0.000 000 000 000 000 013 248 731 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 013 248 731 3(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
0.000 000 000 000 000 013 248 731 3(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: 0.000 000 000 000 000 013 248 731 3.

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) 0.000 000 000 000 000 013 248 731 3 × 2 = 0 + 0.000 000 000 000 000 026 497 462 6;
  • 2) 0.000 000 000 000 000 026 497 462 6 × 2 = 0 + 0.000 000 000 000 000 052 994 925 2;
  • 3) 0.000 000 000 000 000 052 994 925 2 × 2 = 0 + 0.000 000 000 000 000 105 989 850 4;
  • 4) 0.000 000 000 000 000 105 989 850 4 × 2 = 0 + 0.000 000 000 000 000 211 979 700 8;
  • 5) 0.000 000 000 000 000 211 979 700 8 × 2 = 0 + 0.000 000 000 000 000 423 959 401 6;
  • 6) 0.000 000 000 000 000 423 959 401 6 × 2 = 0 + 0.000 000 000 000 000 847 918 803 2;
  • 7) 0.000 000 000 000 000 847 918 803 2 × 2 = 0 + 0.000 000 000 000 001 695 837 606 4;
  • 8) 0.000 000 000 000 001 695 837 606 4 × 2 = 0 + 0.000 000 000 000 003 391 675 212 8;
  • 9) 0.000 000 000 000 003 391 675 212 8 × 2 = 0 + 0.000 000 000 000 006 783 350 425 6;
  • 10) 0.000 000 000 000 006 783 350 425 6 × 2 = 0 + 0.000 000 000 000 013 566 700 851 2;
  • 11) 0.000 000 000 000 013 566 700 851 2 × 2 = 0 + 0.000 000 000 000 027 133 401 702 4;
  • 12) 0.000 000 000 000 027 133 401 702 4 × 2 = 0 + 0.000 000 000 000 054 266 803 404 8;
  • 13) 0.000 000 000 000 054 266 803 404 8 × 2 = 0 + 0.000 000 000 000 108 533 606 809 6;
  • 14) 0.000 000 000 000 108 533 606 809 6 × 2 = 0 + 0.000 000 000 000 217 067 213 619 2;
  • 15) 0.000 000 000 000 217 067 213 619 2 × 2 = 0 + 0.000 000 000 000 434 134 427 238 4;
  • 16) 0.000 000 000 000 434 134 427 238 4 × 2 = 0 + 0.000 000 000 000 868 268 854 476 8;
  • 17) 0.000 000 000 000 868 268 854 476 8 × 2 = 0 + 0.000 000 000 001 736 537 708 953 6;
  • 18) 0.000 000 000 001 736 537 708 953 6 × 2 = 0 + 0.000 000 000 003 473 075 417 907 2;
  • 19) 0.000 000 000 003 473 075 417 907 2 × 2 = 0 + 0.000 000 000 006 946 150 835 814 4;
  • 20) 0.000 000 000 006 946 150 835 814 4 × 2 = 0 + 0.000 000 000 013 892 301 671 628 8;
  • 21) 0.000 000 000 013 892 301 671 628 8 × 2 = 0 + 0.000 000 000 027 784 603 343 257 6;
  • 22) 0.000 000 000 027 784 603 343 257 6 × 2 = 0 + 0.000 000 000 055 569 206 686 515 2;
  • 23) 0.000 000 000 055 569 206 686 515 2 × 2 = 0 + 0.000 000 000 111 138 413 373 030 4;
  • 24) 0.000 000 000 111 138 413 373 030 4 × 2 = 0 + 0.000 000 000 222 276 826 746 060 8;
  • 25) 0.000 000 000 222 276 826 746 060 8 × 2 = 0 + 0.000 000 000 444 553 653 492 121 6;
  • 26) 0.000 000 000 444 553 653 492 121 6 × 2 = 0 + 0.000 000 000 889 107 306 984 243 2;
  • 27) 0.000 000 000 889 107 306 984 243 2 × 2 = 0 + 0.000 000 001 778 214 613 968 486 4;
  • 28) 0.000 000 001 778 214 613 968 486 4 × 2 = 0 + 0.000 000 003 556 429 227 936 972 8;
  • 29) 0.000 000 003 556 429 227 936 972 8 × 2 = 0 + 0.000 000 007 112 858 455 873 945 6;
  • 30) 0.000 000 007 112 858 455 873 945 6 × 2 = 0 + 0.000 000 014 225 716 911 747 891 2;
  • 31) 0.000 000 014 225 716 911 747 891 2 × 2 = 0 + 0.000 000 028 451 433 823 495 782 4;
  • 32) 0.000 000 028 451 433 823 495 782 4 × 2 = 0 + 0.000 000 056 902 867 646 991 564 8;
  • 33) 0.000 000 056 902 867 646 991 564 8 × 2 = 0 + 0.000 000 113 805 735 293 983 129 6;
  • 34) 0.000 000 113 805 735 293 983 129 6 × 2 = 0 + 0.000 000 227 611 470 587 966 259 2;
  • 35) 0.000 000 227 611 470 587 966 259 2 × 2 = 0 + 0.000 000 455 222 941 175 932 518 4;
  • 36) 0.000 000 455 222 941 175 932 518 4 × 2 = 0 + 0.000 000 910 445 882 351 865 036 8;
  • 37) 0.000 000 910 445 882 351 865 036 8 × 2 = 0 + 0.000 001 820 891 764 703 730 073 6;
  • 38) 0.000 001 820 891 764 703 730 073 6 × 2 = 0 + 0.000 003 641 783 529 407 460 147 2;
  • 39) 0.000 003 641 783 529 407 460 147 2 × 2 = 0 + 0.000 007 283 567 058 814 920 294 4;
  • 40) 0.000 007 283 567 058 814 920 294 4 × 2 = 0 + 0.000 014 567 134 117 629 840 588 8;
  • 41) 0.000 014 567 134 117 629 840 588 8 × 2 = 0 + 0.000 029 134 268 235 259 681 177 6;
  • 42) 0.000 029 134 268 235 259 681 177 6 × 2 = 0 + 0.000 058 268 536 470 519 362 355 2;
  • 43) 0.000 058 268 536 470 519 362 355 2 × 2 = 0 + 0.000 116 537 072 941 038 724 710 4;
  • 44) 0.000 116 537 072 941 038 724 710 4 × 2 = 0 + 0.000 233 074 145 882 077 449 420 8;
  • 45) 0.000 233 074 145 882 077 449 420 8 × 2 = 0 + 0.000 466 148 291 764 154 898 841 6;
  • 46) 0.000 466 148 291 764 154 898 841 6 × 2 = 0 + 0.000 932 296 583 528 309 797 683 2;
  • 47) 0.000 932 296 583 528 309 797 683 2 × 2 = 0 + 0.001 864 593 167 056 619 595 366 4;
  • 48) 0.001 864 593 167 056 619 595 366 4 × 2 = 0 + 0.003 729 186 334 113 239 190 732 8;
  • 49) 0.003 729 186 334 113 239 190 732 8 × 2 = 0 + 0.007 458 372 668 226 478 381 465 6;
  • 50) 0.007 458 372 668 226 478 381 465 6 × 2 = 0 + 0.014 916 745 336 452 956 762 931 2;
  • 51) 0.014 916 745 336 452 956 762 931 2 × 2 = 0 + 0.029 833 490 672 905 913 525 862 4;
  • 52) 0.029 833 490 672 905 913 525 862 4 × 2 = 0 + 0.059 666 981 345 811 827 051 724 8;
  • 53) 0.059 666 981 345 811 827 051 724 8 × 2 = 0 + 0.119 333 962 691 623 654 103 449 6;
  • 54) 0.119 333 962 691 623 654 103 449 6 × 2 = 0 + 0.238 667 925 383 247 308 206 899 2;
  • 55) 0.238 667 925 383 247 308 206 899 2 × 2 = 0 + 0.477 335 850 766 494 616 413 798 4;
  • 56) 0.477 335 850 766 494 616 413 798 4 × 2 = 0 + 0.954 671 701 532 989 232 827 596 8;
  • 57) 0.954 671 701 532 989 232 827 596 8 × 2 = 1 + 0.909 343 403 065 978 465 655 193 6;
  • 58) 0.909 343 403 065 978 465 655 193 6 × 2 = 1 + 0.818 686 806 131 956 931 310 387 2;
  • 59) 0.818 686 806 131 956 931 310 387 2 × 2 = 1 + 0.637 373 612 263 913 862 620 774 4;
  • 60) 0.637 373 612 263 913 862 620 774 4 × 2 = 1 + 0.274 747 224 527 827 725 241 548 8;
  • 61) 0.274 747 224 527 827 725 241 548 8 × 2 = 0 + 0.549 494 449 055 655 450 483 097 6;
  • 62) 0.549 494 449 055 655 450 483 097 6 × 2 = 1 + 0.098 988 898 111 310 900 966 195 2;
  • 63) 0.098 988 898 111 310 900 966 195 2 × 2 = 0 + 0.197 977 796 222 621 801 932 390 4;
  • 64) 0.197 977 796 222 621 801 932 390 4 × 2 = 0 + 0.395 955 592 445 243 603 864 780 8;
  • 65) 0.395 955 592 445 243 603 864 780 8 × 2 = 0 + 0.791 911 184 890 487 207 729 561 6;
  • 66) 0.791 911 184 890 487 207 729 561 6 × 2 = 1 + 0.583 822 369 780 974 415 459 123 2;
  • 67) 0.583 822 369 780 974 415 459 123 2 × 2 = 1 + 0.167 644 739 561 948 830 918 246 4;
  • 68) 0.167 644 739 561 948 830 918 246 4 × 2 = 0 + 0.335 289 479 123 897 661 836 492 8;
  • 69) 0.335 289 479 123 897 661 836 492 8 × 2 = 0 + 0.670 578 958 247 795 323 672 985 6;
  • 70) 0.670 578 958 247 795 323 672 985 6 × 2 = 1 + 0.341 157 916 495 590 647 345 971 2;
  • 71) 0.341 157 916 495 590 647 345 971 2 × 2 = 0 + 0.682 315 832 991 181 294 691 942 4;
  • 72) 0.682 315 832 991 181 294 691 942 4 × 2 = 1 + 0.364 631 665 982 362 589 383 884 8;
  • 73) 0.364 631 665 982 362 589 383 884 8 × 2 = 0 + 0.729 263 331 964 725 178 767 769 6;
  • 74) 0.729 263 331 964 725 178 767 769 6 × 2 = 1 + 0.458 526 663 929 450 357 535 539 2;
  • 75) 0.458 526 663 929 450 357 535 539 2 × 2 = 0 + 0.917 053 327 858 900 715 071 078 4;
  • 76) 0.917 053 327 858 900 715 071 078 4 × 2 = 1 + 0.834 106 655 717 801 430 142 156 8;
  • 77) 0.834 106 655 717 801 430 142 156 8 × 2 = 1 + 0.668 213 311 435 602 860 284 313 6;
  • 78) 0.668 213 311 435 602 860 284 313 6 × 2 = 1 + 0.336 426 622 871 205 720 568 627 2;
  • 79) 0.336 426 622 871 205 720 568 627 2 × 2 = 0 + 0.672 853 245 742 411 441 137 254 4;
  • 80) 0.672 853 245 742 411 441 137 254 4 × 2 = 1 + 0.345 706 491 484 822 882 274 508 8;
  • 81) 0.345 706 491 484 822 882 274 508 8 × 2 = 0 + 0.691 412 982 969 645 764 549 017 6;
  • 82) 0.691 412 982 969 645 764 549 017 6 × 2 = 1 + 0.382 825 965 939 291 529 098 035 2;
  • 83) 0.382 825 965 939 291 529 098 035 2 × 2 = 0 + 0.765 651 931 878 583 058 196 070 4;
  • 84) 0.765 651 931 878 583 058 196 070 4 × 2 = 1 + 0.531 303 863 757 166 116 392 140 8;
  • 85) 0.531 303 863 757 166 116 392 140 8 × 2 = 1 + 0.062 607 727 514 332 232 784 281 6;
  • 86) 0.062 607 727 514 332 232 784 281 6 × 2 = 0 + 0.125 215 455 028 664 465 568 563 2;
  • 87) 0.125 215 455 028 664 465 568 563 2 × 2 = 0 + 0.250 430 910 057 328 931 137 126 4;
  • 88) 0.250 430 910 057 328 931 137 126 4 × 2 = 0 + 0.500 861 820 114 657 862 274 252 8;
  • 89) 0.500 861 820 114 657 862 274 252 8 × 2 = 1 + 0.001 723 640 229 315 724 548 505 6;
  • 90) 0.001 723 640 229 315 724 548 505 6 × 2 = 0 + 0.003 447 280 458 631 449 097 011 2;
  • 91) 0.003 447 280 458 631 449 097 011 2 × 2 = 0 + 0.006 894 560 917 262 898 194 022 4;
  • 92) 0.006 894 560 917 262 898 194 022 4 × 2 = 0 + 0.013 789 121 834 525 796 388 044 8;
  • 93) 0.013 789 121 834 525 796 388 044 8 × 2 = 0 + 0.027 578 243 669 051 592 776 089 6;
  • 94) 0.027 578 243 669 051 592 776 089 6 × 2 = 0 + 0.055 156 487 338 103 185 552 179 2;
  • 95) 0.055 156 487 338 103 185 552 179 2 × 2 = 0 + 0.110 312 974 676 206 371 104 358 4;
  • 96) 0.110 312 974 676 206 371 104 358 4 × 2 = 0 + 0.220 625 949 352 412 742 208 716 8;
  • 97) 0.220 625 949 352 412 742 208 716 8 × 2 = 0 + 0.441 251 898 704 825 484 417 433 6;
  • 98) 0.441 251 898 704 825 484 417 433 6 × 2 = 0 + 0.882 503 797 409 650 968 834 867 2;
  • 99) 0.882 503 797 409 650 968 834 867 2 × 2 = 1 + 0.765 007 594 819 301 937 669 734 4;
  • 100) 0.765 007 594 819 301 937 669 734 4 × 2 = 1 + 0.530 015 189 638 603 875 339 468 8;
  • 101) 0.530 015 189 638 603 875 339 468 8 × 2 = 1 + 0.060 030 379 277 207 750 678 937 6;
  • 102) 0.060 030 379 277 207 750 678 937 6 × 2 = 0 + 0.120 060 758 554 415 501 357 875 2;
  • 103) 0.120 060 758 554 415 501 357 875 2 × 2 = 0 + 0.240 121 517 108 831 002 715 750 4;
  • 104) 0.240 121 517 108 831 002 715 750 4 × 2 = 0 + 0.480 243 034 217 662 005 431 500 8;
  • 105) 0.480 243 034 217 662 005 431 500 8 × 2 = 0 + 0.960 486 068 435 324 010 863 001 6;
  • 106) 0.960 486 068 435 324 010 863 001 6 × 2 = 1 + 0.920 972 136 870 648 021 726 003 2;
  • 107) 0.920 972 136 870 648 021 726 003 2 × 2 = 1 + 0.841 944 273 741 296 043 452 006 4;
  • 108) 0.841 944 273 741 296 043 452 006 4 × 2 = 1 + 0.683 888 547 482 592 086 904 012 8;
  • 109) 0.683 888 547 482 592 086 904 012 8 × 2 = 1 + 0.367 777 094 965 184 173 808 025 6;

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:


0.000 000 000 000 000 013 248 731 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0101 1101 0101 1000 1000 0000 0011 1000 0111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 013 248 731 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0101 1101 0101 1000 1000 0000 0011 1000 0111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 013 248 731 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0101 1101 0101 1000 1000 0000 0011 1000 0111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0101 1101 0101 1000 1000 0000 0011 1000 0111 1(2) × 20 =

1.1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111 =

1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111


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 1100 0110

Mantissa (52 bits) =
1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111


Decimal number 0.000 000 000 000 000 013 248 731 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1110 1000 1100 1010 1011 1010 1011 0001 0000 0000 0111 0000 1111

Share

» Convert decimal 0.000 000 000 000 000 013 248 739 9 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 09, 2026 [September]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: September - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. Gemini Notebook YouTube Video of 7.50 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