0.000 000 000 000 000 000 008 523 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 523 5(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
0.000 000 000 000 000 000 008 523 5(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: 0.000 000 000 000 000 000 008 523 5.

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) 0.000 000 000 000 000 000 008 523 5 × 2 = 0 + 0.000 000 000 000 000 000 017 047;
  • 2) 0.000 000 000 000 000 000 017 047 × 2 = 0 + 0.000 000 000 000 000 000 034 094;
  • 3) 0.000 000 000 000 000 000 034 094 × 2 = 0 + 0.000 000 000 000 000 000 068 188;
  • 4) 0.000 000 000 000 000 000 068 188 × 2 = 0 + 0.000 000 000 000 000 000 136 376;
  • 5) 0.000 000 000 000 000 000 136 376 × 2 = 0 + 0.000 000 000 000 000 000 272 752;
  • 6) 0.000 000 000 000 000 000 272 752 × 2 = 0 + 0.000 000 000 000 000 000 545 504;
  • 7) 0.000 000 000 000 000 000 545 504 × 2 = 0 + 0.000 000 000 000 000 001 091 008;
  • 8) 0.000 000 000 000 000 001 091 008 × 2 = 0 + 0.000 000 000 000 000 002 182 016;
  • 9) 0.000 000 000 000 000 002 182 016 × 2 = 0 + 0.000 000 000 000 000 004 364 032;
  • 10) 0.000 000 000 000 000 004 364 032 × 2 = 0 + 0.000 000 000 000 000 008 728 064;
  • 11) 0.000 000 000 000 000 008 728 064 × 2 = 0 + 0.000 000 000 000 000 017 456 128;
  • 12) 0.000 000 000 000 000 017 456 128 × 2 = 0 + 0.000 000 000 000 000 034 912 256;
  • 13) 0.000 000 000 000 000 034 912 256 × 2 = 0 + 0.000 000 000 000 000 069 824 512;
  • 14) 0.000 000 000 000 000 069 824 512 × 2 = 0 + 0.000 000 000 000 000 139 649 024;
  • 15) 0.000 000 000 000 000 139 649 024 × 2 = 0 + 0.000 000 000 000 000 279 298 048;
  • 16) 0.000 000 000 000 000 279 298 048 × 2 = 0 + 0.000 000 000 000 000 558 596 096;
  • 17) 0.000 000 000 000 000 558 596 096 × 2 = 0 + 0.000 000 000 000 001 117 192 192;
  • 18) 0.000 000 000 000 001 117 192 192 × 2 = 0 + 0.000 000 000 000 002 234 384 384;
  • 19) 0.000 000 000 000 002 234 384 384 × 2 = 0 + 0.000 000 000 000 004 468 768 768;
  • 20) 0.000 000 000 000 004 468 768 768 × 2 = 0 + 0.000 000 000 000 008 937 537 536;
  • 21) 0.000 000 000 000 008 937 537 536 × 2 = 0 + 0.000 000 000 000 017 875 075 072;
  • 22) 0.000 000 000 000 017 875 075 072 × 2 = 0 + 0.000 000 000 000 035 750 150 144;
  • 23) 0.000 000 000 000 035 750 150 144 × 2 = 0 + 0.000 000 000 000 071 500 300 288;
  • 24) 0.000 000 000 000 071 500 300 288 × 2 = 0 + 0.000 000 000 000 143 000 600 576;
  • 25) 0.000 000 000 000 143 000 600 576 × 2 = 0 + 0.000 000 000 000 286 001 201 152;
  • 26) 0.000 000 000 000 286 001 201 152 × 2 = 0 + 0.000 000 000 000 572 002 402 304;
  • 27) 0.000 000 000 000 572 002 402 304 × 2 = 0 + 0.000 000 000 001 144 004 804 608;
  • 28) 0.000 000 000 001 144 004 804 608 × 2 = 0 + 0.000 000 000 002 288 009 609 216;
  • 29) 0.000 000 000 002 288 009 609 216 × 2 = 0 + 0.000 000 000 004 576 019 218 432;
  • 30) 0.000 000 000 004 576 019 218 432 × 2 = 0 + 0.000 000 000 009 152 038 436 864;
  • 31) 0.000 000 000 009 152 038 436 864 × 2 = 0 + 0.000 000 000 018 304 076 873 728;
  • 32) 0.000 000 000 018 304 076 873 728 × 2 = 0 + 0.000 000 000 036 608 153 747 456;
  • 33) 0.000 000 000 036 608 153 747 456 × 2 = 0 + 0.000 000 000 073 216 307 494 912;
  • 34) 0.000 000 000 073 216 307 494 912 × 2 = 0 + 0.000 000 000 146 432 614 989 824;
  • 35) 0.000 000 000 146 432 614 989 824 × 2 = 0 + 0.000 000 000 292 865 229 979 648;
  • 36) 0.000 000 000 292 865 229 979 648 × 2 = 0 + 0.000 000 000 585 730 459 959 296;
  • 37) 0.000 000 000 585 730 459 959 296 × 2 = 0 + 0.000 000 001 171 460 919 918 592;
  • 38) 0.000 000 001 171 460 919 918 592 × 2 = 0 + 0.000 000 002 342 921 839 837 184;
  • 39) 0.000 000 002 342 921 839 837 184 × 2 = 0 + 0.000 000 004 685 843 679 674 368;
  • 40) 0.000 000 004 685 843 679 674 368 × 2 = 0 + 0.000 000 009 371 687 359 348 736;
  • 41) 0.000 000 009 371 687 359 348 736 × 2 = 0 + 0.000 000 018 743 374 718 697 472;
  • 42) 0.000 000 018 743 374 718 697 472 × 2 = 0 + 0.000 000 037 486 749 437 394 944;
  • 43) 0.000 000 037 486 749 437 394 944 × 2 = 0 + 0.000 000 074 973 498 874 789 888;
  • 44) 0.000 000 074 973 498 874 789 888 × 2 = 0 + 0.000 000 149 946 997 749 579 776;
  • 45) 0.000 000 149 946 997 749 579 776 × 2 = 0 + 0.000 000 299 893 995 499 159 552;
  • 46) 0.000 000 299 893 995 499 159 552 × 2 = 0 + 0.000 000 599 787 990 998 319 104;
  • 47) 0.000 000 599 787 990 998 319 104 × 2 = 0 + 0.000 001 199 575 981 996 638 208;
  • 48) 0.000 001 199 575 981 996 638 208 × 2 = 0 + 0.000 002 399 151 963 993 276 416;
  • 49) 0.000 002 399 151 963 993 276 416 × 2 = 0 + 0.000 004 798 303 927 986 552 832;
  • 50) 0.000 004 798 303 927 986 552 832 × 2 = 0 + 0.000 009 596 607 855 973 105 664;
  • 51) 0.000 009 596 607 855 973 105 664 × 2 = 0 + 0.000 019 193 215 711 946 211 328;
  • 52) 0.000 019 193 215 711 946 211 328 × 2 = 0 + 0.000 038 386 431 423 892 422 656;
  • 53) 0.000 038 386 431 423 892 422 656 × 2 = 0 + 0.000 076 772 862 847 784 845 312;
  • 54) 0.000 076 772 862 847 784 845 312 × 2 = 0 + 0.000 153 545 725 695 569 690 624;
  • 55) 0.000 153 545 725 695 569 690 624 × 2 = 0 + 0.000 307 091 451 391 139 381 248;
  • 56) 0.000 307 091 451 391 139 381 248 × 2 = 0 + 0.000 614 182 902 782 278 762 496;
  • 57) 0.000 614 182 902 782 278 762 496 × 2 = 0 + 0.001 228 365 805 564 557 524 992;
  • 58) 0.001 228 365 805 564 557 524 992 × 2 = 0 + 0.002 456 731 611 129 115 049 984;
  • 59) 0.002 456 731 611 129 115 049 984 × 2 = 0 + 0.004 913 463 222 258 230 099 968;
  • 60) 0.004 913 463 222 258 230 099 968 × 2 = 0 + 0.009 826 926 444 516 460 199 936;
  • 61) 0.009 826 926 444 516 460 199 936 × 2 = 0 + 0.019 653 852 889 032 920 399 872;
  • 62) 0.019 653 852 889 032 920 399 872 × 2 = 0 + 0.039 307 705 778 065 840 799 744;
  • 63) 0.039 307 705 778 065 840 799 744 × 2 = 0 + 0.078 615 411 556 131 681 599 488;
  • 64) 0.078 615 411 556 131 681 599 488 × 2 = 0 + 0.157 230 823 112 263 363 198 976;
  • 65) 0.157 230 823 112 263 363 198 976 × 2 = 0 + 0.314 461 646 224 526 726 397 952;
  • 66) 0.314 461 646 224 526 726 397 952 × 2 = 0 + 0.628 923 292 449 053 452 795 904;
  • 67) 0.628 923 292 449 053 452 795 904 × 2 = 1 + 0.257 846 584 898 106 905 591 808;
  • 68) 0.257 846 584 898 106 905 591 808 × 2 = 0 + 0.515 693 169 796 213 811 183 616;
  • 69) 0.515 693 169 796 213 811 183 616 × 2 = 1 + 0.031 386 339 592 427 622 367 232;
  • 70) 0.031 386 339 592 427 622 367 232 × 2 = 0 + 0.062 772 679 184 855 244 734 464;
  • 71) 0.062 772 679 184 855 244 734 464 × 2 = 0 + 0.125 545 358 369 710 489 468 928;
  • 72) 0.125 545 358 369 710 489 468 928 × 2 = 0 + 0.251 090 716 739 420 978 937 856;
  • 73) 0.251 090 716 739 420 978 937 856 × 2 = 0 + 0.502 181 433 478 841 957 875 712;
  • 74) 0.502 181 433 478 841 957 875 712 × 2 = 1 + 0.004 362 866 957 683 915 751 424;
  • 75) 0.004 362 866 957 683 915 751 424 × 2 = 0 + 0.008 725 733 915 367 831 502 848;
  • 76) 0.008 725 733 915 367 831 502 848 × 2 = 0 + 0.017 451 467 830 735 663 005 696;
  • 77) 0.017 451 467 830 735 663 005 696 × 2 = 0 + 0.034 902 935 661 471 326 011 392;
  • 78) 0.034 902 935 661 471 326 011 392 × 2 = 0 + 0.069 805 871 322 942 652 022 784;
  • 79) 0.069 805 871 322 942 652 022 784 × 2 = 0 + 0.139 611 742 645 885 304 045 568;
  • 80) 0.139 611 742 645 885 304 045 568 × 2 = 0 + 0.279 223 485 291 770 608 091 136;
  • 81) 0.279 223 485 291 770 608 091 136 × 2 = 0 + 0.558 446 970 583 541 216 182 272;
  • 82) 0.558 446 970 583 541 216 182 272 × 2 = 1 + 0.116 893 941 167 082 432 364 544;
  • 83) 0.116 893 941 167 082 432 364 544 × 2 = 0 + 0.233 787 882 334 164 864 729 088;
  • 84) 0.233 787 882 334 164 864 729 088 × 2 = 0 + 0.467 575 764 668 329 729 458 176;
  • 85) 0.467 575 764 668 329 729 458 176 × 2 = 0 + 0.935 151 529 336 659 458 916 352;
  • 86) 0.935 151 529 336 659 458 916 352 × 2 = 1 + 0.870 303 058 673 318 917 832 704;
  • 87) 0.870 303 058 673 318 917 832 704 × 2 = 1 + 0.740 606 117 346 637 835 665 408;
  • 88) 0.740 606 117 346 637 835 665 408 × 2 = 1 + 0.481 212 234 693 275 671 330 816;
  • 89) 0.481 212 234 693 275 671 330 816 × 2 = 0 + 0.962 424 469 386 551 342 661 632;
  • 90) 0.962 424 469 386 551 342 661 632 × 2 = 1 + 0.924 848 938 773 102 685 323 264;
  • 91) 0.924 848 938 773 102 685 323 264 × 2 = 1 + 0.849 697 877 546 205 370 646 528;
  • 92) 0.849 697 877 546 205 370 646 528 × 2 = 1 + 0.699 395 755 092 410 741 293 056;
  • 93) 0.699 395 755 092 410 741 293 056 × 2 = 1 + 0.398 791 510 184 821 482 586 112;
  • 94) 0.398 791 510 184 821 482 586 112 × 2 = 0 + 0.797 583 020 369 642 965 172 224;
  • 95) 0.797 583 020 369 642 965 172 224 × 2 = 1 + 0.595 166 040 739 285 930 344 448;
  • 96) 0.595 166 040 739 285 930 344 448 × 2 = 1 + 0.190 332 081 478 571 860 688 896;
  • 97) 0.190 332 081 478 571 860 688 896 × 2 = 0 + 0.380 664 162 957 143 721 377 792;
  • 98) 0.380 664 162 957 143 721 377 792 × 2 = 0 + 0.761 328 325 914 287 442 755 584;
  • 99) 0.761 328 325 914 287 442 755 584 × 2 = 1 + 0.522 656 651 828 574 885 511 168;
  • 100) 0.522 656 651 828 574 885 511 168 × 2 = 1 + 0.045 313 303 657 149 771 022 336;
  • 101) 0.045 313 303 657 149 771 022 336 × 2 = 0 + 0.090 626 607 314 299 542 044 672;
  • 102) 0.090 626 607 314 299 542 044 672 × 2 = 0 + 0.181 253 214 628 599 084 089 344;
  • 103) 0.181 253 214 628 599 084 089 344 × 2 = 0 + 0.362 506 429 257 198 168 178 688;
  • 104) 0.362 506 429 257 198 168 178 688 × 2 = 0 + 0.725 012 858 514 396 336 357 376;
  • 105) 0.725 012 858 514 396 336 357 376 × 2 = 1 + 0.450 025 717 028 792 672 714 752;
  • 106) 0.450 025 717 028 792 672 714 752 × 2 = 0 + 0.900 051 434 057 585 345 429 504;
  • 107) 0.900 051 434 057 585 345 429 504 × 2 = 1 + 0.800 102 868 115 170 690 859 008;
  • 108) 0.800 102 868 115 170 690 859 008 × 2 = 1 + 0.600 205 736 230 341 381 718 016;
  • 109) 0.600 205 736 230 341 381 718 016 × 2 = 1 + 0.200 411 472 460 682 763 436 032;
  • 110) 0.200 411 472 460 682 763 436 032 × 2 = 0 + 0.400 822 944 921 365 526 872 064;
  • 111) 0.400 822 944 921 365 526 872 064 × 2 = 0 + 0.801 645 889 842 731 053 744 128;
  • 112) 0.801 645 889 842 731 053 744 128 × 2 = 1 + 0.603 291 779 685 462 107 488 256;
  • 113) 0.603 291 779 685 462 107 488 256 × 2 = 1 + 0.206 583 559 370 924 214 976 512;
  • 114) 0.206 583 559 370 924 214 976 512 × 2 = 0 + 0.413 167 118 741 848 429 953 024;
  • 115) 0.413 167 118 741 848 429 953 024 × 2 = 0 + 0.826 334 237 483 696 859 906 048;
  • 116) 0.826 334 237 483 696 859 906 048 × 2 = 1 + 0.652 668 474 967 393 719 812 096;
  • 117) 0.652 668 474 967 393 719 812 096 × 2 = 1 + 0.305 336 949 934 787 439 624 192;
  • 118) 0.305 336 949 934 787 439 624 192 × 2 = 0 + 0.610 673 899 869 574 879 248 384;
  • 119) 0.610 673 899 869 574 879 248 384 × 2 = 1 + 0.221 347 799 739 149 758 496 768;

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:


0.000 000 000 000 000 000 008 523 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0000 0100 0111 0111 1011 0011 0000 1011 1001 1001 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 523 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0000 0100 0111 0111 1011 0011 0000 1011 1001 1001 101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 523 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0000 0100 0111 0111 1011 0011 0000 1011 1001 1001 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0000 0100 0111 0111 1011 0011 0000 1011 1001 1001 101(2) × 20 =

1.0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

956(10) =

011 1011 1100(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. 0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101 =

0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101


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 1011 1100

Mantissa (52 bits) =
0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101


Decimal number 0.000 000 000 000 000 000 008 523 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0000 0010 0011 1011 1101 1001 1000 0101 1100 1100 1101

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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