0.000 000 000 000 000 000 009 23 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 009 23(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
0.000 000 000 000 000 000 009 23(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: 0.000 000 000 000 000 000 009 23.

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) 0.000 000 000 000 000 000 009 23 × 2 = 0 + 0.000 000 000 000 000 000 018 46;
  • 2) 0.000 000 000 000 000 000 018 46 × 2 = 0 + 0.000 000 000 000 000 000 036 92;
  • 3) 0.000 000 000 000 000 000 036 92 × 2 = 0 + 0.000 000 000 000 000 000 073 84;
  • 4) 0.000 000 000 000 000 000 073 84 × 2 = 0 + 0.000 000 000 000 000 000 147 68;
  • 5) 0.000 000 000 000 000 000 147 68 × 2 = 0 + 0.000 000 000 000 000 000 295 36;
  • 6) 0.000 000 000 000 000 000 295 36 × 2 = 0 + 0.000 000 000 000 000 000 590 72;
  • 7) 0.000 000 000 000 000 000 590 72 × 2 = 0 + 0.000 000 000 000 000 001 181 44;
  • 8) 0.000 000 000 000 000 001 181 44 × 2 = 0 + 0.000 000 000 000 000 002 362 88;
  • 9) 0.000 000 000 000 000 002 362 88 × 2 = 0 + 0.000 000 000 000 000 004 725 76;
  • 10) 0.000 000 000 000 000 004 725 76 × 2 = 0 + 0.000 000 000 000 000 009 451 52;
  • 11) 0.000 000 000 000 000 009 451 52 × 2 = 0 + 0.000 000 000 000 000 018 903 04;
  • 12) 0.000 000 000 000 000 018 903 04 × 2 = 0 + 0.000 000 000 000 000 037 806 08;
  • 13) 0.000 000 000 000 000 037 806 08 × 2 = 0 + 0.000 000 000 000 000 075 612 16;
  • 14) 0.000 000 000 000 000 075 612 16 × 2 = 0 + 0.000 000 000 000 000 151 224 32;
  • 15) 0.000 000 000 000 000 151 224 32 × 2 = 0 + 0.000 000 000 000 000 302 448 64;
  • 16) 0.000 000 000 000 000 302 448 64 × 2 = 0 + 0.000 000 000 000 000 604 897 28;
  • 17) 0.000 000 000 000 000 604 897 28 × 2 = 0 + 0.000 000 000 000 001 209 794 56;
  • 18) 0.000 000 000 000 001 209 794 56 × 2 = 0 + 0.000 000 000 000 002 419 589 12;
  • 19) 0.000 000 000 000 002 419 589 12 × 2 = 0 + 0.000 000 000 000 004 839 178 24;
  • 20) 0.000 000 000 000 004 839 178 24 × 2 = 0 + 0.000 000 000 000 009 678 356 48;
  • 21) 0.000 000 000 000 009 678 356 48 × 2 = 0 + 0.000 000 000 000 019 356 712 96;
  • 22) 0.000 000 000 000 019 356 712 96 × 2 = 0 + 0.000 000 000 000 038 713 425 92;
  • 23) 0.000 000 000 000 038 713 425 92 × 2 = 0 + 0.000 000 000 000 077 426 851 84;
  • 24) 0.000 000 000 000 077 426 851 84 × 2 = 0 + 0.000 000 000 000 154 853 703 68;
  • 25) 0.000 000 000 000 154 853 703 68 × 2 = 0 + 0.000 000 000 000 309 707 407 36;
  • 26) 0.000 000 000 000 309 707 407 36 × 2 = 0 + 0.000 000 000 000 619 414 814 72;
  • 27) 0.000 000 000 000 619 414 814 72 × 2 = 0 + 0.000 000 000 001 238 829 629 44;
  • 28) 0.000 000 000 001 238 829 629 44 × 2 = 0 + 0.000 000 000 002 477 659 258 88;
  • 29) 0.000 000 000 002 477 659 258 88 × 2 = 0 + 0.000 000 000 004 955 318 517 76;
  • 30) 0.000 000 000 004 955 318 517 76 × 2 = 0 + 0.000 000 000 009 910 637 035 52;
  • 31) 0.000 000 000 009 910 637 035 52 × 2 = 0 + 0.000 000 000 019 821 274 071 04;
  • 32) 0.000 000 000 019 821 274 071 04 × 2 = 0 + 0.000 000 000 039 642 548 142 08;
  • 33) 0.000 000 000 039 642 548 142 08 × 2 = 0 + 0.000 000 000 079 285 096 284 16;
  • 34) 0.000 000 000 079 285 096 284 16 × 2 = 0 + 0.000 000 000 158 570 192 568 32;
  • 35) 0.000 000 000 158 570 192 568 32 × 2 = 0 + 0.000 000 000 317 140 385 136 64;
  • 36) 0.000 000 000 317 140 385 136 64 × 2 = 0 + 0.000 000 000 634 280 770 273 28;
  • 37) 0.000 000 000 634 280 770 273 28 × 2 = 0 + 0.000 000 001 268 561 540 546 56;
  • 38) 0.000 000 001 268 561 540 546 56 × 2 = 0 + 0.000 000 002 537 123 081 093 12;
  • 39) 0.000 000 002 537 123 081 093 12 × 2 = 0 + 0.000 000 005 074 246 162 186 24;
  • 40) 0.000 000 005 074 246 162 186 24 × 2 = 0 + 0.000 000 010 148 492 324 372 48;
  • 41) 0.000 000 010 148 492 324 372 48 × 2 = 0 + 0.000 000 020 296 984 648 744 96;
  • 42) 0.000 000 020 296 984 648 744 96 × 2 = 0 + 0.000 000 040 593 969 297 489 92;
  • 43) 0.000 000 040 593 969 297 489 92 × 2 = 0 + 0.000 000 081 187 938 594 979 84;
  • 44) 0.000 000 081 187 938 594 979 84 × 2 = 0 + 0.000 000 162 375 877 189 959 68;
  • 45) 0.000 000 162 375 877 189 959 68 × 2 = 0 + 0.000 000 324 751 754 379 919 36;
  • 46) 0.000 000 324 751 754 379 919 36 × 2 = 0 + 0.000 000 649 503 508 759 838 72;
  • 47) 0.000 000 649 503 508 759 838 72 × 2 = 0 + 0.000 001 299 007 017 519 677 44;
  • 48) 0.000 001 299 007 017 519 677 44 × 2 = 0 + 0.000 002 598 014 035 039 354 88;
  • 49) 0.000 002 598 014 035 039 354 88 × 2 = 0 + 0.000 005 196 028 070 078 709 76;
  • 50) 0.000 005 196 028 070 078 709 76 × 2 = 0 + 0.000 010 392 056 140 157 419 52;
  • 51) 0.000 010 392 056 140 157 419 52 × 2 = 0 + 0.000 020 784 112 280 314 839 04;
  • 52) 0.000 020 784 112 280 314 839 04 × 2 = 0 + 0.000 041 568 224 560 629 678 08;
  • 53) 0.000 041 568 224 560 629 678 08 × 2 = 0 + 0.000 083 136 449 121 259 356 16;
  • 54) 0.000 083 136 449 121 259 356 16 × 2 = 0 + 0.000 166 272 898 242 518 712 32;
  • 55) 0.000 166 272 898 242 518 712 32 × 2 = 0 + 0.000 332 545 796 485 037 424 64;
  • 56) 0.000 332 545 796 485 037 424 64 × 2 = 0 + 0.000 665 091 592 970 074 849 28;
  • 57) 0.000 665 091 592 970 074 849 28 × 2 = 0 + 0.001 330 183 185 940 149 698 56;
  • 58) 0.001 330 183 185 940 149 698 56 × 2 = 0 + 0.002 660 366 371 880 299 397 12;
  • 59) 0.002 660 366 371 880 299 397 12 × 2 = 0 + 0.005 320 732 743 760 598 794 24;
  • 60) 0.005 320 732 743 760 598 794 24 × 2 = 0 + 0.010 641 465 487 521 197 588 48;
  • 61) 0.010 641 465 487 521 197 588 48 × 2 = 0 + 0.021 282 930 975 042 395 176 96;
  • 62) 0.021 282 930 975 042 395 176 96 × 2 = 0 + 0.042 565 861 950 084 790 353 92;
  • 63) 0.042 565 861 950 084 790 353 92 × 2 = 0 + 0.085 131 723 900 169 580 707 84;
  • 64) 0.085 131 723 900 169 580 707 84 × 2 = 0 + 0.170 263 447 800 339 161 415 68;
  • 65) 0.170 263 447 800 339 161 415 68 × 2 = 0 + 0.340 526 895 600 678 322 831 36;
  • 66) 0.340 526 895 600 678 322 831 36 × 2 = 0 + 0.681 053 791 201 356 645 662 72;
  • 67) 0.681 053 791 201 356 645 662 72 × 2 = 1 + 0.362 107 582 402 713 291 325 44;
  • 68) 0.362 107 582 402 713 291 325 44 × 2 = 0 + 0.724 215 164 805 426 582 650 88;
  • 69) 0.724 215 164 805 426 582 650 88 × 2 = 1 + 0.448 430 329 610 853 165 301 76;
  • 70) 0.448 430 329 610 853 165 301 76 × 2 = 0 + 0.896 860 659 221 706 330 603 52;
  • 71) 0.896 860 659 221 706 330 603 52 × 2 = 1 + 0.793 721 318 443 412 661 207 04;
  • 72) 0.793 721 318 443 412 661 207 04 × 2 = 1 + 0.587 442 636 886 825 322 414 08;
  • 73) 0.587 442 636 886 825 322 414 08 × 2 = 1 + 0.174 885 273 773 650 644 828 16;
  • 74) 0.174 885 273 773 650 644 828 16 × 2 = 0 + 0.349 770 547 547 301 289 656 32;
  • 75) 0.349 770 547 547 301 289 656 32 × 2 = 0 + 0.699 541 095 094 602 579 312 64;
  • 76) 0.699 541 095 094 602 579 312 64 × 2 = 1 + 0.399 082 190 189 205 158 625 28;
  • 77) 0.399 082 190 189 205 158 625 28 × 2 = 0 + 0.798 164 380 378 410 317 250 56;
  • 78) 0.798 164 380 378 410 317 250 56 × 2 = 1 + 0.596 328 760 756 820 634 501 12;
  • 79) 0.596 328 760 756 820 634 501 12 × 2 = 1 + 0.192 657 521 513 641 269 002 24;
  • 80) 0.192 657 521 513 641 269 002 24 × 2 = 0 + 0.385 315 043 027 282 538 004 48;
  • 81) 0.385 315 043 027 282 538 004 48 × 2 = 0 + 0.770 630 086 054 565 076 008 96;
  • 82) 0.770 630 086 054 565 076 008 96 × 2 = 1 + 0.541 260 172 109 130 152 017 92;
  • 83) 0.541 260 172 109 130 152 017 92 × 2 = 1 + 0.082 520 344 218 260 304 035 84;
  • 84) 0.082 520 344 218 260 304 035 84 × 2 = 0 + 0.165 040 688 436 520 608 071 68;
  • 85) 0.165 040 688 436 520 608 071 68 × 2 = 0 + 0.330 081 376 873 041 216 143 36;
  • 86) 0.330 081 376 873 041 216 143 36 × 2 = 0 + 0.660 162 753 746 082 432 286 72;
  • 87) 0.660 162 753 746 082 432 286 72 × 2 = 1 + 0.320 325 507 492 164 864 573 44;
  • 88) 0.320 325 507 492 164 864 573 44 × 2 = 0 + 0.640 651 014 984 329 729 146 88;
  • 89) 0.640 651 014 984 329 729 146 88 × 2 = 1 + 0.281 302 029 968 659 458 293 76;
  • 90) 0.281 302 029 968 659 458 293 76 × 2 = 0 + 0.562 604 059 937 318 916 587 52;
  • 91) 0.562 604 059 937 318 916 587 52 × 2 = 1 + 0.125 208 119 874 637 833 175 04;
  • 92) 0.125 208 119 874 637 833 175 04 × 2 = 0 + 0.250 416 239 749 275 666 350 08;
  • 93) 0.250 416 239 749 275 666 350 08 × 2 = 0 + 0.500 832 479 498 551 332 700 16;
  • 94) 0.500 832 479 498 551 332 700 16 × 2 = 1 + 0.001 664 958 997 102 665 400 32;
  • 95) 0.001 664 958 997 102 665 400 32 × 2 = 0 + 0.003 329 917 994 205 330 800 64;
  • 96) 0.003 329 917 994 205 330 800 64 × 2 = 0 + 0.006 659 835 988 410 661 601 28;
  • 97) 0.006 659 835 988 410 661 601 28 × 2 = 0 + 0.013 319 671 976 821 323 202 56;
  • 98) 0.013 319 671 976 821 323 202 56 × 2 = 0 + 0.026 639 343 953 642 646 405 12;
  • 99) 0.026 639 343 953 642 646 405 12 × 2 = 0 + 0.053 278 687 907 285 292 810 24;
  • 100) 0.053 278 687 907 285 292 810 24 × 2 = 0 + 0.106 557 375 814 570 585 620 48;
  • 101) 0.106 557 375 814 570 585 620 48 × 2 = 0 + 0.213 114 751 629 141 171 240 96;
  • 102) 0.213 114 751 629 141 171 240 96 × 2 = 0 + 0.426 229 503 258 282 342 481 92;
  • 103) 0.426 229 503 258 282 342 481 92 × 2 = 0 + 0.852 459 006 516 564 684 963 84;
  • 104) 0.852 459 006 516 564 684 963 84 × 2 = 1 + 0.704 918 013 033 129 369 927 68;
  • 105) 0.704 918 013 033 129 369 927 68 × 2 = 1 + 0.409 836 026 066 258 739 855 36;
  • 106) 0.409 836 026 066 258 739 855 36 × 2 = 0 + 0.819 672 052 132 517 479 710 72;
  • 107) 0.819 672 052 132 517 479 710 72 × 2 = 1 + 0.639 344 104 265 034 959 421 44;
  • 108) 0.639 344 104 265 034 959 421 44 × 2 = 1 + 0.278 688 208 530 069 918 842 88;
  • 109) 0.278 688 208 530 069 918 842 88 × 2 = 0 + 0.557 376 417 060 139 837 685 76;
  • 110) 0.557 376 417 060 139 837 685 76 × 2 = 1 + 0.114 752 834 120 279 675 371 52;
  • 111) 0.114 752 834 120 279 675 371 52 × 2 = 0 + 0.229 505 668 240 559 350 743 04;
  • 112) 0.229 505 668 240 559 350 743 04 × 2 = 0 + 0.459 011 336 481 118 701 486 08;
  • 113) 0.459 011 336 481 118 701 486 08 × 2 = 0 + 0.918 022 672 962 237 402 972 16;
  • 114) 0.918 022 672 962 237 402 972 16 × 2 = 1 + 0.836 045 345 924 474 805 944 32;
  • 115) 0.836 045 345 924 474 805 944 32 × 2 = 1 + 0.672 090 691 848 949 611 888 64;
  • 116) 0.672 090 691 848 949 611 888 64 × 2 = 1 + 0.344 181 383 697 899 223 777 28;
  • 117) 0.344 181 383 697 899 223 777 28 × 2 = 0 + 0.688 362 767 395 798 447 554 56;
  • 118) 0.688 362 767 395 798 447 554 56 × 2 = 1 + 0.376 725 534 791 596 895 109 12;
  • 119) 0.376 725 534 791 596 895 109 12 × 2 = 0 + 0.753 451 069 583 193 790 218 24;

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:


0.000 000 000 000 000 000 009 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 0110 0110 0010 1010 0100 0000 0001 1011 0100 0111 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 009 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 0110 0110 0010 1010 0100 0000 0001 1011 0100 0111 010(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:


0.000 000 000 000 000 000 009 23(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 0110 0110 0010 1010 0100 0000 0001 1011 0100 0111 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 0110 0110 0010 1010 0100 0000 0001 1011 0100 0111 010(2) × 20 =

1.0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010(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):
1.0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010


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. 0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010 =

0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010


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) =
0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010


Decimal number 0.000 000 000 000 000 000 009 23 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0101 1100 1011 0011 0001 0101 0010 0000 0000 1101 1010 0011 1010

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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