0.000 000 000 000 000 013 248 736 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 013 248 736 2(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
0.000 000 000 000 000 013 248 736 2(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: 0.000 000 000 000 000 013 248 736 2.

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) 0.000 000 000 000 000 013 248 736 2 × 2 = 0 + 0.000 000 000 000 000 026 497 472 4;
  • 2) 0.000 000 000 000 000 026 497 472 4 × 2 = 0 + 0.000 000 000 000 000 052 994 944 8;
  • 3) 0.000 000 000 000 000 052 994 944 8 × 2 = 0 + 0.000 000 000 000 000 105 989 889 6;
  • 4) 0.000 000 000 000 000 105 989 889 6 × 2 = 0 + 0.000 000 000 000 000 211 979 779 2;
  • 5) 0.000 000 000 000 000 211 979 779 2 × 2 = 0 + 0.000 000 000 000 000 423 959 558 4;
  • 6) 0.000 000 000 000 000 423 959 558 4 × 2 = 0 + 0.000 000 000 000 000 847 919 116 8;
  • 7) 0.000 000 000 000 000 847 919 116 8 × 2 = 0 + 0.000 000 000 000 001 695 838 233 6;
  • 8) 0.000 000 000 000 001 695 838 233 6 × 2 = 0 + 0.000 000 000 000 003 391 676 467 2;
  • 9) 0.000 000 000 000 003 391 676 467 2 × 2 = 0 + 0.000 000 000 000 006 783 352 934 4;
  • 10) 0.000 000 000 000 006 783 352 934 4 × 2 = 0 + 0.000 000 000 000 013 566 705 868 8;
  • 11) 0.000 000 000 000 013 566 705 868 8 × 2 = 0 + 0.000 000 000 000 027 133 411 737 6;
  • 12) 0.000 000 000 000 027 133 411 737 6 × 2 = 0 + 0.000 000 000 000 054 266 823 475 2;
  • 13) 0.000 000 000 000 054 266 823 475 2 × 2 = 0 + 0.000 000 000 000 108 533 646 950 4;
  • 14) 0.000 000 000 000 108 533 646 950 4 × 2 = 0 + 0.000 000 000 000 217 067 293 900 8;
  • 15) 0.000 000 000 000 217 067 293 900 8 × 2 = 0 + 0.000 000 000 000 434 134 587 801 6;
  • 16) 0.000 000 000 000 434 134 587 801 6 × 2 = 0 + 0.000 000 000 000 868 269 175 603 2;
  • 17) 0.000 000 000 000 868 269 175 603 2 × 2 = 0 + 0.000 000 000 001 736 538 351 206 4;
  • 18) 0.000 000 000 001 736 538 351 206 4 × 2 = 0 + 0.000 000 000 003 473 076 702 412 8;
  • 19) 0.000 000 000 003 473 076 702 412 8 × 2 = 0 + 0.000 000 000 006 946 153 404 825 6;
  • 20) 0.000 000 000 006 946 153 404 825 6 × 2 = 0 + 0.000 000 000 013 892 306 809 651 2;
  • 21) 0.000 000 000 013 892 306 809 651 2 × 2 = 0 + 0.000 000 000 027 784 613 619 302 4;
  • 22) 0.000 000 000 027 784 613 619 302 4 × 2 = 0 + 0.000 000 000 055 569 227 238 604 8;
  • 23) 0.000 000 000 055 569 227 238 604 8 × 2 = 0 + 0.000 000 000 111 138 454 477 209 6;
  • 24) 0.000 000 000 111 138 454 477 209 6 × 2 = 0 + 0.000 000 000 222 276 908 954 419 2;
  • 25) 0.000 000 000 222 276 908 954 419 2 × 2 = 0 + 0.000 000 000 444 553 817 908 838 4;
  • 26) 0.000 000 000 444 553 817 908 838 4 × 2 = 0 + 0.000 000 000 889 107 635 817 676 8;
  • 27) 0.000 000 000 889 107 635 817 676 8 × 2 = 0 + 0.000 000 001 778 215 271 635 353 6;
  • 28) 0.000 000 001 778 215 271 635 353 6 × 2 = 0 + 0.000 000 003 556 430 543 270 707 2;
  • 29) 0.000 000 003 556 430 543 270 707 2 × 2 = 0 + 0.000 000 007 112 861 086 541 414 4;
  • 30) 0.000 000 007 112 861 086 541 414 4 × 2 = 0 + 0.000 000 014 225 722 173 082 828 8;
  • 31) 0.000 000 014 225 722 173 082 828 8 × 2 = 0 + 0.000 000 028 451 444 346 165 657 6;
  • 32) 0.000 000 028 451 444 346 165 657 6 × 2 = 0 + 0.000 000 056 902 888 692 331 315 2;
  • 33) 0.000 000 056 902 888 692 331 315 2 × 2 = 0 + 0.000 000 113 805 777 384 662 630 4;
  • 34) 0.000 000 113 805 777 384 662 630 4 × 2 = 0 + 0.000 000 227 611 554 769 325 260 8;
  • 35) 0.000 000 227 611 554 769 325 260 8 × 2 = 0 + 0.000 000 455 223 109 538 650 521 6;
  • 36) 0.000 000 455 223 109 538 650 521 6 × 2 = 0 + 0.000 000 910 446 219 077 301 043 2;
  • 37) 0.000 000 910 446 219 077 301 043 2 × 2 = 0 + 0.000 001 820 892 438 154 602 086 4;
  • 38) 0.000 001 820 892 438 154 602 086 4 × 2 = 0 + 0.000 003 641 784 876 309 204 172 8;
  • 39) 0.000 003 641 784 876 309 204 172 8 × 2 = 0 + 0.000 007 283 569 752 618 408 345 6;
  • 40) 0.000 007 283 569 752 618 408 345 6 × 2 = 0 + 0.000 014 567 139 505 236 816 691 2;
  • 41) 0.000 014 567 139 505 236 816 691 2 × 2 = 0 + 0.000 029 134 279 010 473 633 382 4;
  • 42) 0.000 029 134 279 010 473 633 382 4 × 2 = 0 + 0.000 058 268 558 020 947 266 764 8;
  • 43) 0.000 058 268 558 020 947 266 764 8 × 2 = 0 + 0.000 116 537 116 041 894 533 529 6;
  • 44) 0.000 116 537 116 041 894 533 529 6 × 2 = 0 + 0.000 233 074 232 083 789 067 059 2;
  • 45) 0.000 233 074 232 083 789 067 059 2 × 2 = 0 + 0.000 466 148 464 167 578 134 118 4;
  • 46) 0.000 466 148 464 167 578 134 118 4 × 2 = 0 + 0.000 932 296 928 335 156 268 236 8;
  • 47) 0.000 932 296 928 335 156 268 236 8 × 2 = 0 + 0.001 864 593 856 670 312 536 473 6;
  • 48) 0.001 864 593 856 670 312 536 473 6 × 2 = 0 + 0.003 729 187 713 340 625 072 947 2;
  • 49) 0.003 729 187 713 340 625 072 947 2 × 2 = 0 + 0.007 458 375 426 681 250 145 894 4;
  • 50) 0.007 458 375 426 681 250 145 894 4 × 2 = 0 + 0.014 916 750 853 362 500 291 788 8;
  • 51) 0.014 916 750 853 362 500 291 788 8 × 2 = 0 + 0.029 833 501 706 725 000 583 577 6;
  • 52) 0.029 833 501 706 725 000 583 577 6 × 2 = 0 + 0.059 667 003 413 450 001 167 155 2;
  • 53) 0.059 667 003 413 450 001 167 155 2 × 2 = 0 + 0.119 334 006 826 900 002 334 310 4;
  • 54) 0.119 334 006 826 900 002 334 310 4 × 2 = 0 + 0.238 668 013 653 800 004 668 620 8;
  • 55) 0.238 668 013 653 800 004 668 620 8 × 2 = 0 + 0.477 336 027 307 600 009 337 241 6;
  • 56) 0.477 336 027 307 600 009 337 241 6 × 2 = 0 + 0.954 672 054 615 200 018 674 483 2;
  • 57) 0.954 672 054 615 200 018 674 483 2 × 2 = 1 + 0.909 344 109 230 400 037 348 966 4;
  • 58) 0.909 344 109 230 400 037 348 966 4 × 2 = 1 + 0.818 688 218 460 800 074 697 932 8;
  • 59) 0.818 688 218 460 800 074 697 932 8 × 2 = 1 + 0.637 376 436 921 600 149 395 865 6;
  • 60) 0.637 376 436 921 600 149 395 865 6 × 2 = 1 + 0.274 752 873 843 200 298 791 731 2;
  • 61) 0.274 752 873 843 200 298 791 731 2 × 2 = 0 + 0.549 505 747 686 400 597 583 462 4;
  • 62) 0.549 505 747 686 400 597 583 462 4 × 2 = 1 + 0.099 011 495 372 801 195 166 924 8;
  • 63) 0.099 011 495 372 801 195 166 924 8 × 2 = 0 + 0.198 022 990 745 602 390 333 849 6;
  • 64) 0.198 022 990 745 602 390 333 849 6 × 2 = 0 + 0.396 045 981 491 204 780 667 699 2;
  • 65) 0.396 045 981 491 204 780 667 699 2 × 2 = 0 + 0.792 091 962 982 409 561 335 398 4;
  • 66) 0.792 091 962 982 409 561 335 398 4 × 2 = 1 + 0.584 183 925 964 819 122 670 796 8;
  • 67) 0.584 183 925 964 819 122 670 796 8 × 2 = 1 + 0.168 367 851 929 638 245 341 593 6;
  • 68) 0.168 367 851 929 638 245 341 593 6 × 2 = 0 + 0.336 735 703 859 276 490 683 187 2;
  • 69) 0.336 735 703 859 276 490 683 187 2 × 2 = 0 + 0.673 471 407 718 552 981 366 374 4;
  • 70) 0.673 471 407 718 552 981 366 374 4 × 2 = 1 + 0.346 942 815 437 105 962 732 748 8;
  • 71) 0.346 942 815 437 105 962 732 748 8 × 2 = 0 + 0.693 885 630 874 211 925 465 497 6;
  • 72) 0.693 885 630 874 211 925 465 497 6 × 2 = 1 + 0.387 771 261 748 423 850 930 995 2;
  • 73) 0.387 771 261 748 423 850 930 995 2 × 2 = 0 + 0.775 542 523 496 847 701 861 990 4;
  • 74) 0.775 542 523 496 847 701 861 990 4 × 2 = 1 + 0.551 085 046 993 695 403 723 980 8;
  • 75) 0.551 085 046 993 695 403 723 980 8 × 2 = 1 + 0.102 170 093 987 390 807 447 961 6;
  • 76) 0.102 170 093 987 390 807 447 961 6 × 2 = 0 + 0.204 340 187 974 781 614 895 923 2;
  • 77) 0.204 340 187 974 781 614 895 923 2 × 2 = 0 + 0.408 680 375 949 563 229 791 846 4;
  • 78) 0.408 680 375 949 563 229 791 846 4 × 2 = 0 + 0.817 360 751 899 126 459 583 692 8;
  • 79) 0.817 360 751 899 126 459 583 692 8 × 2 = 1 + 0.634 721 503 798 252 919 167 385 6;
  • 80) 0.634 721 503 798 252 919 167 385 6 × 2 = 1 + 0.269 443 007 596 505 838 334 771 2;
  • 81) 0.269 443 007 596 505 838 334 771 2 × 2 = 0 + 0.538 886 015 193 011 676 669 542 4;
  • 82) 0.538 886 015 193 011 676 669 542 4 × 2 = 1 + 0.077 772 030 386 023 353 339 084 8;
  • 83) 0.077 772 030 386 023 353 339 084 8 × 2 = 0 + 0.155 544 060 772 046 706 678 169 6;
  • 84) 0.155 544 060 772 046 706 678 169 6 × 2 = 0 + 0.311 088 121 544 093 413 356 339 2;
  • 85) 0.311 088 121 544 093 413 356 339 2 × 2 = 0 + 0.622 176 243 088 186 826 712 678 4;
  • 86) 0.622 176 243 088 186 826 712 678 4 × 2 = 1 + 0.244 352 486 176 373 653 425 356 8;
  • 87) 0.244 352 486 176 373 653 425 356 8 × 2 = 0 + 0.488 704 972 352 747 306 850 713 6;
  • 88) 0.488 704 972 352 747 306 850 713 6 × 2 = 0 + 0.977 409 944 705 494 613 701 427 2;
  • 89) 0.977 409 944 705 494 613 701 427 2 × 2 = 1 + 0.954 819 889 410 989 227 402 854 4;
  • 90) 0.954 819 889 410 989 227 402 854 4 × 2 = 1 + 0.909 639 778 821 978 454 805 708 8;
  • 91) 0.909 639 778 821 978 454 805 708 8 × 2 = 1 + 0.819 279 557 643 956 909 611 417 6;
  • 92) 0.819 279 557 643 956 909 611 417 6 × 2 = 1 + 0.638 559 115 287 913 819 222 835 2;
  • 93) 0.638 559 115 287 913 819 222 835 2 × 2 = 1 + 0.277 118 230 575 827 638 445 670 4;
  • 94) 0.277 118 230 575 827 638 445 670 4 × 2 = 0 + 0.554 236 461 151 655 276 891 340 8;
  • 95) 0.554 236 461 151 655 276 891 340 8 × 2 = 1 + 0.108 472 922 303 310 553 782 681 6;
  • 96) 0.108 472 922 303 310 553 782 681 6 × 2 = 0 + 0.216 945 844 606 621 107 565 363 2;
  • 97) 0.216 945 844 606 621 107 565 363 2 × 2 = 0 + 0.433 891 689 213 242 215 130 726 4;
  • 98) 0.433 891 689 213 242 215 130 726 4 × 2 = 0 + 0.867 783 378 426 484 430 261 452 8;
  • 99) 0.867 783 378 426 484 430 261 452 8 × 2 = 1 + 0.735 566 756 852 968 860 522 905 6;
  • 100) 0.735 566 756 852 968 860 522 905 6 × 2 = 1 + 0.471 133 513 705 937 721 045 811 2;
  • 101) 0.471 133 513 705 937 721 045 811 2 × 2 = 0 + 0.942 267 027 411 875 442 091 622 4;
  • 102) 0.942 267 027 411 875 442 091 622 4 × 2 = 1 + 0.884 534 054 823 750 884 183 244 8;
  • 103) 0.884 534 054 823 750 884 183 244 8 × 2 = 1 + 0.769 068 109 647 501 768 366 489 6;
  • 104) 0.769 068 109 647 501 768 366 489 6 × 2 = 1 + 0.538 136 219 295 003 536 732 979 2;
  • 105) 0.538 136 219 295 003 536 732 979 2 × 2 = 1 + 0.076 272 438 590 007 073 465 958 4;
  • 106) 0.076 272 438 590 007 073 465 958 4 × 2 = 0 + 0.152 544 877 180 014 146 931 916 8;
  • 107) 0.152 544 877 180 014 146 931 916 8 × 2 = 0 + 0.305 089 754 360 028 293 863 833 6;
  • 108) 0.305 089 754 360 028 293 863 833 6 × 2 = 0 + 0.610 179 508 720 056 587 727 667 2;
  • 109) 0.610 179 508 720 056 587 727 667 2 × 2 = 1 + 0.220 359 017 440 113 175 455 334 4;

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:


0.000 000 000 000 000 013 248 736 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0100 0100 1111 1010 0011 0111 1000 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 013 248 736 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0100 0100 1111 1010 0011 0111 1000 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:


0.000 000 000 000 000 013 248 736 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0100 0100 1111 1010 0011 0111 1000 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0100 0100 1111 1010 0011 0111 1000 1(2) × 20 =

1.1110 1000 1100 1010 1100 0110 1000 1001 1111 0100 0110 1111 0001(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 1100 0110 1000 1001 1111 0100 0110 1111 0001


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 1100 0110 1000 1001 1111 0100 0110 1111 0001 =

1110 1000 1100 1010 1100 0110 1000 1001 1111 0100 0110 1111 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 1100 0110

Mantissa (52 bits) =
1110 1000 1100 1010 1100 0110 1000 1001 1111 0100 0110 1111 0001


Decimal number 0.000 000 000 000 000 013 248 736 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1110 1000 1100 1010 1100 0110 1000 1001 1111 0100 0110 1111 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