0.000 000 000 000 000 000 008 327 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 327(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
0.000 000 000 000 000 000 008 327(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: 0.000 000 000 000 000 000 008 327.

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) 0.000 000 000 000 000 000 008 327 × 2 = 0 + 0.000 000 000 000 000 000 016 654;
  • 2) 0.000 000 000 000 000 000 016 654 × 2 = 0 + 0.000 000 000 000 000 000 033 308;
  • 3) 0.000 000 000 000 000 000 033 308 × 2 = 0 + 0.000 000 000 000 000 000 066 616;
  • 4) 0.000 000 000 000 000 000 066 616 × 2 = 0 + 0.000 000 000 000 000 000 133 232;
  • 5) 0.000 000 000 000 000 000 133 232 × 2 = 0 + 0.000 000 000 000 000 000 266 464;
  • 6) 0.000 000 000 000 000 000 266 464 × 2 = 0 + 0.000 000 000 000 000 000 532 928;
  • 7) 0.000 000 000 000 000 000 532 928 × 2 = 0 + 0.000 000 000 000 000 001 065 856;
  • 8) 0.000 000 000 000 000 001 065 856 × 2 = 0 + 0.000 000 000 000 000 002 131 712;
  • 9) 0.000 000 000 000 000 002 131 712 × 2 = 0 + 0.000 000 000 000 000 004 263 424;
  • 10) 0.000 000 000 000 000 004 263 424 × 2 = 0 + 0.000 000 000 000 000 008 526 848;
  • 11) 0.000 000 000 000 000 008 526 848 × 2 = 0 + 0.000 000 000 000 000 017 053 696;
  • 12) 0.000 000 000 000 000 017 053 696 × 2 = 0 + 0.000 000 000 000 000 034 107 392;
  • 13) 0.000 000 000 000 000 034 107 392 × 2 = 0 + 0.000 000 000 000 000 068 214 784;
  • 14) 0.000 000 000 000 000 068 214 784 × 2 = 0 + 0.000 000 000 000 000 136 429 568;
  • 15) 0.000 000 000 000 000 136 429 568 × 2 = 0 + 0.000 000 000 000 000 272 859 136;
  • 16) 0.000 000 000 000 000 272 859 136 × 2 = 0 + 0.000 000 000 000 000 545 718 272;
  • 17) 0.000 000 000 000 000 545 718 272 × 2 = 0 + 0.000 000 000 000 001 091 436 544;
  • 18) 0.000 000 000 000 001 091 436 544 × 2 = 0 + 0.000 000 000 000 002 182 873 088;
  • 19) 0.000 000 000 000 002 182 873 088 × 2 = 0 + 0.000 000 000 000 004 365 746 176;
  • 20) 0.000 000 000 000 004 365 746 176 × 2 = 0 + 0.000 000 000 000 008 731 492 352;
  • 21) 0.000 000 000 000 008 731 492 352 × 2 = 0 + 0.000 000 000 000 017 462 984 704;
  • 22) 0.000 000 000 000 017 462 984 704 × 2 = 0 + 0.000 000 000 000 034 925 969 408;
  • 23) 0.000 000 000 000 034 925 969 408 × 2 = 0 + 0.000 000 000 000 069 851 938 816;
  • 24) 0.000 000 000 000 069 851 938 816 × 2 = 0 + 0.000 000 000 000 139 703 877 632;
  • 25) 0.000 000 000 000 139 703 877 632 × 2 = 0 + 0.000 000 000 000 279 407 755 264;
  • 26) 0.000 000 000 000 279 407 755 264 × 2 = 0 + 0.000 000 000 000 558 815 510 528;
  • 27) 0.000 000 000 000 558 815 510 528 × 2 = 0 + 0.000 000 000 001 117 631 021 056;
  • 28) 0.000 000 000 001 117 631 021 056 × 2 = 0 + 0.000 000 000 002 235 262 042 112;
  • 29) 0.000 000 000 002 235 262 042 112 × 2 = 0 + 0.000 000 000 004 470 524 084 224;
  • 30) 0.000 000 000 004 470 524 084 224 × 2 = 0 + 0.000 000 000 008 941 048 168 448;
  • 31) 0.000 000 000 008 941 048 168 448 × 2 = 0 + 0.000 000 000 017 882 096 336 896;
  • 32) 0.000 000 000 017 882 096 336 896 × 2 = 0 + 0.000 000 000 035 764 192 673 792;
  • 33) 0.000 000 000 035 764 192 673 792 × 2 = 0 + 0.000 000 000 071 528 385 347 584;
  • 34) 0.000 000 000 071 528 385 347 584 × 2 = 0 + 0.000 000 000 143 056 770 695 168;
  • 35) 0.000 000 000 143 056 770 695 168 × 2 = 0 + 0.000 000 000 286 113 541 390 336;
  • 36) 0.000 000 000 286 113 541 390 336 × 2 = 0 + 0.000 000 000 572 227 082 780 672;
  • 37) 0.000 000 000 572 227 082 780 672 × 2 = 0 + 0.000 000 001 144 454 165 561 344;
  • 38) 0.000 000 001 144 454 165 561 344 × 2 = 0 + 0.000 000 002 288 908 331 122 688;
  • 39) 0.000 000 002 288 908 331 122 688 × 2 = 0 + 0.000 000 004 577 816 662 245 376;
  • 40) 0.000 000 004 577 816 662 245 376 × 2 = 0 + 0.000 000 009 155 633 324 490 752;
  • 41) 0.000 000 009 155 633 324 490 752 × 2 = 0 + 0.000 000 018 311 266 648 981 504;
  • 42) 0.000 000 018 311 266 648 981 504 × 2 = 0 + 0.000 000 036 622 533 297 963 008;
  • 43) 0.000 000 036 622 533 297 963 008 × 2 = 0 + 0.000 000 073 245 066 595 926 016;
  • 44) 0.000 000 073 245 066 595 926 016 × 2 = 0 + 0.000 000 146 490 133 191 852 032;
  • 45) 0.000 000 146 490 133 191 852 032 × 2 = 0 + 0.000 000 292 980 266 383 704 064;
  • 46) 0.000 000 292 980 266 383 704 064 × 2 = 0 + 0.000 000 585 960 532 767 408 128;
  • 47) 0.000 000 585 960 532 767 408 128 × 2 = 0 + 0.000 001 171 921 065 534 816 256;
  • 48) 0.000 001 171 921 065 534 816 256 × 2 = 0 + 0.000 002 343 842 131 069 632 512;
  • 49) 0.000 002 343 842 131 069 632 512 × 2 = 0 + 0.000 004 687 684 262 139 265 024;
  • 50) 0.000 004 687 684 262 139 265 024 × 2 = 0 + 0.000 009 375 368 524 278 530 048;
  • 51) 0.000 009 375 368 524 278 530 048 × 2 = 0 + 0.000 018 750 737 048 557 060 096;
  • 52) 0.000 018 750 737 048 557 060 096 × 2 = 0 + 0.000 037 501 474 097 114 120 192;
  • 53) 0.000 037 501 474 097 114 120 192 × 2 = 0 + 0.000 075 002 948 194 228 240 384;
  • 54) 0.000 075 002 948 194 228 240 384 × 2 = 0 + 0.000 150 005 896 388 456 480 768;
  • 55) 0.000 150 005 896 388 456 480 768 × 2 = 0 + 0.000 300 011 792 776 912 961 536;
  • 56) 0.000 300 011 792 776 912 961 536 × 2 = 0 + 0.000 600 023 585 553 825 923 072;
  • 57) 0.000 600 023 585 553 825 923 072 × 2 = 0 + 0.001 200 047 171 107 651 846 144;
  • 58) 0.001 200 047 171 107 651 846 144 × 2 = 0 + 0.002 400 094 342 215 303 692 288;
  • 59) 0.002 400 094 342 215 303 692 288 × 2 = 0 + 0.004 800 188 684 430 607 384 576;
  • 60) 0.004 800 188 684 430 607 384 576 × 2 = 0 + 0.009 600 377 368 861 214 769 152;
  • 61) 0.009 600 377 368 861 214 769 152 × 2 = 0 + 0.019 200 754 737 722 429 538 304;
  • 62) 0.019 200 754 737 722 429 538 304 × 2 = 0 + 0.038 401 509 475 444 859 076 608;
  • 63) 0.038 401 509 475 444 859 076 608 × 2 = 0 + 0.076 803 018 950 889 718 153 216;
  • 64) 0.076 803 018 950 889 718 153 216 × 2 = 0 + 0.153 606 037 901 779 436 306 432;
  • 65) 0.153 606 037 901 779 436 306 432 × 2 = 0 + 0.307 212 075 803 558 872 612 864;
  • 66) 0.307 212 075 803 558 872 612 864 × 2 = 0 + 0.614 424 151 607 117 745 225 728;
  • 67) 0.614 424 151 607 117 745 225 728 × 2 = 1 + 0.228 848 303 214 235 490 451 456;
  • 68) 0.228 848 303 214 235 490 451 456 × 2 = 0 + 0.457 696 606 428 470 980 902 912;
  • 69) 0.457 696 606 428 470 980 902 912 × 2 = 0 + 0.915 393 212 856 941 961 805 824;
  • 70) 0.915 393 212 856 941 961 805 824 × 2 = 1 + 0.830 786 425 713 883 923 611 648;
  • 71) 0.830 786 425 713 883 923 611 648 × 2 = 1 + 0.661 572 851 427 767 847 223 296;
  • 72) 0.661 572 851 427 767 847 223 296 × 2 = 1 + 0.323 145 702 855 535 694 446 592;
  • 73) 0.323 145 702 855 535 694 446 592 × 2 = 0 + 0.646 291 405 711 071 388 893 184;
  • 74) 0.646 291 405 711 071 388 893 184 × 2 = 1 + 0.292 582 811 422 142 777 786 368;
  • 75) 0.292 582 811 422 142 777 786 368 × 2 = 0 + 0.585 165 622 844 285 555 572 736;
  • 76) 0.585 165 622 844 285 555 572 736 × 2 = 1 + 0.170 331 245 688 571 111 145 472;
  • 77) 0.170 331 245 688 571 111 145 472 × 2 = 0 + 0.340 662 491 377 142 222 290 944;
  • 78) 0.340 662 491 377 142 222 290 944 × 2 = 0 + 0.681 324 982 754 284 444 581 888;
  • 79) 0.681 324 982 754 284 444 581 888 × 2 = 1 + 0.362 649 965 508 568 889 163 776;
  • 80) 0.362 649 965 508 568 889 163 776 × 2 = 0 + 0.725 299 931 017 137 778 327 552;
  • 81) 0.725 299 931 017 137 778 327 552 × 2 = 1 + 0.450 599 862 034 275 556 655 104;
  • 82) 0.450 599 862 034 275 556 655 104 × 2 = 0 + 0.901 199 724 068 551 113 310 208;
  • 83) 0.901 199 724 068 551 113 310 208 × 2 = 1 + 0.802 399 448 137 102 226 620 416;
  • 84) 0.802 399 448 137 102 226 620 416 × 2 = 1 + 0.604 798 896 274 204 453 240 832;
  • 85) 0.604 798 896 274 204 453 240 832 × 2 = 1 + 0.209 597 792 548 408 906 481 664;
  • 86) 0.209 597 792 548 408 906 481 664 × 2 = 0 + 0.419 195 585 096 817 812 963 328;
  • 87) 0.419 195 585 096 817 812 963 328 × 2 = 0 + 0.838 391 170 193 635 625 926 656;
  • 88) 0.838 391 170 193 635 625 926 656 × 2 = 1 + 0.676 782 340 387 271 251 853 312;
  • 89) 0.676 782 340 387 271 251 853 312 × 2 = 1 + 0.353 564 680 774 542 503 706 624;
  • 90) 0.353 564 680 774 542 503 706 624 × 2 = 0 + 0.707 129 361 549 085 007 413 248;
  • 91) 0.707 129 361 549 085 007 413 248 × 2 = 1 + 0.414 258 723 098 170 014 826 496;
  • 92) 0.414 258 723 098 170 014 826 496 × 2 = 0 + 0.828 517 446 196 340 029 652 992;
  • 93) 0.828 517 446 196 340 029 652 992 × 2 = 1 + 0.657 034 892 392 680 059 305 984;
  • 94) 0.657 034 892 392 680 059 305 984 × 2 = 1 + 0.314 069 784 785 360 118 611 968;
  • 95) 0.314 069 784 785 360 118 611 968 × 2 = 0 + 0.628 139 569 570 720 237 223 936;
  • 96) 0.628 139 569 570 720 237 223 936 × 2 = 1 + 0.256 279 139 141 440 474 447 872;
  • 97) 0.256 279 139 141 440 474 447 872 × 2 = 0 + 0.512 558 278 282 880 948 895 744;
  • 98) 0.512 558 278 282 880 948 895 744 × 2 = 1 + 0.025 116 556 565 761 897 791 488;
  • 99) 0.025 116 556 565 761 897 791 488 × 2 = 0 + 0.050 233 113 131 523 795 582 976;
  • 100) 0.050 233 113 131 523 795 582 976 × 2 = 0 + 0.100 466 226 263 047 591 165 952;
  • 101) 0.100 466 226 263 047 591 165 952 × 2 = 0 + 0.200 932 452 526 095 182 331 904;
  • 102) 0.200 932 452 526 095 182 331 904 × 2 = 0 + 0.401 864 905 052 190 364 663 808;
  • 103) 0.401 864 905 052 190 364 663 808 × 2 = 0 + 0.803 729 810 104 380 729 327 616;
  • 104) 0.803 729 810 104 380 729 327 616 × 2 = 1 + 0.607 459 620 208 761 458 655 232;
  • 105) 0.607 459 620 208 761 458 655 232 × 2 = 1 + 0.214 919 240 417 522 917 310 464;
  • 106) 0.214 919 240 417 522 917 310 464 × 2 = 0 + 0.429 838 480 835 045 834 620 928;
  • 107) 0.429 838 480 835 045 834 620 928 × 2 = 0 + 0.859 676 961 670 091 669 241 856;
  • 108) 0.859 676 961 670 091 669 241 856 × 2 = 1 + 0.719 353 923 340 183 338 483 712;
  • 109) 0.719 353 923 340 183 338 483 712 × 2 = 1 + 0.438 707 846 680 366 676 967 424;
  • 110) 0.438 707 846 680 366 676 967 424 × 2 = 0 + 0.877 415 693 360 733 353 934 848;
  • 111) 0.877 415 693 360 733 353 934 848 × 2 = 1 + 0.754 831 386 721 466 707 869 696;
  • 112) 0.754 831 386 721 466 707 869 696 × 2 = 1 + 0.509 662 773 442 933 415 739 392;
  • 113) 0.509 662 773 442 933 415 739 392 × 2 = 1 + 0.019 325 546 885 866 831 478 784;
  • 114) 0.019 325 546 885 866 831 478 784 × 2 = 0 + 0.038 651 093 771 733 662 957 568;
  • 115) 0.038 651 093 771 733 662 957 568 × 2 = 0 + 0.077 302 187 543 467 325 915 136;
  • 116) 0.077 302 187 543 467 325 915 136 × 2 = 0 + 0.154 604 375 086 934 651 830 272;
  • 117) 0.154 604 375 086 934 651 830 272 × 2 = 0 + 0.309 208 750 173 869 303 660 544;
  • 118) 0.309 208 750 173 869 303 660 544 × 2 = 0 + 0.618 417 500 347 738 607 321 088;
  • 119) 0.618 417 500 347 738 607 321 088 × 2 = 1 + 0.236 835 000 695 477 214 642 176;

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:


0.000 000 000 000 000 000 008 327(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 0101 0010 1011 1001 1010 1101 0100 0001 1001 1011 1000 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 327(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 0101 0010 1011 1001 1010 1101 0100 0001 1001 1011 1000 001(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:


0.000 000 000 000 000 000 008 327(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 0101 0010 1011 1001 1010 1101 0100 0001 1001 1011 1000 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 0101 0010 1011 1001 1010 1101 0100 0001 1001 1011 1000 001(2) × 20 =

1.0011 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 0001(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 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 0001


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 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 0001 =

0011 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 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 1100

Mantissa (52 bits) =
0011 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 0001


Decimal number 0.000 000 000 000 000 000 008 327 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0011 1010 1001 0101 1100 1101 0110 1010 0000 1100 1101 1100 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