Decimal to 64 Bit IEEE 754 Binary: Convert Number 11 111 100 111 111 001 111 110 001 111 100 999 958 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 11 111 100 111 111 001 111 110 001 111 100 999 958(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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;
  • 11 111 100 111 111 001 111 110 001 111 100 999 958 ÷ 2 = 5 555 550 055 555 500 555 555 000 555 550 499 979 + 0;
  • 5 555 550 055 555 500 555 555 000 555 550 499 979 ÷ 2 = 2 777 775 027 777 750 277 777 500 277 775 249 989 + 1;
  • 2 777 775 027 777 750 277 777 500 277 775 249 989 ÷ 2 = 1 388 887 513 888 875 138 888 750 138 887 624 994 + 1;
  • 1 388 887 513 888 875 138 888 750 138 887 624 994 ÷ 2 = 694 443 756 944 437 569 444 375 069 443 812 497 + 0;
  • 694 443 756 944 437 569 444 375 069 443 812 497 ÷ 2 = 347 221 878 472 218 784 722 187 534 721 906 248 + 1;
  • 347 221 878 472 218 784 722 187 534 721 906 248 ÷ 2 = 173 610 939 236 109 392 361 093 767 360 953 124 + 0;
  • 173 610 939 236 109 392 361 093 767 360 953 124 ÷ 2 = 86 805 469 618 054 696 180 546 883 680 476 562 + 0;
  • 86 805 469 618 054 696 180 546 883 680 476 562 ÷ 2 = 43 402 734 809 027 348 090 273 441 840 238 281 + 0;
  • 43 402 734 809 027 348 090 273 441 840 238 281 ÷ 2 = 21 701 367 404 513 674 045 136 720 920 119 140 + 1;
  • 21 701 367 404 513 674 045 136 720 920 119 140 ÷ 2 = 10 850 683 702 256 837 022 568 360 460 059 570 + 0;
  • 10 850 683 702 256 837 022 568 360 460 059 570 ÷ 2 = 5 425 341 851 128 418 511 284 180 230 029 785 + 0;
  • 5 425 341 851 128 418 511 284 180 230 029 785 ÷ 2 = 2 712 670 925 564 209 255 642 090 115 014 892 + 1;
  • 2 712 670 925 564 209 255 642 090 115 014 892 ÷ 2 = 1 356 335 462 782 104 627 821 045 057 507 446 + 0;
  • 1 356 335 462 782 104 627 821 045 057 507 446 ÷ 2 = 678 167 731 391 052 313 910 522 528 753 723 + 0;
  • 678 167 731 391 052 313 910 522 528 753 723 ÷ 2 = 339 083 865 695 526 156 955 261 264 376 861 + 1;
  • 339 083 865 695 526 156 955 261 264 376 861 ÷ 2 = 169 541 932 847 763 078 477 630 632 188 430 + 1;
  • 169 541 932 847 763 078 477 630 632 188 430 ÷ 2 = 84 770 966 423 881 539 238 815 316 094 215 + 0;
  • 84 770 966 423 881 539 238 815 316 094 215 ÷ 2 = 42 385 483 211 940 769 619 407 658 047 107 + 1;
  • 42 385 483 211 940 769 619 407 658 047 107 ÷ 2 = 21 192 741 605 970 384 809 703 829 023 553 + 1;
  • 21 192 741 605 970 384 809 703 829 023 553 ÷ 2 = 10 596 370 802 985 192 404 851 914 511 776 + 1;
  • 10 596 370 802 985 192 404 851 914 511 776 ÷ 2 = 5 298 185 401 492 596 202 425 957 255 888 + 0;
  • 5 298 185 401 492 596 202 425 957 255 888 ÷ 2 = 2 649 092 700 746 298 101 212 978 627 944 + 0;
  • 2 649 092 700 746 298 101 212 978 627 944 ÷ 2 = 1 324 546 350 373 149 050 606 489 313 972 + 0;
  • 1 324 546 350 373 149 050 606 489 313 972 ÷ 2 = 662 273 175 186 574 525 303 244 656 986 + 0;
  • 662 273 175 186 574 525 303 244 656 986 ÷ 2 = 331 136 587 593 287 262 651 622 328 493 + 0;
  • 331 136 587 593 287 262 651 622 328 493 ÷ 2 = 165 568 293 796 643 631 325 811 164 246 + 1;
  • 165 568 293 796 643 631 325 811 164 246 ÷ 2 = 82 784 146 898 321 815 662 905 582 123 + 0;
  • 82 784 146 898 321 815 662 905 582 123 ÷ 2 = 41 392 073 449 160 907 831 452 791 061 + 1;
  • 41 392 073 449 160 907 831 452 791 061 ÷ 2 = 20 696 036 724 580 453 915 726 395 530 + 1;
  • 20 696 036 724 580 453 915 726 395 530 ÷ 2 = 10 348 018 362 290 226 957 863 197 765 + 0;
  • 10 348 018 362 290 226 957 863 197 765 ÷ 2 = 5 174 009 181 145 113 478 931 598 882 + 1;
  • 5 174 009 181 145 113 478 931 598 882 ÷ 2 = 2 587 004 590 572 556 739 465 799 441 + 0;
  • 2 587 004 590 572 556 739 465 799 441 ÷ 2 = 1 293 502 295 286 278 369 732 899 720 + 1;
  • 1 293 502 295 286 278 369 732 899 720 ÷ 2 = 646 751 147 643 139 184 866 449 860 + 0;
  • 646 751 147 643 139 184 866 449 860 ÷ 2 = 323 375 573 821 569 592 433 224 930 + 0;
  • 323 375 573 821 569 592 433 224 930 ÷ 2 = 161 687 786 910 784 796 216 612 465 + 0;
  • 161 687 786 910 784 796 216 612 465 ÷ 2 = 80 843 893 455 392 398 108 306 232 + 1;
  • 80 843 893 455 392 398 108 306 232 ÷ 2 = 40 421 946 727 696 199 054 153 116 + 0;
  • 40 421 946 727 696 199 054 153 116 ÷ 2 = 20 210 973 363 848 099 527 076 558 + 0;
  • 20 210 973 363 848 099 527 076 558 ÷ 2 = 10 105 486 681 924 049 763 538 279 + 0;
  • 10 105 486 681 924 049 763 538 279 ÷ 2 = 5 052 743 340 962 024 881 769 139 + 1;
  • 5 052 743 340 962 024 881 769 139 ÷ 2 = 2 526 371 670 481 012 440 884 569 + 1;
  • 2 526 371 670 481 012 440 884 569 ÷ 2 = 1 263 185 835 240 506 220 442 284 + 1;
  • 1 263 185 835 240 506 220 442 284 ÷ 2 = 631 592 917 620 253 110 221 142 + 0;
  • 631 592 917 620 253 110 221 142 ÷ 2 = 315 796 458 810 126 555 110 571 + 0;
  • 315 796 458 810 126 555 110 571 ÷ 2 = 157 898 229 405 063 277 555 285 + 1;
  • 157 898 229 405 063 277 555 285 ÷ 2 = 78 949 114 702 531 638 777 642 + 1;
  • 78 949 114 702 531 638 777 642 ÷ 2 = 39 474 557 351 265 819 388 821 + 0;
  • 39 474 557 351 265 819 388 821 ÷ 2 = 19 737 278 675 632 909 694 410 + 1;
  • 19 737 278 675 632 909 694 410 ÷ 2 = 9 868 639 337 816 454 847 205 + 0;
  • 9 868 639 337 816 454 847 205 ÷ 2 = 4 934 319 668 908 227 423 602 + 1;
  • 4 934 319 668 908 227 423 602 ÷ 2 = 2 467 159 834 454 113 711 801 + 0;
  • 2 467 159 834 454 113 711 801 ÷ 2 = 1 233 579 917 227 056 855 900 + 1;
  • 1 233 579 917 227 056 855 900 ÷ 2 = 616 789 958 613 528 427 950 + 0;
  • 616 789 958 613 528 427 950 ÷ 2 = 308 394 979 306 764 213 975 + 0;
  • 308 394 979 306 764 213 975 ÷ 2 = 154 197 489 653 382 106 987 + 1;
  • 154 197 489 653 382 106 987 ÷ 2 = 77 098 744 826 691 053 493 + 1;
  • 77 098 744 826 691 053 493 ÷ 2 = 38 549 372 413 345 526 746 + 1;
  • 38 549 372 413 345 526 746 ÷ 2 = 19 274 686 206 672 763 373 + 0;
  • 19 274 686 206 672 763 373 ÷ 2 = 9 637 343 103 336 381 686 + 1;
  • 9 637 343 103 336 381 686 ÷ 2 = 4 818 671 551 668 190 843 + 0;
  • 4 818 671 551 668 190 843 ÷ 2 = 2 409 335 775 834 095 421 + 1;
  • 2 409 335 775 834 095 421 ÷ 2 = 1 204 667 887 917 047 710 + 1;
  • 1 204 667 887 917 047 710 ÷ 2 = 602 333 943 958 523 855 + 0;
  • 602 333 943 958 523 855 ÷ 2 = 301 166 971 979 261 927 + 1;
  • 301 166 971 979 261 927 ÷ 2 = 150 583 485 989 630 963 + 1;
  • 150 583 485 989 630 963 ÷ 2 = 75 291 742 994 815 481 + 1;
  • 75 291 742 994 815 481 ÷ 2 = 37 645 871 497 407 740 + 1;
  • 37 645 871 497 407 740 ÷ 2 = 18 822 935 748 703 870 + 0;
  • 18 822 935 748 703 870 ÷ 2 = 9 411 467 874 351 935 + 0;
  • 9 411 467 874 351 935 ÷ 2 = 4 705 733 937 175 967 + 1;
  • 4 705 733 937 175 967 ÷ 2 = 2 352 866 968 587 983 + 1;
  • 2 352 866 968 587 983 ÷ 2 = 1 176 433 484 293 991 + 1;
  • 1 176 433 484 293 991 ÷ 2 = 588 216 742 146 995 + 1;
  • 588 216 742 146 995 ÷ 2 = 294 108 371 073 497 + 1;
  • 294 108 371 073 497 ÷ 2 = 147 054 185 536 748 + 1;
  • 147 054 185 536 748 ÷ 2 = 73 527 092 768 374 + 0;
  • 73 527 092 768 374 ÷ 2 = 36 763 546 384 187 + 0;
  • 36 763 546 384 187 ÷ 2 = 18 381 773 192 093 + 1;
  • 18 381 773 192 093 ÷ 2 = 9 190 886 596 046 + 1;
  • 9 190 886 596 046 ÷ 2 = 4 595 443 298 023 + 0;
  • 4 595 443 298 023 ÷ 2 = 2 297 721 649 011 + 1;
  • 2 297 721 649 011 ÷ 2 = 1 148 860 824 505 + 1;
  • 1 148 860 824 505 ÷ 2 = 574 430 412 252 + 1;
  • 574 430 412 252 ÷ 2 = 287 215 206 126 + 0;
  • 287 215 206 126 ÷ 2 = 143 607 603 063 + 0;
  • 143 607 603 063 ÷ 2 = 71 803 801 531 + 1;
  • 71 803 801 531 ÷ 2 = 35 901 900 765 + 1;
  • 35 901 900 765 ÷ 2 = 17 950 950 382 + 1;
  • 17 950 950 382 ÷ 2 = 8 975 475 191 + 0;
  • 8 975 475 191 ÷ 2 = 4 487 737 595 + 1;
  • 4 487 737 595 ÷ 2 = 2 243 868 797 + 1;
  • 2 243 868 797 ÷ 2 = 1 121 934 398 + 1;
  • 1 121 934 398 ÷ 2 = 560 967 199 + 0;
  • 560 967 199 ÷ 2 = 280 483 599 + 1;
  • 280 483 599 ÷ 2 = 140 241 799 + 1;
  • 140 241 799 ÷ 2 = 70 120 899 + 1;
  • 70 120 899 ÷ 2 = 35 060 449 + 1;
  • 35 060 449 ÷ 2 = 17 530 224 + 1;
  • 17 530 224 ÷ 2 = 8 765 112 + 0;
  • 8 765 112 ÷ 2 = 4 382 556 + 0;
  • 4 382 556 ÷ 2 = 2 191 278 + 0;
  • 2 191 278 ÷ 2 = 1 095 639 + 0;
  • 1 095 639 ÷ 2 = 547 819 + 1;
  • 547 819 ÷ 2 = 273 909 + 1;
  • 273 909 ÷ 2 = 136 954 + 1;
  • 136 954 ÷ 2 = 68 477 + 0;
  • 68 477 ÷ 2 = 34 238 + 1;
  • 34 238 ÷ 2 = 17 119 + 0;
  • 17 119 ÷ 2 = 8 559 + 1;
  • 8 559 ÷ 2 = 4 279 + 1;
  • 4 279 ÷ 2 = 2 139 + 1;
  • 2 139 ÷ 2 = 1 069 + 1;
  • 1 069 ÷ 2 = 534 + 1;
  • 534 ÷ 2 = 267 + 0;
  • 267 ÷ 2 = 133 + 1;
  • 133 ÷ 2 = 66 + 1;
  • 66 ÷ 2 = 33 + 0;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.

11 111 100 111 111 001 111 110 001 111 100 999 958(10) =


1000 0101 1011 1110 1011 1000 0111 1101 1101 1100 1110 1100 1111 1100 1111 0110 1011 1001 0101 0110 0111 0001 0001 0101 1010 0000 1110 1100 1001 0001 0110(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 123 positions to the left, so that only one non zero digit remains to the left of it:


11 111 100 111 111 001 111 110 001 111 100 999 958(10) =


1000 0101 1011 1110 1011 1000 0111 1101 1101 1100 1110 1100 1111 1100 1111 0110 1011 1001 0101 0110 0111 0001 0001 0101 1010 0000 1110 1100 1001 0001 0110(2) =


1000 0101 1011 1110 1011 1000 0111 1101 1101 1100 1110 1100 1111 1100 1111 0110 1011 1001 0101 0110 0111 0001 0001 0101 1010 0000 1110 1100 1001 0001 0110(2) × 20 =


1.0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111 1001 1110 1101 0111 0010 1010 1100 1110 0010 0010 1011 0100 0001 1101 1001 0010 0010 110(2) × 2123


4. 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): 123


Mantissa (not normalized):
1.0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111 1001 1110 1101 0111 0010 1010 1100 1110 0010 0010 1011 0100 0001 1101 1001 0010 0010 110


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


123 + 2(11-1) - 1 =


(123 + 1 023)(10) =


1 146(10)


6. 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;
  • 1 146 ÷ 2 = 573 + 0;
  • 573 ÷ 2 = 286 + 1;
  • 286 ÷ 2 = 143 + 0;
  • 143 ÷ 2 = 71 + 1;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1146(10) =


100 0111 1010(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111 100 1111 0110 1011 1001 0101 0110 0111 0001 0001 0101 1010 0000 1110 1100 1001 0001 0110 =


0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111


9. 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) =
100 0111 1010


Mantissa (52 bits) =
0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111


The base ten decimal number 11 111 100 111 111 001 111 110 001 111 100 999 958 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0111 1010 - 0000 1011 0111 1101 0111 0000 1111 1011 1011 1001 1101 1001 1111

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