0.000 000 000 000 000 000 000 083 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 083(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
0.000 000 000 000 000 000 000 083(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: 0.000 000 000 000 000 000 000 083.

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) 0.000 000 000 000 000 000 000 083 × 2 = 0 + 0.000 000 000 000 000 000 000 166;
  • 2) 0.000 000 000 000 000 000 000 166 × 2 = 0 + 0.000 000 000 000 000 000 000 332;
  • 3) 0.000 000 000 000 000 000 000 332 × 2 = 0 + 0.000 000 000 000 000 000 000 664;
  • 4) 0.000 000 000 000 000 000 000 664 × 2 = 0 + 0.000 000 000 000 000 000 001 328;
  • 5) 0.000 000 000 000 000 000 001 328 × 2 = 0 + 0.000 000 000 000 000 000 002 656;
  • 6) 0.000 000 000 000 000 000 002 656 × 2 = 0 + 0.000 000 000 000 000 000 005 312;
  • 7) 0.000 000 000 000 000 000 005 312 × 2 = 0 + 0.000 000 000 000 000 000 010 624;
  • 8) 0.000 000 000 000 000 000 010 624 × 2 = 0 + 0.000 000 000 000 000 000 021 248;
  • 9) 0.000 000 000 000 000 000 021 248 × 2 = 0 + 0.000 000 000 000 000 000 042 496;
  • 10) 0.000 000 000 000 000 000 042 496 × 2 = 0 + 0.000 000 000 000 000 000 084 992;
  • 11) 0.000 000 000 000 000 000 084 992 × 2 = 0 + 0.000 000 000 000 000 000 169 984;
  • 12) 0.000 000 000 000 000 000 169 984 × 2 = 0 + 0.000 000 000 000 000 000 339 968;
  • 13) 0.000 000 000 000 000 000 339 968 × 2 = 0 + 0.000 000 000 000 000 000 679 936;
  • 14) 0.000 000 000 000 000 000 679 936 × 2 = 0 + 0.000 000 000 000 000 001 359 872;
  • 15) 0.000 000 000 000 000 001 359 872 × 2 = 0 + 0.000 000 000 000 000 002 719 744;
  • 16) 0.000 000 000 000 000 002 719 744 × 2 = 0 + 0.000 000 000 000 000 005 439 488;
  • 17) 0.000 000 000 000 000 005 439 488 × 2 = 0 + 0.000 000 000 000 000 010 878 976;
  • 18) 0.000 000 000 000 000 010 878 976 × 2 = 0 + 0.000 000 000 000 000 021 757 952;
  • 19) 0.000 000 000 000 000 021 757 952 × 2 = 0 + 0.000 000 000 000 000 043 515 904;
  • 20) 0.000 000 000 000 000 043 515 904 × 2 = 0 + 0.000 000 000 000 000 087 031 808;
  • 21) 0.000 000 000 000 000 087 031 808 × 2 = 0 + 0.000 000 000 000 000 174 063 616;
  • 22) 0.000 000 000 000 000 174 063 616 × 2 = 0 + 0.000 000 000 000 000 348 127 232;
  • 23) 0.000 000 000 000 000 348 127 232 × 2 = 0 + 0.000 000 000 000 000 696 254 464;
  • 24) 0.000 000 000 000 000 696 254 464 × 2 = 0 + 0.000 000 000 000 001 392 508 928;
  • 25) 0.000 000 000 000 001 392 508 928 × 2 = 0 + 0.000 000 000 000 002 785 017 856;
  • 26) 0.000 000 000 000 002 785 017 856 × 2 = 0 + 0.000 000 000 000 005 570 035 712;
  • 27) 0.000 000 000 000 005 570 035 712 × 2 = 0 + 0.000 000 000 000 011 140 071 424;
  • 28) 0.000 000 000 000 011 140 071 424 × 2 = 0 + 0.000 000 000 000 022 280 142 848;
  • 29) 0.000 000 000 000 022 280 142 848 × 2 = 0 + 0.000 000 000 000 044 560 285 696;
  • 30) 0.000 000 000 000 044 560 285 696 × 2 = 0 + 0.000 000 000 000 089 120 571 392;
  • 31) 0.000 000 000 000 089 120 571 392 × 2 = 0 + 0.000 000 000 000 178 241 142 784;
  • 32) 0.000 000 000 000 178 241 142 784 × 2 = 0 + 0.000 000 000 000 356 482 285 568;
  • 33) 0.000 000 000 000 356 482 285 568 × 2 = 0 + 0.000 000 000 000 712 964 571 136;
  • 34) 0.000 000 000 000 712 964 571 136 × 2 = 0 + 0.000 000 000 001 425 929 142 272;
  • 35) 0.000 000 000 001 425 929 142 272 × 2 = 0 + 0.000 000 000 002 851 858 284 544;
  • 36) 0.000 000 000 002 851 858 284 544 × 2 = 0 + 0.000 000 000 005 703 716 569 088;
  • 37) 0.000 000 000 005 703 716 569 088 × 2 = 0 + 0.000 000 000 011 407 433 138 176;
  • 38) 0.000 000 000 011 407 433 138 176 × 2 = 0 + 0.000 000 000 022 814 866 276 352;
  • 39) 0.000 000 000 022 814 866 276 352 × 2 = 0 + 0.000 000 000 045 629 732 552 704;
  • 40) 0.000 000 000 045 629 732 552 704 × 2 = 0 + 0.000 000 000 091 259 465 105 408;
  • 41) 0.000 000 000 091 259 465 105 408 × 2 = 0 + 0.000 000 000 182 518 930 210 816;
  • 42) 0.000 000 000 182 518 930 210 816 × 2 = 0 + 0.000 000 000 365 037 860 421 632;
  • 43) 0.000 000 000 365 037 860 421 632 × 2 = 0 + 0.000 000 000 730 075 720 843 264;
  • 44) 0.000 000 000 730 075 720 843 264 × 2 = 0 + 0.000 000 001 460 151 441 686 528;
  • 45) 0.000 000 001 460 151 441 686 528 × 2 = 0 + 0.000 000 002 920 302 883 373 056;
  • 46) 0.000 000 002 920 302 883 373 056 × 2 = 0 + 0.000 000 005 840 605 766 746 112;
  • 47) 0.000 000 005 840 605 766 746 112 × 2 = 0 + 0.000 000 011 681 211 533 492 224;
  • 48) 0.000 000 011 681 211 533 492 224 × 2 = 0 + 0.000 000 023 362 423 066 984 448;
  • 49) 0.000 000 023 362 423 066 984 448 × 2 = 0 + 0.000 000 046 724 846 133 968 896;
  • 50) 0.000 000 046 724 846 133 968 896 × 2 = 0 + 0.000 000 093 449 692 267 937 792;
  • 51) 0.000 000 093 449 692 267 937 792 × 2 = 0 + 0.000 000 186 899 384 535 875 584;
  • 52) 0.000 000 186 899 384 535 875 584 × 2 = 0 + 0.000 000 373 798 769 071 751 168;
  • 53) 0.000 000 373 798 769 071 751 168 × 2 = 0 + 0.000 000 747 597 538 143 502 336;
  • 54) 0.000 000 747 597 538 143 502 336 × 2 = 0 + 0.000 001 495 195 076 287 004 672;
  • 55) 0.000 001 495 195 076 287 004 672 × 2 = 0 + 0.000 002 990 390 152 574 009 344;
  • 56) 0.000 002 990 390 152 574 009 344 × 2 = 0 + 0.000 005 980 780 305 148 018 688;
  • 57) 0.000 005 980 780 305 148 018 688 × 2 = 0 + 0.000 011 961 560 610 296 037 376;
  • 58) 0.000 011 961 560 610 296 037 376 × 2 = 0 + 0.000 023 923 121 220 592 074 752;
  • 59) 0.000 023 923 121 220 592 074 752 × 2 = 0 + 0.000 047 846 242 441 184 149 504;
  • 60) 0.000 047 846 242 441 184 149 504 × 2 = 0 + 0.000 095 692 484 882 368 299 008;
  • 61) 0.000 095 692 484 882 368 299 008 × 2 = 0 + 0.000 191 384 969 764 736 598 016;
  • 62) 0.000 191 384 969 764 736 598 016 × 2 = 0 + 0.000 382 769 939 529 473 196 032;
  • 63) 0.000 382 769 939 529 473 196 032 × 2 = 0 + 0.000 765 539 879 058 946 392 064;
  • 64) 0.000 765 539 879 058 946 392 064 × 2 = 0 + 0.001 531 079 758 117 892 784 128;
  • 65) 0.001 531 079 758 117 892 784 128 × 2 = 0 + 0.003 062 159 516 235 785 568 256;
  • 66) 0.003 062 159 516 235 785 568 256 × 2 = 0 + 0.006 124 319 032 471 571 136 512;
  • 67) 0.006 124 319 032 471 571 136 512 × 2 = 0 + 0.012 248 638 064 943 142 273 024;
  • 68) 0.012 248 638 064 943 142 273 024 × 2 = 0 + 0.024 497 276 129 886 284 546 048;
  • 69) 0.024 497 276 129 886 284 546 048 × 2 = 0 + 0.048 994 552 259 772 569 092 096;
  • 70) 0.048 994 552 259 772 569 092 096 × 2 = 0 + 0.097 989 104 519 545 138 184 192;
  • 71) 0.097 989 104 519 545 138 184 192 × 2 = 0 + 0.195 978 209 039 090 276 368 384;
  • 72) 0.195 978 209 039 090 276 368 384 × 2 = 0 + 0.391 956 418 078 180 552 736 768;
  • 73) 0.391 956 418 078 180 552 736 768 × 2 = 0 + 0.783 912 836 156 361 105 473 536;
  • 74) 0.783 912 836 156 361 105 473 536 × 2 = 1 + 0.567 825 672 312 722 210 947 072;
  • 75) 0.567 825 672 312 722 210 947 072 × 2 = 1 + 0.135 651 344 625 444 421 894 144;
  • 76) 0.135 651 344 625 444 421 894 144 × 2 = 0 + 0.271 302 689 250 888 843 788 288;
  • 77) 0.271 302 689 250 888 843 788 288 × 2 = 0 + 0.542 605 378 501 777 687 576 576;
  • 78) 0.542 605 378 501 777 687 576 576 × 2 = 1 + 0.085 210 757 003 555 375 153 152;
  • 79) 0.085 210 757 003 555 375 153 152 × 2 = 0 + 0.170 421 514 007 110 750 306 304;
  • 80) 0.170 421 514 007 110 750 306 304 × 2 = 0 + 0.340 843 028 014 221 500 612 608;
  • 81) 0.340 843 028 014 221 500 612 608 × 2 = 0 + 0.681 686 056 028 443 001 225 216;
  • 82) 0.681 686 056 028 443 001 225 216 × 2 = 1 + 0.363 372 112 056 886 002 450 432;
  • 83) 0.363 372 112 056 886 002 450 432 × 2 = 0 + 0.726 744 224 113 772 004 900 864;
  • 84) 0.726 744 224 113 772 004 900 864 × 2 = 1 + 0.453 488 448 227 544 009 801 728;
  • 85) 0.453 488 448 227 544 009 801 728 × 2 = 0 + 0.906 976 896 455 088 019 603 456;
  • 86) 0.906 976 896 455 088 019 603 456 × 2 = 1 + 0.813 953 792 910 176 039 206 912;
  • 87) 0.813 953 792 910 176 039 206 912 × 2 = 1 + 0.627 907 585 820 352 078 413 824;
  • 88) 0.627 907 585 820 352 078 413 824 × 2 = 1 + 0.255 815 171 640 704 156 827 648;
  • 89) 0.255 815 171 640 704 156 827 648 × 2 = 0 + 0.511 630 343 281 408 313 655 296;
  • 90) 0.511 630 343 281 408 313 655 296 × 2 = 1 + 0.023 260 686 562 816 627 310 592;
  • 91) 0.023 260 686 562 816 627 310 592 × 2 = 0 + 0.046 521 373 125 633 254 621 184;
  • 92) 0.046 521 373 125 633 254 621 184 × 2 = 0 + 0.093 042 746 251 266 509 242 368;
  • 93) 0.093 042 746 251 266 509 242 368 × 2 = 0 + 0.186 085 492 502 533 018 484 736;
  • 94) 0.186 085 492 502 533 018 484 736 × 2 = 0 + 0.372 170 985 005 066 036 969 472;
  • 95) 0.372 170 985 005 066 036 969 472 × 2 = 0 + 0.744 341 970 010 132 073 938 944;
  • 96) 0.744 341 970 010 132 073 938 944 × 2 = 1 + 0.488 683 940 020 264 147 877 888;
  • 97) 0.488 683 940 020 264 147 877 888 × 2 = 0 + 0.977 367 880 040 528 295 755 776;
  • 98) 0.977 367 880 040 528 295 755 776 × 2 = 1 + 0.954 735 760 081 056 591 511 552;
  • 99) 0.954 735 760 081 056 591 511 552 × 2 = 1 + 0.909 471 520 162 113 183 023 104;
  • 100) 0.909 471 520 162 113 183 023 104 × 2 = 1 + 0.818 943 040 324 226 366 046 208;
  • 101) 0.818 943 040 324 226 366 046 208 × 2 = 1 + 0.637 886 080 648 452 732 092 416;
  • 102) 0.637 886 080 648 452 732 092 416 × 2 = 1 + 0.275 772 161 296 905 464 184 832;
  • 103) 0.275 772 161 296 905 464 184 832 × 2 = 0 + 0.551 544 322 593 810 928 369 664;
  • 104) 0.551 544 322 593 810 928 369 664 × 2 = 1 + 0.103 088 645 187 621 856 739 328;
  • 105) 0.103 088 645 187 621 856 739 328 × 2 = 0 + 0.206 177 290 375 243 713 478 656;
  • 106) 0.206 177 290 375 243 713 478 656 × 2 = 0 + 0.412 354 580 750 487 426 957 312;
  • 107) 0.412 354 580 750 487 426 957 312 × 2 = 0 + 0.824 709 161 500 974 853 914 624;
  • 108) 0.824 709 161 500 974 853 914 624 × 2 = 1 + 0.649 418 323 001 949 707 829 248;
  • 109) 0.649 418 323 001 949 707 829 248 × 2 = 1 + 0.298 836 646 003 899 415 658 496;
  • 110) 0.298 836 646 003 899 415 658 496 × 2 = 0 + 0.597 673 292 007 798 831 316 992;
  • 111) 0.597 673 292 007 798 831 316 992 × 2 = 1 + 0.195 346 584 015 597 662 633 984;
  • 112) 0.195 346 584 015 597 662 633 984 × 2 = 0 + 0.390 693 168 031 195 325 267 968;
  • 113) 0.390 693 168 031 195 325 267 968 × 2 = 0 + 0.781 386 336 062 390 650 535 936;
  • 114) 0.781 386 336 062 390 650 535 936 × 2 = 1 + 0.562 772 672 124 781 301 071 872;
  • 115) 0.562 772 672 124 781 301 071 872 × 2 = 1 + 0.125 545 344 249 562 602 143 744;
  • 116) 0.125 545 344 249 562 602 143 744 × 2 = 0 + 0.251 090 688 499 125 204 287 488;
  • 117) 0.251 090 688 499 125 204 287 488 × 2 = 0 + 0.502 181 376 998 250 408 574 976;
  • 118) 0.502 181 376 998 250 408 574 976 × 2 = 1 + 0.004 362 753 996 500 817 149 952;
  • 119) 0.004 362 753 996 500 817 149 952 × 2 = 0 + 0.008 725 507 993 001 634 299 904;
  • 120) 0.008 725 507 993 001 634 299 904 × 2 = 0 + 0.017 451 015 986 003 268 599 808;
  • 121) 0.017 451 015 986 003 268 599 808 × 2 = 0 + 0.034 902 031 972 006 537 199 616;
  • 122) 0.034 902 031 972 006 537 199 616 × 2 = 0 + 0.069 804 063 944 013 074 399 232;
  • 123) 0.069 804 063 944 013 074 399 232 × 2 = 0 + 0.139 608 127 888 026 148 798 464;
  • 124) 0.139 608 127 888 026 148 798 464 × 2 = 0 + 0.279 216 255 776 052 297 596 928;
  • 125) 0.279 216 255 776 052 297 596 928 × 2 = 0 + 0.558 432 511 552 104 595 193 856;
  • 126) 0.558 432 511 552 104 595 193 856 × 2 = 1 + 0.116 865 023 104 209 190 387 712;

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:


0.000 000 000 000 000 000 000 083(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0100 0101 0111 0100 0001 0111 1101 0001 1010 0110 0100 0000 01(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 083(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0100 0101 0111 0100 0001 0111 1101 0001 1010 0110 0100 0000 01(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 74 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 083(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0100 0101 0111 0100 0001 0111 1101 0001 1010 0110 0100 0000 01(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0100 0101 0111 0100 0001 0111 1101 0001 1010 0110 0100 0000 01(2) × 20 =

1.1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 0001(2) × 2-74


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): -74

Mantissa (not normalized):
1.1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-74 + 2(11-1) - 1 =

(-74 + 1 023)(10) =

949(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;
  • 949 ÷ 2 = 474 + 1;
  • 474 ÷ 2 = 237 + 0;
  • 237 ÷ 2 = 118 + 1;
  • 118 ÷ 2 = 59 + 0;
  • 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) =

949(10) =

011 1011 0101(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. 1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 0001 =

1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 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 1011 0101

Mantissa (52 bits) =
1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 0001


Decimal number 0.000 000 000 000 000 000 000 083 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 0101 - 1001 0001 0101 1101 0000 0101 1111 0100 0110 1001 1001 0000 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