0.000 000 000 000 000 000 008 448 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 448(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
0.000 000 000 000 000 000 008 448(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: 0.000 000 000 000 000 000 008 448.

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) 0.000 000 000 000 000 000 008 448 × 2 = 0 + 0.000 000 000 000 000 000 016 896;
  • 2) 0.000 000 000 000 000 000 016 896 × 2 = 0 + 0.000 000 000 000 000 000 033 792;
  • 3) 0.000 000 000 000 000 000 033 792 × 2 = 0 + 0.000 000 000 000 000 000 067 584;
  • 4) 0.000 000 000 000 000 000 067 584 × 2 = 0 + 0.000 000 000 000 000 000 135 168;
  • 5) 0.000 000 000 000 000 000 135 168 × 2 = 0 + 0.000 000 000 000 000 000 270 336;
  • 6) 0.000 000 000 000 000 000 270 336 × 2 = 0 + 0.000 000 000 000 000 000 540 672;
  • 7) 0.000 000 000 000 000 000 540 672 × 2 = 0 + 0.000 000 000 000 000 001 081 344;
  • 8) 0.000 000 000 000 000 001 081 344 × 2 = 0 + 0.000 000 000 000 000 002 162 688;
  • 9) 0.000 000 000 000 000 002 162 688 × 2 = 0 + 0.000 000 000 000 000 004 325 376;
  • 10) 0.000 000 000 000 000 004 325 376 × 2 = 0 + 0.000 000 000 000 000 008 650 752;
  • 11) 0.000 000 000 000 000 008 650 752 × 2 = 0 + 0.000 000 000 000 000 017 301 504;
  • 12) 0.000 000 000 000 000 017 301 504 × 2 = 0 + 0.000 000 000 000 000 034 603 008;
  • 13) 0.000 000 000 000 000 034 603 008 × 2 = 0 + 0.000 000 000 000 000 069 206 016;
  • 14) 0.000 000 000 000 000 069 206 016 × 2 = 0 + 0.000 000 000 000 000 138 412 032;
  • 15) 0.000 000 000 000 000 138 412 032 × 2 = 0 + 0.000 000 000 000 000 276 824 064;
  • 16) 0.000 000 000 000 000 276 824 064 × 2 = 0 + 0.000 000 000 000 000 553 648 128;
  • 17) 0.000 000 000 000 000 553 648 128 × 2 = 0 + 0.000 000 000 000 001 107 296 256;
  • 18) 0.000 000 000 000 001 107 296 256 × 2 = 0 + 0.000 000 000 000 002 214 592 512;
  • 19) 0.000 000 000 000 002 214 592 512 × 2 = 0 + 0.000 000 000 000 004 429 185 024;
  • 20) 0.000 000 000 000 004 429 185 024 × 2 = 0 + 0.000 000 000 000 008 858 370 048;
  • 21) 0.000 000 000 000 008 858 370 048 × 2 = 0 + 0.000 000 000 000 017 716 740 096;
  • 22) 0.000 000 000 000 017 716 740 096 × 2 = 0 + 0.000 000 000 000 035 433 480 192;
  • 23) 0.000 000 000 000 035 433 480 192 × 2 = 0 + 0.000 000 000 000 070 866 960 384;
  • 24) 0.000 000 000 000 070 866 960 384 × 2 = 0 + 0.000 000 000 000 141 733 920 768;
  • 25) 0.000 000 000 000 141 733 920 768 × 2 = 0 + 0.000 000 000 000 283 467 841 536;
  • 26) 0.000 000 000 000 283 467 841 536 × 2 = 0 + 0.000 000 000 000 566 935 683 072;
  • 27) 0.000 000 000 000 566 935 683 072 × 2 = 0 + 0.000 000 000 001 133 871 366 144;
  • 28) 0.000 000 000 001 133 871 366 144 × 2 = 0 + 0.000 000 000 002 267 742 732 288;
  • 29) 0.000 000 000 002 267 742 732 288 × 2 = 0 + 0.000 000 000 004 535 485 464 576;
  • 30) 0.000 000 000 004 535 485 464 576 × 2 = 0 + 0.000 000 000 009 070 970 929 152;
  • 31) 0.000 000 000 009 070 970 929 152 × 2 = 0 + 0.000 000 000 018 141 941 858 304;
  • 32) 0.000 000 000 018 141 941 858 304 × 2 = 0 + 0.000 000 000 036 283 883 716 608;
  • 33) 0.000 000 000 036 283 883 716 608 × 2 = 0 + 0.000 000 000 072 567 767 433 216;
  • 34) 0.000 000 000 072 567 767 433 216 × 2 = 0 + 0.000 000 000 145 135 534 866 432;
  • 35) 0.000 000 000 145 135 534 866 432 × 2 = 0 + 0.000 000 000 290 271 069 732 864;
  • 36) 0.000 000 000 290 271 069 732 864 × 2 = 0 + 0.000 000 000 580 542 139 465 728;
  • 37) 0.000 000 000 580 542 139 465 728 × 2 = 0 + 0.000 000 001 161 084 278 931 456;
  • 38) 0.000 000 001 161 084 278 931 456 × 2 = 0 + 0.000 000 002 322 168 557 862 912;
  • 39) 0.000 000 002 322 168 557 862 912 × 2 = 0 + 0.000 000 004 644 337 115 725 824;
  • 40) 0.000 000 004 644 337 115 725 824 × 2 = 0 + 0.000 000 009 288 674 231 451 648;
  • 41) 0.000 000 009 288 674 231 451 648 × 2 = 0 + 0.000 000 018 577 348 462 903 296;
  • 42) 0.000 000 018 577 348 462 903 296 × 2 = 0 + 0.000 000 037 154 696 925 806 592;
  • 43) 0.000 000 037 154 696 925 806 592 × 2 = 0 + 0.000 000 074 309 393 851 613 184;
  • 44) 0.000 000 074 309 393 851 613 184 × 2 = 0 + 0.000 000 148 618 787 703 226 368;
  • 45) 0.000 000 148 618 787 703 226 368 × 2 = 0 + 0.000 000 297 237 575 406 452 736;
  • 46) 0.000 000 297 237 575 406 452 736 × 2 = 0 + 0.000 000 594 475 150 812 905 472;
  • 47) 0.000 000 594 475 150 812 905 472 × 2 = 0 + 0.000 001 188 950 301 625 810 944;
  • 48) 0.000 001 188 950 301 625 810 944 × 2 = 0 + 0.000 002 377 900 603 251 621 888;
  • 49) 0.000 002 377 900 603 251 621 888 × 2 = 0 + 0.000 004 755 801 206 503 243 776;
  • 50) 0.000 004 755 801 206 503 243 776 × 2 = 0 + 0.000 009 511 602 413 006 487 552;
  • 51) 0.000 009 511 602 413 006 487 552 × 2 = 0 + 0.000 019 023 204 826 012 975 104;
  • 52) 0.000 019 023 204 826 012 975 104 × 2 = 0 + 0.000 038 046 409 652 025 950 208;
  • 53) 0.000 038 046 409 652 025 950 208 × 2 = 0 + 0.000 076 092 819 304 051 900 416;
  • 54) 0.000 076 092 819 304 051 900 416 × 2 = 0 + 0.000 152 185 638 608 103 800 832;
  • 55) 0.000 152 185 638 608 103 800 832 × 2 = 0 + 0.000 304 371 277 216 207 601 664;
  • 56) 0.000 304 371 277 216 207 601 664 × 2 = 0 + 0.000 608 742 554 432 415 203 328;
  • 57) 0.000 608 742 554 432 415 203 328 × 2 = 0 + 0.001 217 485 108 864 830 406 656;
  • 58) 0.001 217 485 108 864 830 406 656 × 2 = 0 + 0.002 434 970 217 729 660 813 312;
  • 59) 0.002 434 970 217 729 660 813 312 × 2 = 0 + 0.004 869 940 435 459 321 626 624;
  • 60) 0.004 869 940 435 459 321 626 624 × 2 = 0 + 0.009 739 880 870 918 643 253 248;
  • 61) 0.009 739 880 870 918 643 253 248 × 2 = 0 + 0.019 479 761 741 837 286 506 496;
  • 62) 0.019 479 761 741 837 286 506 496 × 2 = 0 + 0.038 959 523 483 674 573 012 992;
  • 63) 0.038 959 523 483 674 573 012 992 × 2 = 0 + 0.077 919 046 967 349 146 025 984;
  • 64) 0.077 919 046 967 349 146 025 984 × 2 = 0 + 0.155 838 093 934 698 292 051 968;
  • 65) 0.155 838 093 934 698 292 051 968 × 2 = 0 + 0.311 676 187 869 396 584 103 936;
  • 66) 0.311 676 187 869 396 584 103 936 × 2 = 0 + 0.623 352 375 738 793 168 207 872;
  • 67) 0.623 352 375 738 793 168 207 872 × 2 = 1 + 0.246 704 751 477 586 336 415 744;
  • 68) 0.246 704 751 477 586 336 415 744 × 2 = 0 + 0.493 409 502 955 172 672 831 488;
  • 69) 0.493 409 502 955 172 672 831 488 × 2 = 0 + 0.986 819 005 910 345 345 662 976;
  • 70) 0.986 819 005 910 345 345 662 976 × 2 = 1 + 0.973 638 011 820 690 691 325 952;
  • 71) 0.973 638 011 820 690 691 325 952 × 2 = 1 + 0.947 276 023 641 381 382 651 904;
  • 72) 0.947 276 023 641 381 382 651 904 × 2 = 1 + 0.894 552 047 282 762 765 303 808;
  • 73) 0.894 552 047 282 762 765 303 808 × 2 = 1 + 0.789 104 094 565 525 530 607 616;
  • 74) 0.789 104 094 565 525 530 607 616 × 2 = 1 + 0.578 208 189 131 051 061 215 232;
  • 75) 0.578 208 189 131 051 061 215 232 × 2 = 1 + 0.156 416 378 262 102 122 430 464;
  • 76) 0.156 416 378 262 102 122 430 464 × 2 = 0 + 0.312 832 756 524 204 244 860 928;
  • 77) 0.312 832 756 524 204 244 860 928 × 2 = 0 + 0.625 665 513 048 408 489 721 856;
  • 78) 0.625 665 513 048 408 489 721 856 × 2 = 1 + 0.251 331 026 096 816 979 443 712;
  • 79) 0.251 331 026 096 816 979 443 712 × 2 = 0 + 0.502 662 052 193 633 958 887 424;
  • 80) 0.502 662 052 193 633 958 887 424 × 2 = 1 + 0.005 324 104 387 267 917 774 848;
  • 81) 0.005 324 104 387 267 917 774 848 × 2 = 0 + 0.010 648 208 774 535 835 549 696;
  • 82) 0.010 648 208 774 535 835 549 696 × 2 = 0 + 0.021 296 417 549 071 671 099 392;
  • 83) 0.021 296 417 549 071 671 099 392 × 2 = 0 + 0.042 592 835 098 143 342 198 784;
  • 84) 0.042 592 835 098 143 342 198 784 × 2 = 0 + 0.085 185 670 196 286 684 397 568;
  • 85) 0.085 185 670 196 286 684 397 568 × 2 = 0 + 0.170 371 340 392 573 368 795 136;
  • 86) 0.170 371 340 392 573 368 795 136 × 2 = 0 + 0.340 742 680 785 146 737 590 272;
  • 87) 0.340 742 680 785 146 737 590 272 × 2 = 0 + 0.681 485 361 570 293 475 180 544;
  • 88) 0.681 485 361 570 293 475 180 544 × 2 = 1 + 0.362 970 723 140 586 950 361 088;
  • 89) 0.362 970 723 140 586 950 361 088 × 2 = 0 + 0.725 941 446 281 173 900 722 176;
  • 90) 0.725 941 446 281 173 900 722 176 × 2 = 1 + 0.451 882 892 562 347 801 444 352;
  • 91) 0.451 882 892 562 347 801 444 352 × 2 = 0 + 0.903 765 785 124 695 602 888 704;
  • 92) 0.903 765 785 124 695 602 888 704 × 2 = 1 + 0.807 531 570 249 391 205 777 408;
  • 93) 0.807 531 570 249 391 205 777 408 × 2 = 1 + 0.615 063 140 498 782 411 554 816;
  • 94) 0.615 063 140 498 782 411 554 816 × 2 = 1 + 0.230 126 280 997 564 823 109 632;
  • 95) 0.230 126 280 997 564 823 109 632 × 2 = 0 + 0.460 252 561 995 129 646 219 264;
  • 96) 0.460 252 561 995 129 646 219 264 × 2 = 0 + 0.920 505 123 990 259 292 438 528;
  • 97) 0.920 505 123 990 259 292 438 528 × 2 = 1 + 0.841 010 247 980 518 584 877 056;
  • 98) 0.841 010 247 980 518 584 877 056 × 2 = 1 + 0.682 020 495 961 037 169 754 112;
  • 99) 0.682 020 495 961 037 169 754 112 × 2 = 1 + 0.364 040 991 922 074 339 508 224;
  • 100) 0.364 040 991 922 074 339 508 224 × 2 = 0 + 0.728 081 983 844 148 679 016 448;
  • 101) 0.728 081 983 844 148 679 016 448 × 2 = 1 + 0.456 163 967 688 297 358 032 896;
  • 102) 0.456 163 967 688 297 358 032 896 × 2 = 0 + 0.912 327 935 376 594 716 065 792;
  • 103) 0.912 327 935 376 594 716 065 792 × 2 = 1 + 0.824 655 870 753 189 432 131 584;
  • 104) 0.824 655 870 753 189 432 131 584 × 2 = 1 + 0.649 311 741 506 378 864 263 168;
  • 105) 0.649 311 741 506 378 864 263 168 × 2 = 1 + 0.298 623 483 012 757 728 526 336;
  • 106) 0.298 623 483 012 757 728 526 336 × 2 = 0 + 0.597 246 966 025 515 457 052 672;
  • 107) 0.597 246 966 025 515 457 052 672 × 2 = 1 + 0.194 493 932 051 030 914 105 344;
  • 108) 0.194 493 932 051 030 914 105 344 × 2 = 0 + 0.388 987 864 102 061 828 210 688;
  • 109) 0.388 987 864 102 061 828 210 688 × 2 = 0 + 0.777 975 728 204 123 656 421 376;
  • 110) 0.777 975 728 204 123 656 421 376 × 2 = 1 + 0.555 951 456 408 247 312 842 752;
  • 111) 0.555 951 456 408 247 312 842 752 × 2 = 1 + 0.111 902 912 816 494 625 685 504;
  • 112) 0.111 902 912 816 494 625 685 504 × 2 = 0 + 0.223 805 825 632 989 251 371 008;
  • 113) 0.223 805 825 632 989 251 371 008 × 2 = 0 + 0.447 611 651 265 978 502 742 016;
  • 114) 0.447 611 651 265 978 502 742 016 × 2 = 0 + 0.895 223 302 531 957 005 484 032;
  • 115) 0.895 223 302 531 957 005 484 032 × 2 = 1 + 0.790 446 605 063 914 010 968 064;
  • 116) 0.790 446 605 063 914 010 968 064 × 2 = 1 + 0.580 893 210 127 828 021 936 128;
  • 117) 0.580 893 210 127 828 021 936 128 × 2 = 1 + 0.161 786 420 255 656 043 872 256;
  • 118) 0.161 786 420 255 656 043 872 256 × 2 = 0 + 0.323 572 840 511 312 087 744 512;
  • 119) 0.323 572 840 511 312 087 744 512 × 2 = 0 + 0.647 145 681 022 624 175 489 024;

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:


0.000 000 000 000 000 000 008 448(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1110 0101 0000 0001 0101 1100 1110 1011 1010 0110 0011 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 448(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1110 0101 0000 0001 0101 1100 1110 1011 1010 0110 0011 100(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:


0.000 000 000 000 000 000 008 448(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1110 0101 0000 0001 0101 1100 1110 1011 1010 0110 0011 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1110 0101 0000 0001 0101 1100 1110 1011 1010 0110 0011 100(2) × 20 =

1.0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 1100(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):
1.0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 1100


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. 0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 1100 =

0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 1100


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) =
0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 1100


Decimal number 0.000 000 000 000 000 000 008 448 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0011 1111 0010 1000 0000 1010 1110 0111 0101 1101 0011 0001 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