1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282(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
1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282(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: 1 100 011.
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;
  • 1 100 011 ÷ 2 = 550 005 + 1;
  • 550 005 ÷ 2 = 275 002 + 1;
  • 275 002 ÷ 2 = 137 501 + 0;
  • 137 501 ÷ 2 = 68 750 + 1;
  • 68 750 ÷ 2 = 34 375 + 0;
  • 34 375 ÷ 2 = 17 187 + 1;
  • 17 187 ÷ 2 = 8 593 + 1;
  • 8 593 ÷ 2 = 4 296 + 1;
  • 4 296 ÷ 2 = 2 148 + 0;
  • 2 148 ÷ 2 = 1 074 + 0;
  • 1 074 ÷ 2 = 537 + 0;
  • 537 ÷ 2 = 268 + 1;
  • 268 ÷ 2 = 134 + 0;
  • 134 ÷ 2 = 67 + 0;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 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.

1 100 011(10) =

1 0000 1100 1000 1110 1011(2)


3. Convert to binary (base 2) the fractional part: 0.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282.

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) 0.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282 × 2 = 0 + 0.222 222 020 222 000 020 200 022 220 202 220 000 202 000 222 564;
  • 2) 0.222 222 020 222 000 020 200 022 220 202 220 000 202 000 222 564 × 2 = 0 + 0.444 444 040 444 000 040 400 044 440 404 440 000 404 000 445 128;
  • 3) 0.444 444 040 444 000 040 400 044 440 404 440 000 404 000 445 128 × 2 = 0 + 0.888 888 080 888 000 080 800 088 880 808 880 000 808 000 890 256;
  • 4) 0.888 888 080 888 000 080 800 088 880 808 880 000 808 000 890 256 × 2 = 1 + 0.777 776 161 776 000 161 600 177 761 617 760 001 616 001 780 512;
  • 5) 0.777 776 161 776 000 161 600 177 761 617 760 001 616 001 780 512 × 2 = 1 + 0.555 552 323 552 000 323 200 355 523 235 520 003 232 003 561 024;
  • 6) 0.555 552 323 552 000 323 200 355 523 235 520 003 232 003 561 024 × 2 = 1 + 0.111 104 647 104 000 646 400 711 046 471 040 006 464 007 122 048;
  • 7) 0.111 104 647 104 000 646 400 711 046 471 040 006 464 007 122 048 × 2 = 0 + 0.222 209 294 208 001 292 801 422 092 942 080 012 928 014 244 096;
  • 8) 0.222 209 294 208 001 292 801 422 092 942 080 012 928 014 244 096 × 2 = 0 + 0.444 418 588 416 002 585 602 844 185 884 160 025 856 028 488 192;
  • 9) 0.444 418 588 416 002 585 602 844 185 884 160 025 856 028 488 192 × 2 = 0 + 0.888 837 176 832 005 171 205 688 371 768 320 051 712 056 976 384;
  • 10) 0.888 837 176 832 005 171 205 688 371 768 320 051 712 056 976 384 × 2 = 1 + 0.777 674 353 664 010 342 411 376 743 536 640 103 424 113 952 768;
  • 11) 0.777 674 353 664 010 342 411 376 743 536 640 103 424 113 952 768 × 2 = 1 + 0.555 348 707 328 020 684 822 753 487 073 280 206 848 227 905 536;
  • 12) 0.555 348 707 328 020 684 822 753 487 073 280 206 848 227 905 536 × 2 = 1 + 0.110 697 414 656 041 369 645 506 974 146 560 413 696 455 811 072;
  • 13) 0.110 697 414 656 041 369 645 506 974 146 560 413 696 455 811 072 × 2 = 0 + 0.221 394 829 312 082 739 291 013 948 293 120 827 392 911 622 144;
  • 14) 0.221 394 829 312 082 739 291 013 948 293 120 827 392 911 622 144 × 2 = 0 + 0.442 789 658 624 165 478 582 027 896 586 241 654 785 823 244 288;
  • 15) 0.442 789 658 624 165 478 582 027 896 586 241 654 785 823 244 288 × 2 = 0 + 0.885 579 317 248 330 957 164 055 793 172 483 309 571 646 488 576;
  • 16) 0.885 579 317 248 330 957 164 055 793 172 483 309 571 646 488 576 × 2 = 1 + 0.771 158 634 496 661 914 328 111 586 344 966 619 143 292 977 152;
  • 17) 0.771 158 634 496 661 914 328 111 586 344 966 619 143 292 977 152 × 2 = 1 + 0.542 317 268 993 323 828 656 223 172 689 933 238 286 585 954 304;
  • 18) 0.542 317 268 993 323 828 656 223 172 689 933 238 286 585 954 304 × 2 = 1 + 0.084 634 537 986 647 657 312 446 345 379 866 476 573 171 908 608;
  • 19) 0.084 634 537 986 647 657 312 446 345 379 866 476 573 171 908 608 × 2 = 0 + 0.169 269 075 973 295 314 624 892 690 759 732 953 146 343 817 216;
  • 20) 0.169 269 075 973 295 314 624 892 690 759 732 953 146 343 817 216 × 2 = 0 + 0.338 538 151 946 590 629 249 785 381 519 465 906 292 687 634 432;
  • 21) 0.338 538 151 946 590 629 249 785 381 519 465 906 292 687 634 432 × 2 = 0 + 0.677 076 303 893 181 258 499 570 763 038 931 812 585 375 268 864;
  • 22) 0.677 076 303 893 181 258 499 570 763 038 931 812 585 375 268 864 × 2 = 1 + 0.354 152 607 786 362 516 999 141 526 077 863 625 170 750 537 728;
  • 23) 0.354 152 607 786 362 516 999 141 526 077 863 625 170 750 537 728 × 2 = 0 + 0.708 305 215 572 725 033 998 283 052 155 727 250 341 501 075 456;
  • 24) 0.708 305 215 572 725 033 998 283 052 155 727 250 341 501 075 456 × 2 = 1 + 0.416 610 431 145 450 067 996 566 104 311 454 500 683 002 150 912;
  • 25) 0.416 610 431 145 450 067 996 566 104 311 454 500 683 002 150 912 × 2 = 0 + 0.833 220 862 290 900 135 993 132 208 622 909 001 366 004 301 824;
  • 26) 0.833 220 862 290 900 135 993 132 208 622 909 001 366 004 301 824 × 2 = 1 + 0.666 441 724 581 800 271 986 264 417 245 818 002 732 008 603 648;
  • 27) 0.666 441 724 581 800 271 986 264 417 245 818 002 732 008 603 648 × 2 = 1 + 0.332 883 449 163 600 543 972 528 834 491 636 005 464 017 207 296;
  • 28) 0.332 883 449 163 600 543 972 528 834 491 636 005 464 017 207 296 × 2 = 0 + 0.665 766 898 327 201 087 945 057 668 983 272 010 928 034 414 592;
  • 29) 0.665 766 898 327 201 087 945 057 668 983 272 010 928 034 414 592 × 2 = 1 + 0.331 533 796 654 402 175 890 115 337 966 544 021 856 068 829 184;
  • 30) 0.331 533 796 654 402 175 890 115 337 966 544 021 856 068 829 184 × 2 = 0 + 0.663 067 593 308 804 351 780 230 675 933 088 043 712 137 658 368;
  • 31) 0.663 067 593 308 804 351 780 230 675 933 088 043 712 137 658 368 × 2 = 1 + 0.326 135 186 617 608 703 560 461 351 866 176 087 424 275 316 736;
  • 32) 0.326 135 186 617 608 703 560 461 351 866 176 087 424 275 316 736 × 2 = 0 + 0.652 270 373 235 217 407 120 922 703 732 352 174 848 550 633 472;
  • 33) 0.652 270 373 235 217 407 120 922 703 732 352 174 848 550 633 472 × 2 = 1 + 0.304 540 746 470 434 814 241 845 407 464 704 349 697 101 266 944;
  • 34) 0.304 540 746 470 434 814 241 845 407 464 704 349 697 101 266 944 × 2 = 0 + 0.609 081 492 940 869 628 483 690 814 929 408 699 394 202 533 888;
  • 35) 0.609 081 492 940 869 628 483 690 814 929 408 699 394 202 533 888 × 2 = 1 + 0.218 162 985 881 739 256 967 381 629 858 817 398 788 405 067 776;
  • 36) 0.218 162 985 881 739 256 967 381 629 858 817 398 788 405 067 776 × 2 = 0 + 0.436 325 971 763 478 513 934 763 259 717 634 797 576 810 135 552;
  • 37) 0.436 325 971 763 478 513 934 763 259 717 634 797 576 810 135 552 × 2 = 0 + 0.872 651 943 526 957 027 869 526 519 435 269 595 153 620 271 104;
  • 38) 0.872 651 943 526 957 027 869 526 519 435 269 595 153 620 271 104 × 2 = 1 + 0.745 303 887 053 914 055 739 053 038 870 539 190 307 240 542 208;
  • 39) 0.745 303 887 053 914 055 739 053 038 870 539 190 307 240 542 208 × 2 = 1 + 0.490 607 774 107 828 111 478 106 077 741 078 380 614 481 084 416;
  • 40) 0.490 607 774 107 828 111 478 106 077 741 078 380 614 481 084 416 × 2 = 0 + 0.981 215 548 215 656 222 956 212 155 482 156 761 228 962 168 832;
  • 41) 0.981 215 548 215 656 222 956 212 155 482 156 761 228 962 168 832 × 2 = 1 + 0.962 431 096 431 312 445 912 424 310 964 313 522 457 924 337 664;
  • 42) 0.962 431 096 431 312 445 912 424 310 964 313 522 457 924 337 664 × 2 = 1 + 0.924 862 192 862 624 891 824 848 621 928 627 044 915 848 675 328;
  • 43) 0.924 862 192 862 624 891 824 848 621 928 627 044 915 848 675 328 × 2 = 1 + 0.849 724 385 725 249 783 649 697 243 857 254 089 831 697 350 656;
  • 44) 0.849 724 385 725 249 783 649 697 243 857 254 089 831 697 350 656 × 2 = 1 + 0.699 448 771 450 499 567 299 394 487 714 508 179 663 394 701 312;
  • 45) 0.699 448 771 450 499 567 299 394 487 714 508 179 663 394 701 312 × 2 = 1 + 0.398 897 542 900 999 134 598 788 975 429 016 359 326 789 402 624;
  • 46) 0.398 897 542 900 999 134 598 788 975 429 016 359 326 789 402 624 × 2 = 0 + 0.797 795 085 801 998 269 197 577 950 858 032 718 653 578 805 248;
  • 47) 0.797 795 085 801 998 269 197 577 950 858 032 718 653 578 805 248 × 2 = 1 + 0.595 590 171 603 996 538 395 155 901 716 065 437 307 157 610 496;
  • 48) 0.595 590 171 603 996 538 395 155 901 716 065 437 307 157 610 496 × 2 = 1 + 0.191 180 343 207 993 076 790 311 803 432 130 874 614 315 220 992;
  • 49) 0.191 180 343 207 993 076 790 311 803 432 130 874 614 315 220 992 × 2 = 0 + 0.382 360 686 415 986 153 580 623 606 864 261 749 228 630 441 984;
  • 50) 0.382 360 686 415 986 153 580 623 606 864 261 749 228 630 441 984 × 2 = 0 + 0.764 721 372 831 972 307 161 247 213 728 523 498 457 260 883 968;
  • 51) 0.764 721 372 831 972 307 161 247 213 728 523 498 457 260 883 968 × 2 = 1 + 0.529 442 745 663 944 614 322 494 427 457 046 996 914 521 767 936;
  • 52) 0.529 442 745 663 944 614 322 494 427 457 046 996 914 521 767 936 × 2 = 1 + 0.058 885 491 327 889 228 644 988 854 914 093 993 829 043 535 872;
  • 53) 0.058 885 491 327 889 228 644 988 854 914 093 993 829 043 535 872 × 2 = 0 + 0.117 770 982 655 778 457 289 977 709 828 187 987 658 087 071 744;

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:


0.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282(10) =

0.0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0(2)

5. Positive number before normalization:

1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282(10) =

1 0000 1100 1000 1110 1011.0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 20 positions to the left, so that only one non zero digit remains to the left of it:


1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282(10) =

1 0000 1100 1000 1110 1011.0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0(2) =

1 0000 1100 1000 1110 1011.0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0(2) × 20 =

1.0000 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0(2) × 220


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): 20

Mantissa (not normalized):
1.0000 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010 1010 0110 1111 1011 0011 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

20 + 2(11-1) - 1 =

(20 + 1 023)(10) =

1 043(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;
  • 1 043 ÷ 2 = 521 + 1;
  • 521 ÷ 2 = 260 + 1;
  • 260 ÷ 2 = 130 + 0;
  • 130 ÷ 2 = 65 + 0;
  • 65 ÷ 2 = 32 + 1;
  • 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) =

1043(10) =

100 0001 0011(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 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010 1 0100 1101 1111 0110 0110 =

0000 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010


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 0001 0011

Mantissa (52 bits) =
0000 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010


Decimal number 1 100 011.111 111 010 111 000 010 100 011 110 101 110 000 101 000 111 282 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0001 0011 - 0000 1100 1000 1110 1011 0001 1100 0111 0001 1100 0101 0110 1010

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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