 563 322 135 835 793 424 409 660 416 327 68;
  • 27) 0.624 251 552 889 105 891 563 322 135 835 793 424 409 660 416 327 68 × 2 = 1 + 0.248 503 105 778 211 783 126 644 271 671 586 848 819 320 832 655 36;
  • 28) 0.248 503 105 778 211 783 126 644 271 671 586 848 819 320 832 655 36 × 2 = 0 + 0.497 006 211 556 423 566 253 288 543 343 173 697 638 641 665 310 72;
  • 29) 0.497 006 211 556 423 566 253 288 543 343 173 697 638 641 665 310 72 × 2 = 0 + 0.994 012 423 112 847 132 506 577 086 686 347 395 277 283 330 621 44;
  • 30) 0.994 012 423 112 847 132 506 577 086 686 347 395 277 283 330 621 44 × 2 = 1 + 0.988 024 846 225 694 265 013 154 173 372 694 790 554 566 661 242 88;
  • 31) 0.988 024 846 225 694 265 013 154 173 372 694 790 554 566 661 242 88 × 2 = 1 + 0.976 049 692 451 388 530 026 308 346 745 389 581 109 133 322 485 76;
  • 32) 0.976 049 692 451 388 530 026 308 346 745 389 581 109 133 322 485 76 × 2 = 1 + 0.952 099 384 902 777 060 052 616 693 490 779 162 218 266 644 971 52;
  • 33) 0.952 099 384 902 777 060 052 616 693 490 779 162 218 266 644 971 52 × 2 = 1 + 0.904 198 769 805 554 120 105 233 386 981 558 324 436 533 289 943 04;
  • 34) 0.904 198 769 805 554 120 105 233 386 981 558 324 436 533 289 943 04 × 2 = 1 + 0.808 397 539 611 108 240 210 466 773 963 116 648 873 066 579 886 08;
  • 35) 0.808 397 539 611 108 240 210 466 773 963 116 648 873 066 579 886 08 × 2 = 1 + 0.616 795 079 222 216 480 420 933 547 926 233 297 746 133 159 772 16;
  • 36) 0.616 795 079 222 216 480 420 933 547 926 233 297 746 133 159 772 16 × 2 = 1 + 0.233 590 158 444 432 960 841 867 095 852 466 595 492 266 319 544 32;
  • 37) 0.233 590 158 444 432 960 841 867 095 852 466 595 492 266 319 544 32 × 2 = 0 + 0.467 180 316 888 865 921 683 734 191 704 933 190 984 532 639 088 64;
  • 38) 0.467 180 316 888 865 921 683 734 191 704 933 190 984 532 639 088 64 × 2 = 0 + 0.934 360 633 777 731 843 367 468 383 409 866 381 969 065 278 177 28;
  • 39) 0.934 360 633 777 731 843 367 468 383 409 866 381 969 065 278 177 28 × 2 = 1 + 0.868 721 267 555 463 686 734 936 766 819 732 763 938 130 556 354 56;
  • 40) 0.868 721 267 555 463 686 734 936 766 819 732 763 938 130 556 354 56 × 2 = 1 + 0.737 442 535 110 927 373 469 873 533 639 465 527 876 261 112 709 12;
  • 41) 0.737 442 535 110 927 373 469 873 533 639 465 527 876 261 112 709 12 × 2 = 1 + 0.474 885 070 221 854 746 939 747 067 278 931 055 752 522 225 418 24;
  • 42) 0.474 885 070 221 854 746 939 747 067 278 931 055 752 522 225 418 24 × 2 = 0 + 0.949 770 140 443 709 493 879 494 134 557 862 111 505 044 450 836 48;
  • 43) 0.949 770 140 443 709 493 879 494 134 557 862 111 505 044 450 836 48 × 2 = 1 + 0.899 540 280 887 418 987 758 988 269 115 724 223 010 088 901 672 96;
  • 44) 0.899 540 280 887 418 987 758 988 269 115 724 223 010 088 901 672 96 × 2 = 1 + 0.799 080 561 774 837 975 517 976 538 231 448 446 020 177 803 345 92;
  • 45) 0.799 080 561 774 837 975 517 976 538 231 448 446 020 177 803 345 92 × 2 = 1 + 0.598 161 123 549 675 951 035 953 076 462 896 892 040 355 606 691 84;
  • 46) 0.598 161 123 549 675 951 035 953 076 462 896 892 040 355 606 691 84 × 2 = 1 + 0.196 322 247 099 351 902 071 906 152 925 793 784 080 711 213 383 68;
  • 47) 0.196 322 247 099 351 902 071 906 152 925 793 784 080 711 213 383 68 × 2 = 0 + 0.392 644 494 198 703 804 143 812 305 851 587 568 161 422 426 767 36;
  • 48) 0.392 644 494 198 703 804 143 812 305 851 587 568 161 422 426 767 36 × 2 = 0 + 0.785 288 988 397 407 608 287 624 611 703 175 136 322 844 853 534 72;
  • 49) 0.785 288 988 397 407 608 287 624 611 703 175 136 322 844 853 534 72 × 2 = 1 + 0.570 577 976 794 815 216 575 249 223 406 350 272 645 689 707 069 44;
  • 50) 0.570 577 976 794 815 216 575 249 223 406 350 272 645 689 707 069 44 × 2 = 1 + 0.141 155 953 589 630 433 150 498 446 812 700 545 291 379 414 138 88;
  • 51) 0.141 155 953 589 630 433 150 498 446 812 700 545 291 379 414 138 88 × 2 = 0 + 0.282 311 907 179 260 866 300 996 893 625 401 090 582 758 828 277 76;
  • 52) 0.282 311 907 179 260 866 300 996 893 625 401 090 582 758 828 277 76 × 2 = 0 + 0.564 623 814 358 521 732 601 993 787 250 802 181 165 517 656 555 52;
  • 53) 0.564 623 814 358 521 732 601 993 787 250 802 181 165 517 656 555 52 × 2 = 1 + 0.129 247 628 717 043 465 203 987 574 501 604 362 331 035 313 111 04;

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:


0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37(10) =

0.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2)

5. Positive number before normalization:

1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37(10) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 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:


1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37(10) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 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):
1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 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. 0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1 =

0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 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) =
011 1111 1111

Mantissa (52 bits) =
0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100


Decimal number 1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 382 37 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100

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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