0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02(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
0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02(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: 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02.

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) 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02 × 2 = 0 + 0.018 469 135 621 974 469 134 196 665 975 308 642 001 975 397 530 04;
  • 2) 0.018 469 135 621 974 469 134 196 665 975 308 642 001 975 397 530 04 × 2 = 0 + 0.036 938 271 243 948 938 268 393 331 950 617 284 003 950 795 060 08;
  • 3) 0.036 938 271 243 948 938 268 393 331 950 617 284 003 950 795 060 08 × 2 = 0 + 0.073 876 542 487 897 876 536 786 663 901 234 568 007 901 590 120 16;
  • 4) 0.073 876 542 487 897 876 536 786 663 901 234 568 007 901 590 120 16 × 2 = 0 + 0.147 753 084 975 795 753 073 573 327 802 469 136 015 803 180 240 32;
  • 5) 0.147 753 084 975 795 753 073 573 327 802 469 136 015 803 180 240 32 × 2 = 0 + 0.295 506 169 951 591 506 147 146 655 604 938 272 031 606 360 480 64;
  • 6) 0.295 506 169 951 591 506 147 146 655 604 938 272 031 606 360 480 64 × 2 = 0 + 0.591 012 339 903 183 012 294 293 311 209 876 544 063 212 720 961 28;
  • 7) 0.591 012 339 903 183 012 294 293 311 209 876 544 063 212 720 961 28 × 2 = 1 + 0.182 024 679 806 366 024 588 586 622 419 753 088 126 425 441 922 56;
  • 8) 0.182 024 679 806 366 024 588 586 622 419 753 088 126 425 441 922 56 × 2 = 0 + 0.364 049 359 612 732 049 177 173 244 839 506 176 252 850 883 845 12;
  • 9) 0.364 049 359 612 732 049 177 173 244 839 506 176 252 850 883 845 12 × 2 = 0 + 0.728 098 719 225 464 098 354 346 489 679 012 352 505 701 767 690 24;
  • 10) 0.728 098 719 225 464 098 354 346 489 679 012 352 505 701 767 690 24 × 2 = 1 + 0.456 197 438 450 928 196 708 692 979 358 024 705 011 403 535 380 48;
  • 11) 0.456 197 438 450 928 196 708 692 979 358 024 705 011 403 535 380 48 × 2 = 0 + 0.912 394 876 901 856 393 417 385 958 716 049 410 022 807 070 760 96;
  • 12) 0.912 394 876 901 856 393 417 385 958 716 049 410 022 807 070 760 96 × 2 = 1 + 0.824 789 753 803 712 786 834 771 917 432 098 820 045 614 141 521 92;
  • 13) 0.824 789 753 803 712 786 834 771 917 432 098 820 045 614 141 521 92 × 2 = 1 + 0.649 579 507 607 425 573 669 543 834 864 197 640 091 228 283 043 84;
  • 14) 0.649 579 507 607 425 573 669 543 834 864 197 640 091 228 283 043 84 × 2 = 1 + 0.299 159 015 214 851 147 339 087 669 728 395 280 182 456 566 087 68;
  • 15) 0.299 159 015 214 851 147 339 087 669 728 395 280 182 456 566 087 68 × 2 = 0 + 0.598 318 030 429 702 294 678 175 339 456 790 560 364 913 132 175 36;
  • 16) 0.598 318 030 429 702 294 678 175 339 456 790 560 364 913 132 175 36 × 2 = 1 + 0.196 636 060 859 404 589 356 350 678 913 581 120 729 826 264 350 72;
  • 17) 0.196 636 060 859 404 589 356 350 678 913 581 120 729 826 264 350 72 × 2 = 0 + 0.393 272 121 718 809 178 712 701 357 827 162 241 459 652 528 701 44;
  • 18) 0.393 272 121 718 809 178 712 701 357 827 162 241 459 652 528 701 44 × 2 = 0 + 0.786 544 243 437 618 357 425 402 715 654 324 482 919 305 057 402 88;
  • 19) 0.786 544 243 437 618 357 425 402 715 654 324 482 919 305 057 402 88 × 2 = 1 + 0.573 088 486 875 236 714 850 805 431 308 648 965 838 610 114 805 76;
  • 20) 0.573 088 486 875 236 714 850 805 431 308 648 965 838 610 114 805 76 × 2 = 1 + 0.146 176 973 750 473 429 701 610 862 617 297 931 677 220 229 611 52;
  • 21) 0.146 176 973 750 473 429 701 610 862 617 297 931 677 220 229 611 52 × 2 = 0 + 0.292 353 947 500 946 859 403 221 725 234 595 863 354 440 459 223 04;
  • 22) 0.292 353 947 500 946 859 403 221 725 234 595 863 354 440 459 223 04 × 2 = 0 + 0.584 707 895 001 893 718 806 443 450 469 191 726 708 880 918 446 08;
  • 23) 0.584 707 895 001 893 718 806 443 450 469 191 726 708 880 918 446 08 × 2 = 1 + 0.169 415 790 003 787 437 612 886 900 938 383 453 417 761 836 892 16;
  • 24) 0.169 415 790 003 787 437 612 886 900 938 383 453 417 761 836 892 16 × 2 = 0 + 0.338 831 580 007 574 875 225 773 801 876 766 906 835 523 673 784 32;
  • 25) 0.338 831 580 007 574 875 225 773 801 876 766 906 835 523 673 784 32 × 2 = 0 + 0.677 663 160 015 149 750 451 547 603 753 533 813 671 047 347 568 64;
  • 26) 0.677 663 160 015 149 750 451 547 603 753 533 813 671 047 347 568 64 × 2 = 1 + 0.355 326 320 030 299 500 903 095 207 507 067 627 342 094 695 137 28;
  • 27) 0.355 326 320 030 299 500 903 095 207 507 067 627 342 094 695 137 28 × 2 = 0 + 0.710 652 640 060 599 001 806 190 415 014 135 254 684 189 390 274 56;
  • 28) 0.710 652 640 060 599 001 806 190 415 014 135 254 684 189 390 274 56 × 2 = 1 + 0.421 305 280 121 198 003 612 380 830 028 270 509 368 378 780 549 12;
  • 29) 0.421 305 280 121 198 003 612 380 830 028 270 509 368 378 780 549 12 × 2 = 0 + 0.842 610 560 242 396 007 224 761 660 056 541 018 736 757 561 098 24;
  • 30) 0.842 610 560 242 396 007 224 761 660 056 541 018 736 757 561 098 24 × 2 = 1 + 0.685 221 120 484 792 014 449 523 320 113 082 037 473 515 122 196 48;
  • 31) 0.685 221 120 484 792 014 449 523 320 113 082 037 473 515 122 196 48 × 2 = 1 + 0.370 442 240 969 584 028 899 046 640 226 164 074 947 030 244 392 96;
  • 32) 0.370 442 240 969 584 028 899 046 640 226 164 074 947 030 244 392 96 × 2 = 0 + 0.740 884 481 939 168 057 798 093 280 452 328 149 894 060 488 785 92;
  • 33) 0.740 884 481 939 168 057 798 093 280 452 328 149 894 060 488 785 92 × 2 = 1 + 0.481 768 963 878 336 115 596 186 560 904 656 299 788 120 977 571 84;
  • 34) 0.481 768 963 878 336 115 596 186 560 904 656 299 788 120 977 571 84 × 2 = 0 + 0.963 537 927 756 672 231 192 373 121 809 312 599 576 241 955 143 68;
  • 35) 0.963 537 927 756 672 231 192 373 121 809 312 599 576 241 955 143 68 × 2 = 1 + 0.927 075 855 513 344 462 384 746 243 618 625 199 152 483 910 287 36;
  • 36) 0.927 075 855 513 344 462 384 746 243 618 625 199 152 483 910 287 36 × 2 = 1 + 0.854 151 711 026 688 924 769 492 487 237 250 398 304 967 820 574 72;
  • 37) 0.854 151 711 026 688 924 769 492 487 237 250 398 304 967 820 574 72 × 2 = 1 + 0.708 303 422 053 377 849 538 984 974 474 500 796 609 935 641 149 44;
  • 38) 0.708 303 422 053 377 849 538 984 974 474 500 796 609 935 641 149 44 × 2 = 1 + 0.416 606 844 106 755 699 077 969 948 949 001 593 219 871 282 298 88;
  • 39) 0.416 606 844 106 755 699 077 969 948 949 001 593 219 871 282 298 88 × 2 = 0 + 0.833 213 688 213 511 398 155 939 897 898 003 186 439 742 564 597 76;
  • 40) 0.833 213 688 213 511 398 155 939 897 898 003 186 439 742 564 597 76 × 2 = 1 + 0.666 427 376 427 022 796 311 879 795 796 006 372 879 485 129 195 52;
  • 41) 0.666 427 376 427 022 796 311 879 795 796 006 372 879 485 129 195 52 × 2 = 1 + 0.332 854 752 854 045 592 623 759 591 592 012 745 758 970 258 391 04;
  • 42) 0.332 854 752 854 045 592 623 759 591 592 012 745 758 970 258 391 04 × 2 = 0 + 0.665 709 505 708 091 185 247 519 183 184 025 491 517 940 516 782 08;
  • 43) 0.665 709 505 708 091 185 247 519 183 184 025 491 517 940 516 782 08 × 2 = 1 + 0.331 419 011 416 182 370 495 038 366 368 050 983 035 881 033 564 16;
  • 44) 0.331 419 011 416 182 370 495 038 366 368 050 983 035 881 033 564 16 × 2 = 0 + 0.662 838 022 832 364 740 990 076 732 736 101 966 071 762 067 128 32;
  • 45) 0.662 838 022 832 364 740 990 076 732 736 101 966 071 762 067 128 32 × 2 = 1 + 0.325 676 045 664 729 481 980 153 465 472 203 932 143 524 134 256 64;
  • 46) 0.325 676 045 664 729 481 980 153 465 472 203 932 143 524 134 256 64 × 2 = 0 + 0.651 352 091 329 458 963 960 306 930 944 407 864 287 048 268 513 28;
  • 47) 0.651 352 091 329 458 963 960 306 930 944 407 864 287 048 268 513 28 × 2 = 1 + 0.302 704 182 658 917 927 920 613 861 888 815 728 574 096 537 026 56;
  • 48) 0.302 704 182 658 917 927 920 613 861 888 815 728 574 096 537 026 56 × 2 = 0 + 0.605 408 365 317 835 855 841 227 723 777 631 457 148 193 074 053 12;
  • 49) 0.605 408 365 317 835 855 841 227 723 777 631 457 148 193 074 053 12 × 2 = 1 + 0.210 816 730 635 671 711 682 455 447 555 262 914 296 386 148 106 24;
  • 50) 0.210 816 730 635 671 711 682 455 447 555 262 914 296 386 148 106 24 × 2 = 0 + 0.421 633 461 271 343 423 364 910 895 110 525 828 592 772 296 212 48;
  • 51) 0.421 633 461 271 343 423 364 910 895 110 525 828 592 772 296 212 48 × 2 = 0 + 0.843 266 922 542 686 846 729 821 790 221 051 657 185 544 592 424 96;
  • 52) 0.843 266 922 542 686 846 729 821 790 221 051 657 185 544 592 424 96 × 2 = 1 + 0.686 533 845 085 373 693 459 643 580 442 103 314 371 089 184 849 92;
  • 53) 0.686 533 845 085 373 693 459 643 580 442 103 314 371 089 184 849 92 × 2 = 1 + 0.373 067 690 170 747 386 919 287 160 884 206 628 742 178 369 699 84;
  • 54) 0.373 067 690 170 747 386 919 287 160 884 206 628 742 178 369 699 84 × 2 = 0 + 0.746 135 380 341 494 773 838 574 321 768 413 257 484 356 739 399 68;
  • 55) 0.746 135 380 341 494 773 838 574 321 768 413 257 484 356 739 399 68 × 2 = 1 + 0.492 270 760 682 989 547 677 148 643 536 826 514 968 713 478 799 36;
  • 56) 0.492 270 760 682 989 547 677 148 643 536 826 514 968 713 478 799 36 × 2 = 0 + 0.984 541 521 365 979 095 354 297 287 073 653 029 937 426 957 598 72;
  • 57) 0.984 541 521 365 979 095 354 297 287 073 653 029 937 426 957 598 72 × 2 = 1 + 0.969 083 042 731 958 190 708 594 574 147 306 059 874 853 915 197 44;
  • 58) 0.969 083 042 731 958 190 708 594 574 147 306 059 874 853 915 197 44 × 2 = 1 + 0.938 166 085 463 916 381 417 189 148 294 612 119 749 707 830 394 88;
  • 59) 0.938 166 085 463 916 381 417 189 148 294 612 119 749 707 830 394 88 × 2 = 1 + 0.876 332 170 927 832 762 834 378 296 589 224 239 499 415 660 789 76;

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:


0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2)

5. Positive number before normalization:

0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02(10) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2) =

0.0000 0010 0101 1101 0011 0010 0101 0110 1011 1101 1010 1010 1001 1010 111(2) × 20 =

1.0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111(2) × 2-7


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): -7

Mantissa (not normalized):
1.0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-7 + 2(11-1) - 1 =

(-7 + 1 023)(10) =

1 016(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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1016(10) =

011 1111 1000(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. 0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111 =

0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


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 1111 1000

Mantissa (52 bits) =
0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111


Decimal number 0.009 234 567 810 987 234 567 098 332 987 654 321 000 987 698 765 02 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1000 - 0010 1110 1001 1001 0010 1011 0101 1110 1101 0101 0100 1101 0111

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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