-0.000 000 000 000 000 051 120 132 432 65 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 051 120 132 432 65(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
-0.000 000 000 000 000 051 120 132 432 65(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:

|-0.000 000 000 000 000 051 120 132 432 65| = 0.000 000 000 000 000 051 120 132 432 65


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: 0.000 000 000 000 000 051 120 132 432 65.

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) 0.000 000 000 000 000 051 120 132 432 65 × 2 = 0 + 0.000 000 000 000 000 102 240 264 865 3;
  • 2) 0.000 000 000 000 000 102 240 264 865 3 × 2 = 0 + 0.000 000 000 000 000 204 480 529 730 6;
  • 3) 0.000 000 000 000 000 204 480 529 730 6 × 2 = 0 + 0.000 000 000 000 000 408 961 059 461 2;
  • 4) 0.000 000 000 000 000 408 961 059 461 2 × 2 = 0 + 0.000 000 000 000 000 817 922 118 922 4;
  • 5) 0.000 000 000 000 000 817 922 118 922 4 × 2 = 0 + 0.000 000 000 000 001 635 844 237 844 8;
  • 6) 0.000 000 000 000 001 635 844 237 844 8 × 2 = 0 + 0.000 000 000 000 003 271 688 475 689 6;
  • 7) 0.000 000 000 000 003 271 688 475 689 6 × 2 = 0 + 0.000 000 000 000 006 543 376 951 379 2;
  • 8) 0.000 000 000 000 006 543 376 951 379 2 × 2 = 0 + 0.000 000 000 000 013 086 753 902 758 4;
  • 9) 0.000 000 000 000 013 086 753 902 758 4 × 2 = 0 + 0.000 000 000 000 026 173 507 805 516 8;
  • 10) 0.000 000 000 000 026 173 507 805 516 8 × 2 = 0 + 0.000 000 000 000 052 347 015 611 033 6;
  • 11) 0.000 000 000 000 052 347 015 611 033 6 × 2 = 0 + 0.000 000 000 000 104 694 031 222 067 2;
  • 12) 0.000 000 000 000 104 694 031 222 067 2 × 2 = 0 + 0.000 000 000 000 209 388 062 444 134 4;
  • 13) 0.000 000 000 000 209 388 062 444 134 4 × 2 = 0 + 0.000 000 000 000 418 776 124 888 268 8;
  • 14) 0.000 000 000 000 418 776 124 888 268 8 × 2 = 0 + 0.000 000 000 000 837 552 249 776 537 6;
  • 15) 0.000 000 000 000 837 552 249 776 537 6 × 2 = 0 + 0.000 000 000 001 675 104 499 553 075 2;
  • 16) 0.000 000 000 001 675 104 499 553 075 2 × 2 = 0 + 0.000 000 000 003 350 208 999 106 150 4;
  • 17) 0.000 000 000 003 350 208 999 106 150 4 × 2 = 0 + 0.000 000 000 006 700 417 998 212 300 8;
  • 18) 0.000 000 000 006 700 417 998 212 300 8 × 2 = 0 + 0.000 000 000 013 400 835 996 424 601 6;
  • 19) 0.000 000 000 013 400 835 996 424 601 6 × 2 = 0 + 0.000 000 000 026 801 671 992 849 203 2;
  • 20) 0.000 000 000 026 801 671 992 849 203 2 × 2 = 0 + 0.000 000 000 053 603 343 985 698 406 4;
  • 21) 0.000 000 000 053 603 343 985 698 406 4 × 2 = 0 + 0.000 000 000 107 206 687 971 396 812 8;
  • 22) 0.000 000 000 107 206 687 971 396 812 8 × 2 = 0 + 0.000 000 000 214 413 375 942 793 625 6;
  • 23) 0.000 000 000 214 413 375 942 793 625 6 × 2 = 0 + 0.000 000 000 428 826 751 885 587 251 2;
  • 24) 0.000 000 000 428 826 751 885 587 251 2 × 2 = 0 + 0.000 000 000 857 653 503 771 174 502 4;
  • 25) 0.000 000 000 857 653 503 771 174 502 4 × 2 = 0 + 0.000 000 001 715 307 007 542 349 004 8;
  • 26) 0.000 000 001 715 307 007 542 349 004 8 × 2 = 0 + 0.000 000 003 430 614 015 084 698 009 6;
  • 27) 0.000 000 003 430 614 015 084 698 009 6 × 2 = 0 + 0.000 000 006 861 228 030 169 396 019 2;
  • 28) 0.000 000 006 861 228 030 169 396 019 2 × 2 = 0 + 0.000 000 013 722 456 060 338 792 038 4;
  • 29) 0.000 000 013 722 456 060 338 792 038 4 × 2 = 0 + 0.000 000 027 444 912 120 677 584 076 8;
  • 30) 0.000 000 027 444 912 120 677 584 076 8 × 2 = 0 + 0.000 000 054 889 824 241 355 168 153 6;
  • 31) 0.000 000 054 889 824 241 355 168 153 6 × 2 = 0 + 0.000 000 109 779 648 482 710 336 307 2;
  • 32) 0.000 000 109 779 648 482 710 336 307 2 × 2 = 0 + 0.000 000 219 559 296 965 420 672 614 4;
  • 33) 0.000 000 219 559 296 965 420 672 614 4 × 2 = 0 + 0.000 000 439 118 593 930 841 345 228 8;
  • 34) 0.000 000 439 118 593 930 841 345 228 8 × 2 = 0 + 0.000 000 878 237 187 861 682 690 457 6;
  • 35) 0.000 000 878 237 187 861 682 690 457 6 × 2 = 0 + 0.000 001 756 474 375 723 365 380 915 2;
  • 36) 0.000 001 756 474 375 723 365 380 915 2 × 2 = 0 + 0.000 003 512 948 751 446 730 761 830 4;
  • 37) 0.000 003 512 948 751 446 730 761 830 4 × 2 = 0 + 0.000 007 025 897 502 893 461 523 660 8;
  • 38) 0.000 007 025 897 502 893 461 523 660 8 × 2 = 0 + 0.000 014 051 795 005 786 923 047 321 6;
  • 39) 0.000 014 051 795 005 786 923 047 321 6 × 2 = 0 + 0.000 028 103 590 011 573 846 094 643 2;
  • 40) 0.000 028 103 590 011 573 846 094 643 2 × 2 = 0 + 0.000 056 207 180 023 147 692 189 286 4;
  • 41) 0.000 056 207 180 023 147 692 189 286 4 × 2 = 0 + 0.000 112 414 360 046 295 384 378 572 8;
  • 42) 0.000 112 414 360 046 295 384 378 572 8 × 2 = 0 + 0.000 224 828 720 092 590 768 757 145 6;
  • 43) 0.000 224 828 720 092 590 768 757 145 6 × 2 = 0 + 0.000 449 657 440 185 181 537 514 291 2;
  • 44) 0.000 449 657 440 185 181 537 514 291 2 × 2 = 0 + 0.000 899 314 880 370 363 075 028 582 4;
  • 45) 0.000 899 314 880 370 363 075 028 582 4 × 2 = 0 + 0.001 798 629 760 740 726 150 057 164 8;
  • 46) 0.001 798 629 760 740 726 150 057 164 8 × 2 = 0 + 0.003 597 259 521 481 452 300 114 329 6;
  • 47) 0.003 597 259 521 481 452 300 114 329 6 × 2 = 0 + 0.007 194 519 042 962 904 600 228 659 2;
  • 48) 0.007 194 519 042 962 904 600 228 659 2 × 2 = 0 + 0.014 389 038 085 925 809 200 457 318 4;
  • 49) 0.014 389 038 085 925 809 200 457 318 4 × 2 = 0 + 0.028 778 076 171 851 618 400 914 636 8;
  • 50) 0.028 778 076 171 851 618 400 914 636 8 × 2 = 0 + 0.057 556 152 343 703 236 801 829 273 6;
  • 51) 0.057 556 152 343 703 236 801 829 273 6 × 2 = 0 + 0.115 112 304 687 406 473 603 658 547 2;
  • 52) 0.115 112 304 687 406 473 603 658 547 2 × 2 = 0 + 0.230 224 609 374 812 947 207 317 094 4;
  • 53) 0.230 224 609 374 812 947 207 317 094 4 × 2 = 0 + 0.460 449 218 749 625 894 414 634 188 8;
  • 54) 0.460 449 218 749 625 894 414 634 188 8 × 2 = 0 + 0.920 898 437 499 251 788 829 268 377 6;
  • 55) 0.920 898 437 499 251 788 829 268 377 6 × 2 = 1 + 0.841 796 874 998 503 577 658 536 755 2;
  • 56) 0.841 796 874 998 503 577 658 536 755 2 × 2 = 1 + 0.683 593 749 997 007 155 317 073 510 4;
  • 57) 0.683 593 749 997 007 155 317 073 510 4 × 2 = 1 + 0.367 187 499 994 014 310 634 147 020 8;
  • 58) 0.367 187 499 994 014 310 634 147 020 8 × 2 = 0 + 0.734 374 999 988 028 621 268 294 041 6;
  • 59) 0.734 374 999 988 028 621 268 294 041 6 × 2 = 1 + 0.468 749 999 976 057 242 536 588 083 2;
  • 60) 0.468 749 999 976 057 242 536 588 083 2 × 2 = 0 + 0.937 499 999 952 114 485 073 176 166 4;
  • 61) 0.937 499 999 952 114 485 073 176 166 4 × 2 = 1 + 0.874 999 999 904 228 970 146 352 332 8;
  • 62) 0.874 999 999 904 228 970 146 352 332 8 × 2 = 1 + 0.749 999 999 808 457 940 292 704 665 6;
  • 63) 0.749 999 999 808 457 940 292 704 665 6 × 2 = 1 + 0.499 999 999 616 915 880 585 409 331 2;
  • 64) 0.499 999 999 616 915 880 585 409 331 2 × 2 = 0 + 0.999 999 999 233 831 761 170 818 662 4;
  • 65) 0.999 999 999 233 831 761 170 818 662 4 × 2 = 1 + 0.999 999 998 467 663 522 341 637 324 8;
  • 66) 0.999 999 998 467 663 522 341 637 324 8 × 2 = 1 + 0.999 999 996 935 327 044 683 274 649 6;
  • 67) 0.999 999 996 935 327 044 683 274 649 6 × 2 = 1 + 0.999 999 993 870 654 089 366 549 299 2;
  • 68) 0.999 999 993 870 654 089 366 549 299 2 × 2 = 1 + 0.999 999 987 741 308 178 733 098 598 4;
  • 69) 0.999 999 987 741 308 178 733 098 598 4 × 2 = 1 + 0.999 999 975 482 616 357 466 197 196 8;
  • 70) 0.999 999 975 482 616 357 466 197 196 8 × 2 = 1 + 0.999 999 950 965 232 714 932 394 393 6;
  • 71) 0.999 999 950 965 232 714 932 394 393 6 × 2 = 1 + 0.999 999 901 930 465 429 864 788 787 2;
  • 72) 0.999 999 901 930 465 429 864 788 787 2 × 2 = 1 + 0.999 999 803 860 930 859 729 577 574 4;
  • 73) 0.999 999 803 860 930 859 729 577 574 4 × 2 = 1 + 0.999 999 607 721 861 719 459 155 148 8;
  • 74) 0.999 999 607 721 861 719 459 155 148 8 × 2 = 1 + 0.999 999 215 443 723 438 918 310 297 6;
  • 75) 0.999 999 215 443 723 438 918 310 297 6 × 2 = 1 + 0.999 998 430 887 446 877 836 620 595 2;
  • 76) 0.999 998 430 887 446 877 836 620 595 2 × 2 = 1 + 0.999 996 861 774 893 755 673 241 190 4;
  • 77) 0.999 996 861 774 893 755 673 241 190 4 × 2 = 1 + 0.999 993 723 549 787 511 346 482 380 8;
  • 78) 0.999 993 723 549 787 511 346 482 380 8 × 2 = 1 + 0.999 987 447 099 575 022 692 964 761 6;
  • 79) 0.999 987 447 099 575 022 692 964 761 6 × 2 = 1 + 0.999 974 894 199 150 045 385 929 523 2;
  • 80) 0.999 974 894 199 150 045 385 929 523 2 × 2 = 1 + 0.999 949 788 398 300 090 771 859 046 4;
  • 81) 0.999 949 788 398 300 090 771 859 046 4 × 2 = 1 + 0.999 899 576 796 600 181 543 718 092 8;
  • 82) 0.999 899 576 796 600 181 543 718 092 8 × 2 = 1 + 0.999 799 153 593 200 363 087 436 185 6;
  • 83) 0.999 799 153 593 200 363 087 436 185 6 × 2 = 1 + 0.999 598 307 186 400 726 174 872 371 2;
  • 84) 0.999 598 307 186 400 726 174 872 371 2 × 2 = 1 + 0.999 196 614 372 801 452 349 744 742 4;
  • 85) 0.999 196 614 372 801 452 349 744 742 4 × 2 = 1 + 0.998 393 228 745 602 904 699 489 484 8;
  • 86) 0.998 393 228 745 602 904 699 489 484 8 × 2 = 1 + 0.996 786 457 491 205 809 398 978 969 6;
  • 87) 0.996 786 457 491 205 809 398 978 969 6 × 2 = 1 + 0.993 572 914 982 411 618 797 957 939 2;
  • 88) 0.993 572 914 982 411 618 797 957 939 2 × 2 = 1 + 0.987 145 829 964 823 237 595 915 878 4;
  • 89) 0.987 145 829 964 823 237 595 915 878 4 × 2 = 1 + 0.974 291 659 929 646 475 191 831 756 8;
  • 90) 0.974 291 659 929 646 475 191 831 756 8 × 2 = 1 + 0.948 583 319 859 292 950 383 663 513 6;
  • 91) 0.948 583 319 859 292 950 383 663 513 6 × 2 = 1 + 0.897 166 639 718 585 900 767 327 027 2;
  • 92) 0.897 166 639 718 585 900 767 327 027 2 × 2 = 1 + 0.794 333 279 437 171 801 534 654 054 4;
  • 93) 0.794 333 279 437 171 801 534 654 054 4 × 2 = 1 + 0.588 666 558 874 343 603 069 308 108 8;
  • 94) 0.588 666 558 874 343 603 069 308 108 8 × 2 = 1 + 0.177 333 117 748 687 206 138 616 217 6;
  • 95) 0.177 333 117 748 687 206 138 616 217 6 × 2 = 0 + 0.354 666 235 497 374 412 277 232 435 2;
  • 96) 0.354 666 235 497 374 412 277 232 435 2 × 2 = 0 + 0.709 332 470 994 748 824 554 464 870 4;
  • 97) 0.709 332 470 994 748 824 554 464 870 4 × 2 = 1 + 0.418 664 941 989 497 649 108 929 740 8;
  • 98) 0.418 664 941 989 497 649 108 929 740 8 × 2 = 0 + 0.837 329 883 978 995 298 217 859 481 6;
  • 99) 0.837 329 883 978 995 298 217 859 481 6 × 2 = 1 + 0.674 659 767 957 990 596 435 718 963 2;
  • 100) 0.674 659 767 957 990 596 435 718 963 2 × 2 = 1 + 0.349 319 535 915 981 192 871 437 926 4;
  • 101) 0.349 319 535 915 981 192 871 437 926 4 × 2 = 0 + 0.698 639 071 831 962 385 742 875 852 8;
  • 102) 0.698 639 071 831 962 385 742 875 852 8 × 2 = 1 + 0.397 278 143 663 924 771 485 751 705 6;
  • 103) 0.397 278 143 663 924 771 485 751 705 6 × 2 = 0 + 0.794 556 287 327 849 542 971 503 411 2;
  • 104) 0.794 556 287 327 849 542 971 503 411 2 × 2 = 1 + 0.589 112 574 655 699 085 943 006 822 4;
  • 105) 0.589 112 574 655 699 085 943 006 822 4 × 2 = 1 + 0.178 225 149 311 398 171 886 013 644 8;
  • 106) 0.178 225 149 311 398 171 886 013 644 8 × 2 = 0 + 0.356 450 298 622 796 343 772 027 289 6;
  • 107) 0.356 450 298 622 796 343 772 027 289 6 × 2 = 0 + 0.712 900 597 245 592 687 544 054 579 2;

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:


0.000 000 000 000 000 051 120 132 432 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1110 1111 1111 1111 1111 1111 1111 1111 1100 1011 0101 100(2)

6. Positive number before normalization:

0.000 000 000 000 000 051 120 132 432 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1110 1111 1111 1111 1111 1111 1111 1111 1100 1011 0101 100(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 55 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 051 120 132 432 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1110 1111 1111 1111 1111 1111 1111 1111 1100 1011 0101 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1110 1111 1111 1111 1111 1111 1111 1111 1100 1011 0101 100(2) × 20 =

1.1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100(2) × 2-55


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): -55

Mantissa (not normalized):
1.1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-55 + 2(11-1) - 1 =

(-55 + 1 023)(10) =

968(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;
  • 968 ÷ 2 = 484 + 0;
  • 484 ÷ 2 = 242 + 0;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

968(10) =

011 1100 1000(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. 1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100 =

1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100


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 1100 1000

Mantissa (52 bits) =
1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100


Decimal number -0.000 000 000 000 000 051 120 132 432 65 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 1000 - 1101 0111 0111 1111 1111 1111 1111 1111 1111 1110 0101 1010 1100

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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