-0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2


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

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.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 718 4;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 718 4 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 436 8;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 436 8 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 873 6;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 873 6 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 747 2;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 747 2 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 099 494 4;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 099 494 4 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 198 988 8;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 198 988 8 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 397 977 6;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 397 977 6 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 795 955 2;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 795 955 2 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 591 910 4;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 591 910 4 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 183 820 8;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 183 820 8 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 367 641 6;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 367 641 6 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 735 283 2;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 735 283 2 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 470 566 4;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 470 566 4 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 941 132 8;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 941 132 8 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 882 265 6;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 882 265 6 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 531 764 531 2;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 531 764 531 2 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 063 529 062 4;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 063 529 062 4 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 127 058 124 8;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 127 058 124 8 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 254 116 249 6;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 254 116 249 6 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 508 232 499 2;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 508 232 499 2 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 385 016 464 998 4;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 385 016 464 998 4 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 770 032 929 996 8;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 770 032 929 996 8 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 540 065 859 993 6;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 540 065 859 993 6 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 080 131 719 987 2;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 080 131 719 987 2 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 160 263 439 974 4;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 160 263 439 974 4 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 320 526 879 948 8;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 320 526 879 948 8 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 641 053 759 897 6;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 641 053 759 897 6 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 282 107 519 795 2;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 282 107 519 795 2 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 564 215 039 590 4;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 564 215 039 590 4 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 128 430 079 180 8;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 128 430 079 180 8 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 256 860 158 361 6;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 256 860 158 361 6 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 513 720 316 723 2;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 513 720 316 723 2 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 601 027 440 633 446 4;
  • 34) 0.380 441 764 524 415 469 331 902 850 601 027 440 633 446 4 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 202 054 881 266 892 8;
  • 35) 0.760 883 529 048 830 938 663 805 701 202 054 881 266 892 8 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 404 109 762 533 785 6;
  • 36) 0.521 767 058 097 661 877 327 611 402 404 109 762 533 785 6 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 808 219 525 067 571 2;
  • 37) 0.043 534 116 195 323 754 655 222 804 808 219 525 067 571 2 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 616 439 050 135 142 4;
  • 38) 0.087 068 232 390 647 509 310 445 609 616 439 050 135 142 4 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 232 878 100 270 284 8;
  • 39) 0.174 136 464 781 295 018 620 891 219 232 878 100 270 284 8 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 465 756 200 540 569 6;
  • 40) 0.348 272 929 562 590 037 241 782 438 465 756 200 540 569 6 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 931 512 401 081 139 2;
  • 41) 0.696 545 859 125 180 074 483 564 876 931 512 401 081 139 2 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 863 024 802 162 278 4;
  • 42) 0.393 091 718 250 360 148 967 129 753 863 024 802 162 278 4 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 726 049 604 324 556 8;
  • 43) 0.786 183 436 500 720 297 934 259 507 726 049 604 324 556 8 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 452 099 208 649 113 6;
  • 44) 0.572 366 873 001 440 595 868 519 015 452 099 208 649 113 6 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 904 198 417 298 227 2;
  • 45) 0.144 733 746 002 881 191 737 038 030 904 198 417 298 227 2 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 808 396 834 596 454 4;
  • 46) 0.289 467 492 005 762 383 474 076 061 808 396 834 596 454 4 × 2 = 0 + 0.578 934 984 011 524 766 948 152 123 616 793 669 192 908 8;
  • 47) 0.578 934 984 011 524 766 948 152 123 616 793 669 192 908 8 × 2 = 1 + 0.157 869 968 023 049 533 896 304 247 233 587 338 385 817 6;
  • 48) 0.157 869 968 023 049 533 896 304 247 233 587 338 385 817 6 × 2 = 0 + 0.315 739 936 046 099 067 792 608 494 467 174 676 771 635 2;
  • 49) 0.315 739 936 046 099 067 792 608 494 467 174 676 771 635 2 × 2 = 0 + 0.631 479 872 092 198 135 585 216 988 934 349 353 543 270 4;
  • 50) 0.631 479 872 092 198 135 585 216 988 934 349 353 543 270 4 × 2 = 1 + 0.262 959 744 184 396 271 170 433 977 868 698 707 086 540 8;
  • 51) 0.262 959 744 184 396 271 170 433 977 868 698 707 086 540 8 × 2 = 0 + 0.525 919 488 368 792 542 340 867 955 737 397 414 173 081 6;
  • 52) 0.525 919 488 368 792 542 340 867 955 737 397 414 173 081 6 × 2 = 1 + 0.051 838 976 737 585 084 681 735 911 474 794 828 346 163 2;
  • 53) 0.051 838 976 737 585 084 681 735 911 474 794 828 346 163 2 × 2 = 0 + 0.103 677 953 475 170 169 363 471 822 949 589 656 692 326 4;
  • 54) 0.103 677 953 475 170 169 363 471 822 949 589 656 692 326 4 × 2 = 0 + 0.207 355 906 950 340 338 726 943 645 899 179 313 384 652 8;
  • 55) 0.207 355 906 950 340 338 726 943 645 899 179 313 384 652 8 × 2 = 0 + 0.414 711 813 900 680 677 453 887 291 798 358 626 769 305 6;
  • 56) 0.414 711 813 900 680 677 453 887 291 798 358 626 769 305 6 × 2 = 0 + 0.829 423 627 801 361 354 907 774 583 596 717 253 538 611 2;
  • 57) 0.829 423 627 801 361 354 907 774 583 596 717 253 538 611 2 × 2 = 1 + 0.658 847 255 602 722 709 815 549 167 193 434 507 077 222 4;
  • 58) 0.658 847 255 602 722 709 815 549 167 193 434 507 077 222 4 × 2 = 1 + 0.317 694 511 205 445 419 631 098 334 386 869 014 154 444 8;
  • 59) 0.317 694 511 205 445 419 631 098 334 386 869 014 154 444 8 × 2 = 0 + 0.635 389 022 410 890 839 262 196 668 773 738 028 308 889 6;
  • 60) 0.635 389 022 410 890 839 262 196 668 773 738 028 308 889 6 × 2 = 1 + 0.270 778 044 821 781 678 524 393 337 547 476 056 617 779 2;
  • 61) 0.270 778 044 821 781 678 524 393 337 547 476 056 617 779 2 × 2 = 0 + 0.541 556 089 643 563 357 048 786 675 094 952 113 235 558 4;
  • 62) 0.541 556 089 643 563 357 048 786 675 094 952 113 235 558 4 × 2 = 1 + 0.083 112 179 287 126 714 097 573 350 189 904 226 471 116 8;
  • 63) 0.083 112 179 287 126 714 097 573 350 189 904 226 471 116 8 × 2 = 0 + 0.166 224 358 574 253 428 195 146 700 379 808 452 942 233 6;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 11 positions to the right, so that only one non zero digit remains to the left of it:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) × 20 =

1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010(2) × 2-11


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): -11

Mantissa (not normalized):
1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-11 + 1 023)(10) =

1 012(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;
  • 1 012 ÷ 2 = 506 + 0;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1012(10) =

011 1111 0100(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010 =

1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


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) =
011 1111 0100

Mantissa (52 bits) =
1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


Decimal number -0.000 806 264 623 585 362 514 063 654 156 856 105 034 359 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0100 - 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 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