0.000 020 830 729 321 671 205 134 999 154 509 659 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 659 37(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
0.000 020 830 729 321 671 205 134 999 154 509 659 37(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: 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;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 020 830 729 321 671 205 134 999 154 509 659 37.

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) 0.000 020 830 729 321 671 205 134 999 154 509 659 37 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 318 74;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 318 74 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 637 48;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 637 48 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 274 96;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 274 96 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 549 92;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 549 92 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 099 84;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 099 84 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 199 68;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 199 68 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 399 36;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 399 36 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 472 798 72;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 472 798 72 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 945 597 44;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 945 597 44 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 891 194 88;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 891 194 88 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 782 389 76;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 782 389 76 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 564 779 52;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 564 779 52 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 129 559 04;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 129 559 04 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 259 118 08;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 259 118 08 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 518 236 16;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 518 236 16 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 036 472 32;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 036 472 32 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 072 944 64;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 072 944 64 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 145 889 28;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 145 889 28 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 291 778 56;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 291 778 56 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 120 583 557 12;
  • 21) 0.842 602 829 200 705 595 636 873 439 120 583 557 12 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 241 167 114 24;
  • 22) 0.685 205 658 401 411 191 273 746 878 241 167 114 24 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 482 334 228 48;
  • 23) 0.370 411 316 802 822 382 547 493 756 482 334 228 48 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 964 668 456 96;
  • 24) 0.740 822 633 605 644 765 094 987 512 964 668 456 96 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 929 336 913 92;
  • 25) 0.481 645 267 211 289 530 189 975 025 929 336 913 92 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 858 673 827 84;
  • 26) 0.963 290 534 422 579 060 379 950 051 858 673 827 84 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 717 347 655 68;
  • 27) 0.926 581 068 845 158 120 759 900 103 717 347 655 68 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 434 695 311 36;
  • 28) 0.853 162 137 690 316 241 519 800 207 434 695 311 36 × 2 = 1 + 0.706 324 275 380 632 483 039 600 414 869 390 622 72;
  • 29) 0.706 324 275 380 632 483 039 600 414 869 390 622 72 × 2 = 1 + 0.412 648 550 761 264 966 079 200 829 738 781 245 44;
  • 30) 0.412 648 550 761 264 966 079 200 829 738 781 245 44 × 2 = 0 + 0.825 297 101 522 529 932 158 401 659 477 562 490 88;
  • 31) 0.825 297 101 522 529 932 158 401 659 477 562 490 88 × 2 = 1 + 0.650 594 203 045 059 864 316 803 318 955 124 981 76;
  • 32) 0.650 594 203 045 059 864 316 803 318 955 124 981 76 × 2 = 1 + 0.301 188 406 090 119 728 633 606 637 910 249 963 52;
  • 33) 0.301 188 406 090 119 728 633 606 637 910 249 963 52 × 2 = 0 + 0.602 376 812 180 239 457 267 213 275 820 499 927 04;
  • 34) 0.602 376 812 180 239 457 267 213 275 820 499 927 04 × 2 = 1 + 0.204 753 624 360 478 914 534 426 551 640 999 854 08;
  • 35) 0.204 753 624 360 478 914 534 426 551 640 999 854 08 × 2 = 0 + 0.409 507 248 720 957 829 068 853 103 281 999 708 16;
  • 36) 0.409 507 248 720 957 829 068 853 103 281 999 708 16 × 2 = 0 + 0.819 014 497 441 915 658 137 706 206 563 999 416 32;
  • 37) 0.819 014 497 441 915 658 137 706 206 563 999 416 32 × 2 = 1 + 0.638 028 994 883 831 316 275 412 413 127 998 832 64;
  • 38) 0.638 028 994 883 831 316 275 412 413 127 998 832 64 × 2 = 1 + 0.276 057 989 767 662 632 550 824 826 255 997 665 28;
  • 39) 0.276 057 989 767 662 632 550 824 826 255 997 665 28 × 2 = 0 + 0.552 115 979 535 325 265 101 649 652 511 995 330 56;
  • 40) 0.552 115 979 535 325 265 101 649 652 511 995 330 56 × 2 = 1 + 0.104 231 959 070 650 530 203 299 305 023 990 661 12;
  • 41) 0.104 231 959 070 650 530 203 299 305 023 990 661 12 × 2 = 0 + 0.208 463 918 141 301 060 406 598 610 047 981 322 24;
  • 42) 0.208 463 918 141 301 060 406 598 610 047 981 322 24 × 2 = 0 + 0.416 927 836 282 602 120 813 197 220 095 962 644 48;
  • 43) 0.416 927 836 282 602 120 813 197 220 095 962 644 48 × 2 = 0 + 0.833 855 672 565 204 241 626 394 440 191 925 288 96;
  • 44) 0.833 855 672 565 204 241 626 394 440 191 925 288 96 × 2 = 1 + 0.667 711 345 130 408 483 252 788 880 383 850 577 92;
  • 45) 0.667 711 345 130 408 483 252 788 880 383 850 577 92 × 2 = 1 + 0.335 422 690 260 816 966 505 577 760 767 701 155 84;
  • 46) 0.335 422 690 260 816 966 505 577 760 767 701 155 84 × 2 = 0 + 0.670 845 380 521 633 933 011 155 521 535 402 311 68;
  • 47) 0.670 845 380 521 633 933 011 155 521 535 402 311 68 × 2 = 1 + 0.341 690 761 043 267 866 022 311 043 070 804 623 36;
  • 48) 0.341 690 761 043 267 866 022 311 043 070 804 623 36 × 2 = 0 + 0.683 381 522 086 535 732 044 622 086 141 609 246 72;
  • 49) 0.683 381 522 086 535 732 044 622 086 141 609 246 72 × 2 = 1 + 0.366 763 044 173 071 464 089 244 172 283 218 493 44;
  • 50) 0.366 763 044 173 071 464 089 244 172 283 218 493 44 × 2 = 0 + 0.733 526 088 346 142 928 178 488 344 566 436 986 88;
  • 51) 0.733 526 088 346 142 928 178 488 344 566 436 986 88 × 2 = 1 + 0.467 052 176 692 285 856 356 976 689 132 873 973 76;
  • 52) 0.467 052 176 692 285 856 356 976 689 132 873 973 76 × 2 = 0 + 0.934 104 353 384 571 712 713 953 378 265 747 947 52;
  • 53) 0.934 104 353 384 571 712 713 953 378 265 747 947 52 × 2 = 1 + 0.868 208 706 769 143 425 427 906 756 531 495 895 04;
  • 54) 0.868 208 706 769 143 425 427 906 756 531 495 895 04 × 2 = 1 + 0.736 417 413 538 286 850 855 813 513 062 991 790 08;
  • 55) 0.736 417 413 538 286 850 855 813 513 062 991 790 08 × 2 = 1 + 0.472 834 827 076 573 701 711 627 026 125 983 580 16;
  • 56) 0.472 834 827 076 573 701 711 627 026 125 983 580 16 × 2 = 0 + 0.945 669 654 153 147 403 423 254 052 251 967 160 32;
  • 57) 0.945 669 654 153 147 403 423 254 052 251 967 160 32 × 2 = 1 + 0.891 339 308 306 294 806 846 508 104 503 934 320 64;
  • 58) 0.891 339 308 306 294 806 846 508 104 503 934 320 64 × 2 = 1 + 0.782 678 616 612 589 613 693 016 209 007 868 641 28;
  • 59) 0.782 678 616 612 589 613 693 016 209 007 868 641 28 × 2 = 1 + 0.565 357 233 225 179 227 386 032 418 015 737 282 56;
  • 60) 0.565 357 233 225 179 227 386 032 418 015 737 282 56 × 2 = 1 + 0.130 714 466 450 358 454 772 064 836 031 474 565 12;
  • 61) 0.130 714 466 450 358 454 772 064 836 031 474 565 12 × 2 = 0 + 0.261 428 932 900 716 909 544 129 672 062 949 130 24;
  • 62) 0.261 428 932 900 716 909 544 129 672 062 949 130 24 × 2 = 0 + 0.522 857 865 801 433 819 088 259 344 125 898 260 48;
  • 63) 0.522 857 865 801 433 819 088 259 344 125 898 260 48 × 2 = 1 + 0.045 715 731 602 867 638 176 518 688 251 796 520 96;
  • 64) 0.045 715 731 602 867 638 176 518 688 251 796 520 96 × 2 = 0 + 0.091 431 463 205 735 276 353 037 376 503 593 041 92;
  • 65) 0.091 431 463 205 735 276 353 037 376 503 593 041 92 × 2 = 0 + 0.182 862 926 411 470 552 706 074 753 007 186 083 84;
  • 66) 0.182 862 926 411 470 552 706 074 753 007 186 083 84 × 2 = 0 + 0.365 725 852 822 941 105 412 149 506 014 372 167 68;
  • 67) 0.365 725 852 822 941 105 412 149 506 014 372 167 68 × 2 = 0 + 0.731 451 705 645 882 210 824 299 012 028 744 335 36;
  • 68) 0.731 451 705 645 882 210 824 299 012 028 744 335 36 × 2 = 1 + 0.462 903 411 291 764 421 648 598 024 057 488 670 72;

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:


0.000 020 830 729 321 671 205 134 999 154 509 659 37(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 154 509 659 37(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 154 509 659 37(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1007(10) =

011 1110 1111(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


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) =
011 1110 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 154 509 659 37 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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