-0.000 806 264 623 585 362 514 063 654 156 856 105 124 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 124(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 124(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 124| = 0.000 806 264 623 585 362 514 063 654 156 856 105 124


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 124.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 124 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 248;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 248 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 496;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 496 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 992;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 992 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 681 984;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 681 984 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 363 968;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 363 968 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 727 936;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 727 936 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 455 872;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 455 872 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 911 744;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 911 744 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 823 488;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 823 488 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 646 976;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 646 976 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 293 952;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 293 952 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 587 904;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 587 904 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 213 175 808;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 213 175 808 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 426 351 616;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 426 351 616 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 852 703 232;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 852 703 232 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 705 406 464;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 705 406 464 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 410 812 928;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 410 812 928 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 821 625 856;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 821 625 856 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 643 251 712;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 643 251 712 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 286 503 424;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 286 503 424 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 573 006 848;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 573 006 848 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 189 146 013 696;
  • 23) 0.718 935 762 580 334 187 240 884 718 189 146 013 696 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 378 292 027 392;
  • 24) 0.437 871 525 160 668 374 481 769 436 378 292 027 392 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 756 584 054 784;
  • 25) 0.875 743 050 321 336 748 963 538 872 756 584 054 784 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 513 168 109 568;
  • 26) 0.751 486 100 642 673 497 927 077 745 513 168 109 568 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 026 336 219 136;
  • 27) 0.502 972 201 285 346 995 854 155 491 026 336 219 136 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 052 672 438 272;
  • 28) 0.005 944 402 570 693 991 708 310 982 052 672 438 272 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 105 344 876 544;
  • 29) 0.011 888 805 141 387 983 416 621 964 105 344 876 544 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 210 689 753 088;
  • 30) 0.023 777 610 282 775 966 833 243 928 210 689 753 088 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 421 379 506 176;
  • 31) 0.047 555 220 565 551 933 666 487 856 421 379 506 176 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 842 759 012 352;
  • 32) 0.095 110 441 131 103 867 332 975 712 842 759 012 352 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 685 518 024 704;
  • 33) 0.190 220 882 262 207 734 665 951 425 685 518 024 704 × 2 = 0 + 0.380 441 764 524 415 469 331 902 851 371 036 049 408;
  • 34) 0.380 441 764 524 415 469 331 902 851 371 036 049 408 × 2 = 0 + 0.760 883 529 048 830 938 663 805 702 742 072 098 816;
  • 35) 0.760 883 529 048 830 938 663 805 702 742 072 098 816 × 2 = 1 + 0.521 767 058 097 661 877 327 611 405 484 144 197 632;
  • 36) 0.521 767 058 097 661 877 327 611 405 484 144 197 632 × 2 = 1 + 0.043 534 116 195 323 754 655 222 810 968 288 395 264;
  • 37) 0.043 534 116 195 323 754 655 222 810 968 288 395 264 × 2 = 0 + 0.087 068 232 390 647 509 310 445 621 936 576 790 528;
  • 38) 0.087 068 232 390 647 509 310 445 621 936 576 790 528 × 2 = 0 + 0.174 136 464 781 295 018 620 891 243 873 153 581 056;
  • 39) 0.174 136 464 781 295 018 620 891 243 873 153 581 056 × 2 = 0 + 0.348 272 929 562 590 037 241 782 487 746 307 162 112;
  • 40) 0.348 272 929 562 590 037 241 782 487 746 307 162 112 × 2 = 0 + 0.696 545 859 125 180 074 483 564 975 492 614 324 224;
  • 41) 0.696 545 859 125 180 074 483 564 975 492 614 324 224 × 2 = 1 + 0.393 091 718 250 360 148 967 129 950 985 228 648 448;
  • 42) 0.393 091 718 250 360 148 967 129 950 985 228 648 448 × 2 = 0 + 0.786 183 436 500 720 297 934 259 901 970 457 296 896;
  • 43) 0.786 183 436 500 720 297 934 259 901 970 457 296 896 × 2 = 1 + 0.572 366 873 001 440 595 868 519 803 940 914 593 792;
  • 44) 0.572 366 873 001 440 595 868 519 803 940 914 593 792 × 2 = 1 + 0.144 733 746 002 881 191 737 039 607 881 829 187 584;
  • 45) 0.144 733 746 002 881 191 737 039 607 881 829 187 584 × 2 = 0 + 0.289 467 492 005 762 383 474 079 215 763 658 375 168;
  • 46) 0.289 467 492 005 762 383 474 079 215 763 658 375 168 × 2 = 0 + 0.578 934 984 011 524 766 948 158 431 527 316 750 336;
  • 47) 0.578 934 984 011 524 766 948 158 431 527 316 750 336 × 2 = 1 + 0.157 869 968 023 049 533 896 316 863 054 633 500 672;
  • 48) 0.157 869 968 023 049 533 896 316 863 054 633 500 672 × 2 = 0 + 0.315 739 936 046 099 067 792 633 726 109 267 001 344;
  • 49) 0.315 739 936 046 099 067 792 633 726 109 267 001 344 × 2 = 0 + 0.631 479 872 092 198 135 585 267 452 218 534 002 688;
  • 50) 0.631 479 872 092 198 135 585 267 452 218 534 002 688 × 2 = 1 + 0.262 959 744 184 396 271 170 534 904 437 068 005 376;
  • 51) 0.262 959 744 184 396 271 170 534 904 437 068 005 376 × 2 = 0 + 0.525 919 488 368 792 542 341 069 808 874 136 010 752;
  • 52) 0.525 919 488 368 792 542 341 069 808 874 136 010 752 × 2 = 1 + 0.051 838 976 737 585 084 682 139 617 748 272 021 504;
  • 53) 0.051 838 976 737 585 084 682 139 617 748 272 021 504 × 2 = 0 + 0.103 677 953 475 170 169 364 279 235 496 544 043 008;
  • 54) 0.103 677 953 475 170 169 364 279 235 496 544 043 008 × 2 = 0 + 0.207 355 906 950 340 338 728 558 470 993 088 086 016;
  • 55) 0.207 355 906 950 340 338 728 558 470 993 088 086 016 × 2 = 0 + 0.414 711 813 900 680 677 457 116 941 986 176 172 032;
  • 56) 0.414 711 813 900 680 677 457 116 941 986 176 172 032 × 2 = 0 + 0.829 423 627 801 361 354 914 233 883 972 352 344 064;
  • 57) 0.829 423 627 801 361 354 914 233 883 972 352 344 064 × 2 = 1 + 0.658 847 255 602 722 709 828 467 767 944 704 688 128;
  • 58) 0.658 847 255 602 722 709 828 467 767 944 704 688 128 × 2 = 1 + 0.317 694 511 205 445 419 656 935 535 889 409 376 256;
  • 59) 0.317 694 511 205 445 419 656 935 535 889 409 376 256 × 2 = 0 + 0.635 389 022 410 890 839 313 871 071 778 818 752 512;
  • 60) 0.635 389 022 410 890 839 313 871 071 778 818 752 512 × 2 = 1 + 0.270 778 044 821 781 678 627 742 143 557 637 505 024;
  • 61) 0.270 778 044 821 781 678 627 742 143 557 637 505 024 × 2 = 0 + 0.541 556 089 643 563 357 255 484 287 115 275 010 048;
  • 62) 0.541 556 089 643 563 357 255 484 287 115 275 010 048 × 2 = 1 + 0.083 112 179 287 126 714 510 968 574 230 550 020 096;
  • 63) 0.083 112 179 287 126 714 510 968 574 230 550 020 096 × 2 = 0 + 0.166 224 358 574 253 429 021 937 148 461 100 040 192;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 124(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 124(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 124(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 124 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

Share

» Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 126 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 minutes

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