-0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 2| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 2 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 604 4;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 604 4 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 208 8;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 208 8 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 417 6;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 417 6 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 548 835 2;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 548 835 2 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 097 670 4;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 097 670 4 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 195 340 8;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 195 340 8 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 390 681 6;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 390 681 6 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 781 363 2;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 781 363 2 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 562 726 4;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 562 726 4 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 125 452 8;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 125 452 8 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 250 905 6;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 250 905 6 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 501 811 2;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 501 811 2 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 003 622 4;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 003 622 4 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 007 244 8;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 007 244 8 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 764 014 489 6;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 764 014 489 6 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 528 028 979 2;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 528 028 979 2 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 056 057 958 4;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 056 057 958 4 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 112 115 916 8;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 112 115 916 8 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 224 231 833 6;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 224 231 833 6 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 448 463 667 2;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 448 463 667 2 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 896 927 334 4;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 896 927 334 4 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 793 854 668 8;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 793 854 668 8 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 587 709 337 6;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 587 709 337 6 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 175 418 675 2;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 175 418 675 2 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 158 350 837 350 4;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 158 350 837 350 4 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 316 701 674 700 8;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 316 701 674 700 8 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 633 403 349 401 6;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 633 403 349 401 6 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 266 806 698 803 2;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 266 806 698 803 2 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 533 613 397 606 4;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 533 613 397 606 4 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 067 226 795 212 8;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 067 226 795 212 8 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 134 453 590 425 6;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 134 453 590 425 6 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 268 907 180 851 2;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 268 907 180 851 2 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 537 814 361 702 4;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 537 814 361 702 4 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 075 628 723 404 8;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 075 628 723 404 8 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 402 151 257 446 809 6;
  • 36) 0.521 767 058 097 661 877 327 611 402 402 151 257 446 809 6 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 804 302 514 893 619 2;
  • 37) 0.043 534 116 195 323 754 655 222 804 804 302 514 893 619 2 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 608 605 029 787 238 4;
  • 38) 0.087 068 232 390 647 509 310 445 609 608 605 029 787 238 4 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 217 210 059 574 476 8;
  • 39) 0.174 136 464 781 295 018 620 891 219 217 210 059 574 476 8 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 434 420 119 148 953 6;
  • 40) 0.348 272 929 562 590 037 241 782 438 434 420 119 148 953 6 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 868 840 238 297 907 2;
  • 41) 0.696 545 859 125 180 074 483 564 876 868 840 238 297 907 2 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 737 680 476 595 814 4;
  • 42) 0.393 091 718 250 360 148 967 129 753 737 680 476 595 814 4 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 475 360 953 191 628 8;
  • 43) 0.786 183 436 500 720 297 934 259 507 475 360 953 191 628 8 × 2 = 1 + 0.572 366 873 001 440 595 868 519 014 950 721 906 383 257 6;
  • 44) 0.572 366 873 001 440 595 868 519 014 950 721 906 383 257 6 × 2 = 1 + 0.144 733 746 002 881 191 737 038 029 901 443 812 766 515 2;
  • 45) 0.144 733 746 002 881 191 737 038 029 901 443 812 766 515 2 × 2 = 0 + 0.289 467 492 005 762 383 474 076 059 802 887 625 533 030 4;
  • 46) 0.289 467 492 005 762 383 474 076 059 802 887 625 533 030 4 × 2 = 0 + 0.578 934 984 011 524 766 948 152 119 605 775 251 066 060 8;
  • 47) 0.578 934 984 011 524 766 948 152 119 605 775 251 066 060 8 × 2 = 1 + 0.157 869 968 023 049 533 896 304 239 211 550 502 132 121 6;
  • 48) 0.157 869 968 023 049 533 896 304 239 211 550 502 132 121 6 × 2 = 0 + 0.315 739 936 046 099 067 792 608 478 423 101 004 264 243 2;
  • 49) 0.315 739 936 046 099 067 792 608 478 423 101 004 264 243 2 × 2 = 0 + 0.631 479 872 092 198 135 585 216 956 846 202 008 528 486 4;
  • 50) 0.631 479 872 092 198 135 585 216 956 846 202 008 528 486 4 × 2 = 1 + 0.262 959 744 184 396 271 170 433 913 692 404 017 056 972 8;
  • 51) 0.262 959 744 184 396 271 170 433 913 692 404 017 056 972 8 × 2 = 0 + 0.525 919 488 368 792 542 340 867 827 384 808 034 113 945 6;
  • 52) 0.525 919 488 368 792 542 340 867 827 384 808 034 113 945 6 × 2 = 1 + 0.051 838 976 737 585 084 681 735 654 769 616 068 227 891 2;
  • 53) 0.051 838 976 737 585 084 681 735 654 769 616 068 227 891 2 × 2 = 0 + 0.103 677 953 475 170 169 363 471 309 539 232 136 455 782 4;
  • 54) 0.103 677 953 475 170 169 363 471 309 539 232 136 455 782 4 × 2 = 0 + 0.207 355 906 950 340 338 726 942 619 078 464 272 911 564 8;
  • 55) 0.207 355 906 950 340 338 726 942 619 078 464 272 911 564 8 × 2 = 0 + 0.414 711 813 900 680 677 453 885 238 156 928 545 823 129 6;
  • 56) 0.414 711 813 900 680 677 453 885 238 156 928 545 823 129 6 × 2 = 0 + 0.829 423 627 801 361 354 907 770 476 313 857 091 646 259 2;
  • 57) 0.829 423 627 801 361 354 907 770 476 313 857 091 646 259 2 × 2 = 1 + 0.658 847 255 602 722 709 815 540 952 627 714 183 292 518 4;
  • 58) 0.658 847 255 602 722 709 815 540 952 627 714 183 292 518 4 × 2 = 1 + 0.317 694 511 205 445 419 631 081 905 255 428 366 585 036 8;
  • 59) 0.317 694 511 205 445 419 631 081 905 255 428 366 585 036 8 × 2 = 0 + 0.635 389 022 410 890 839 262 163 810 510 856 733 170 073 6;
  • 60) 0.635 389 022 410 890 839 262 163 810 510 856 733 170 073 6 × 2 = 1 + 0.270 778 044 821 781 678 524 327 621 021 713 466 340 147 2;
  • 61) 0.270 778 044 821 781 678 524 327 621 021 713 466 340 147 2 × 2 = 0 + 0.541 556 089 643 563 357 048 655 242 043 426 932 680 294 4;
  • 62) 0.541 556 089 643 563 357 048 655 242 043 426 932 680 294 4 × 2 = 1 + 0.083 112 179 287 126 714 097 310 484 086 853 865 360 588 8;
  • 63) 0.083 112 179 287 126 714 097 310 484 086 853 865 360 588 8 × 2 = 0 + 0.166 224 358 574 253 428 194 620 968 173 707 730 721 177 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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 302 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