-0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 545;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 545 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 09;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 09 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 18;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 18 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 548 36;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 548 36 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 096 72;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 096 72 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 193 44;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 193 44 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 386 88;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 386 88 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 773 76;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 773 76 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 547 52;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 547 52 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 095 04;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 095 04 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 190 08;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 190 08 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 380 16;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 380 16 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 440 760 32;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 440 760 32 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 881 520 64;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 881 520 64 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 763 041 28;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 763 041 28 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 526 082 56;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 526 082 56 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 052 165 12;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 052 165 12 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 104 330 24;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 104 330 24 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 208 660 48;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 208 660 48 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 417 320 96;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 417 320 96 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 834 641 92;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 834 641 92 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 669 283 84;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 669 283 84 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 338 567 68;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 338 567 68 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 078 677 135 36;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 078 677 135 36 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 157 354 270 72;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 157 354 270 72 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 314 708 541 44;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 314 708 541 44 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 629 417 082 88;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 629 417 082 88 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 258 834 165 76;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 258 834 165 76 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 517 668 331 52;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 517 668 331 52 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 035 336 663 04;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 035 336 663 04 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 070 673 326 08;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 070 673 326 08 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 141 346 652 16;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 141 346 652 16 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 282 693 304 32;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 282 693 304 32 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 200 565 386 608 64;
  • 35) 0.760 883 529 048 830 938 663 805 701 200 565 386 608 64 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 401 130 773 217 28;
  • 36) 0.521 767 058 097 661 877 327 611 402 401 130 773 217 28 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 802 261 546 434 56;
  • 37) 0.043 534 116 195 323 754 655 222 804 802 261 546 434 56 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 604 523 092 869 12;
  • 38) 0.087 068 232 390 647 509 310 445 609 604 523 092 869 12 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 209 046 185 738 24;
  • 39) 0.174 136 464 781 295 018 620 891 219 209 046 185 738 24 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 418 092 371 476 48;
  • 40) 0.348 272 929 562 590 037 241 782 438 418 092 371 476 48 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 836 184 742 952 96;
  • 41) 0.696 545 859 125 180 074 483 564 876 836 184 742 952 96 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 672 369 485 905 92;
  • 42) 0.393 091 718 250 360 148 967 129 753 672 369 485 905 92 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 344 738 971 811 84;
  • 43) 0.786 183 436 500 720 297 934 259 507 344 738 971 811 84 × 2 = 1 + 0.572 366 873 001 440 595 868 519 014 689 477 943 623 68;
  • 44) 0.572 366 873 001 440 595 868 519 014 689 477 943 623 68 × 2 = 1 + 0.144 733 746 002 881 191 737 038 029 378 955 887 247 36;
  • 45) 0.144 733 746 002 881 191 737 038 029 378 955 887 247 36 × 2 = 0 + 0.289 467 492 005 762 383 474 076 058 757 911 774 494 72;
  • 46) 0.289 467 492 005 762 383 474 076 058 757 911 774 494 72 × 2 = 0 + 0.578 934 984 011 524 766 948 152 117 515 823 548 989 44;
  • 47) 0.578 934 984 011 524 766 948 152 117 515 823 548 989 44 × 2 = 1 + 0.157 869 968 023 049 533 896 304 235 031 647 097 978 88;
  • 48) 0.157 869 968 023 049 533 896 304 235 031 647 097 978 88 × 2 = 0 + 0.315 739 936 046 099 067 792 608 470 063 294 195 957 76;
  • 49) 0.315 739 936 046 099 067 792 608 470 063 294 195 957 76 × 2 = 0 + 0.631 479 872 092 198 135 585 216 940 126 588 391 915 52;
  • 50) 0.631 479 872 092 198 135 585 216 940 126 588 391 915 52 × 2 = 1 + 0.262 959 744 184 396 271 170 433 880 253 176 783 831 04;
  • 51) 0.262 959 744 184 396 271 170 433 880 253 176 783 831 04 × 2 = 0 + 0.525 919 488 368 792 542 340 867 760 506 353 567 662 08;
  • 52) 0.525 919 488 368 792 542 340 867 760 506 353 567 662 08 × 2 = 1 + 0.051 838 976 737 585 084 681 735 521 012 707 135 324 16;
  • 53) 0.051 838 976 737 585 084 681 735 521 012 707 135 324 16 × 2 = 0 + 0.103 677 953 475 170 169 363 471 042 025 414 270 648 32;
  • 54) 0.103 677 953 475 170 169 363 471 042 025 414 270 648 32 × 2 = 0 + 0.207 355 906 950 340 338 726 942 084 050 828 541 296 64;
  • 55) 0.207 355 906 950 340 338 726 942 084 050 828 541 296 64 × 2 = 0 + 0.414 711 813 900 680 677 453 884 168 101 657 082 593 28;
  • 56) 0.414 711 813 900 680 677 453 884 168 101 657 082 593 28 × 2 = 0 + 0.829 423 627 801 361 354 907 768 336 203 314 165 186 56;
  • 57) 0.829 423 627 801 361 354 907 768 336 203 314 165 186 56 × 2 = 1 + 0.658 847 255 602 722 709 815 536 672 406 628 330 373 12;
  • 58) 0.658 847 255 602 722 709 815 536 672 406 628 330 373 12 × 2 = 1 + 0.317 694 511 205 445 419 631 073 344 813 256 660 746 24;
  • 59) 0.317 694 511 205 445 419 631 073 344 813 256 660 746 24 × 2 = 0 + 0.635 389 022 410 890 839 262 146 689 626 513 321 492 48;
  • 60) 0.635 389 022 410 890 839 262 146 689 626 513 321 492 48 × 2 = 1 + 0.270 778 044 821 781 678 524 293 379 253 026 642 984 96;
  • 61) 0.270 778 044 821 781 678 524 293 379 253 026 642 984 96 × 2 = 0 + 0.541 556 089 643 563 357 048 586 758 506 053 285 969 92;
  • 62) 0.541 556 089 643 563 357 048 586 758 506 053 285 969 92 × 2 = 1 + 0.083 112 179 287 126 714 097 173 517 012 106 571 939 84;
  • 63) 0.083 112 179 287 126 714 097 173 517 012 106 571 939 84 × 2 = 0 + 0.166 224 358 574 253 428 194 347 034 024 213 143 879 68;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 272 5 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