0.000 000 000 000 000 000 008 565 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 565 2(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
0.000 000 000 000 000 000 008 565 2(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: 0.000 000 000 000 000 000 008 565 2.

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) 0.000 000 000 000 000 000 008 565 2 × 2 = 0 + 0.000 000 000 000 000 000 017 130 4;
  • 2) 0.000 000 000 000 000 000 017 130 4 × 2 = 0 + 0.000 000 000 000 000 000 034 260 8;
  • 3) 0.000 000 000 000 000 000 034 260 8 × 2 = 0 + 0.000 000 000 000 000 000 068 521 6;
  • 4) 0.000 000 000 000 000 000 068 521 6 × 2 = 0 + 0.000 000 000 000 000 000 137 043 2;
  • 5) 0.000 000 000 000 000 000 137 043 2 × 2 = 0 + 0.000 000 000 000 000 000 274 086 4;
  • 6) 0.000 000 000 000 000 000 274 086 4 × 2 = 0 + 0.000 000 000 000 000 000 548 172 8;
  • 7) 0.000 000 000 000 000 000 548 172 8 × 2 = 0 + 0.000 000 000 000 000 001 096 345 6;
  • 8) 0.000 000 000 000 000 001 096 345 6 × 2 = 0 + 0.000 000 000 000 000 002 192 691 2;
  • 9) 0.000 000 000 000 000 002 192 691 2 × 2 = 0 + 0.000 000 000 000 000 004 385 382 4;
  • 10) 0.000 000 000 000 000 004 385 382 4 × 2 = 0 + 0.000 000 000 000 000 008 770 764 8;
  • 11) 0.000 000 000 000 000 008 770 764 8 × 2 = 0 + 0.000 000 000 000 000 017 541 529 6;
  • 12) 0.000 000 000 000 000 017 541 529 6 × 2 = 0 + 0.000 000 000 000 000 035 083 059 2;
  • 13) 0.000 000 000 000 000 035 083 059 2 × 2 = 0 + 0.000 000 000 000 000 070 166 118 4;
  • 14) 0.000 000 000 000 000 070 166 118 4 × 2 = 0 + 0.000 000 000 000 000 140 332 236 8;
  • 15) 0.000 000 000 000 000 140 332 236 8 × 2 = 0 + 0.000 000 000 000 000 280 664 473 6;
  • 16) 0.000 000 000 000 000 280 664 473 6 × 2 = 0 + 0.000 000 000 000 000 561 328 947 2;
  • 17) 0.000 000 000 000 000 561 328 947 2 × 2 = 0 + 0.000 000 000 000 001 122 657 894 4;
  • 18) 0.000 000 000 000 001 122 657 894 4 × 2 = 0 + 0.000 000 000 000 002 245 315 788 8;
  • 19) 0.000 000 000 000 002 245 315 788 8 × 2 = 0 + 0.000 000 000 000 004 490 631 577 6;
  • 20) 0.000 000 000 000 004 490 631 577 6 × 2 = 0 + 0.000 000 000 000 008 981 263 155 2;
  • 21) 0.000 000 000 000 008 981 263 155 2 × 2 = 0 + 0.000 000 000 000 017 962 526 310 4;
  • 22) 0.000 000 000 000 017 962 526 310 4 × 2 = 0 + 0.000 000 000 000 035 925 052 620 8;
  • 23) 0.000 000 000 000 035 925 052 620 8 × 2 = 0 + 0.000 000 000 000 071 850 105 241 6;
  • 24) 0.000 000 000 000 071 850 105 241 6 × 2 = 0 + 0.000 000 000 000 143 700 210 483 2;
  • 25) 0.000 000 000 000 143 700 210 483 2 × 2 = 0 + 0.000 000 000 000 287 400 420 966 4;
  • 26) 0.000 000 000 000 287 400 420 966 4 × 2 = 0 + 0.000 000 000 000 574 800 841 932 8;
  • 27) 0.000 000 000 000 574 800 841 932 8 × 2 = 0 + 0.000 000 000 001 149 601 683 865 6;
  • 28) 0.000 000 000 001 149 601 683 865 6 × 2 = 0 + 0.000 000 000 002 299 203 367 731 2;
  • 29) 0.000 000 000 002 299 203 367 731 2 × 2 = 0 + 0.000 000 000 004 598 406 735 462 4;
  • 30) 0.000 000 000 004 598 406 735 462 4 × 2 = 0 + 0.000 000 000 009 196 813 470 924 8;
  • 31) 0.000 000 000 009 196 813 470 924 8 × 2 = 0 + 0.000 000 000 018 393 626 941 849 6;
  • 32) 0.000 000 000 018 393 626 941 849 6 × 2 = 0 + 0.000 000 000 036 787 253 883 699 2;
  • 33) 0.000 000 000 036 787 253 883 699 2 × 2 = 0 + 0.000 000 000 073 574 507 767 398 4;
  • 34) 0.000 000 000 073 574 507 767 398 4 × 2 = 0 + 0.000 000 000 147 149 015 534 796 8;
  • 35) 0.000 000 000 147 149 015 534 796 8 × 2 = 0 + 0.000 000 000 294 298 031 069 593 6;
  • 36) 0.000 000 000 294 298 031 069 593 6 × 2 = 0 + 0.000 000 000 588 596 062 139 187 2;
  • 37) 0.000 000 000 588 596 062 139 187 2 × 2 = 0 + 0.000 000 001 177 192 124 278 374 4;
  • 38) 0.000 000 001 177 192 124 278 374 4 × 2 = 0 + 0.000 000 002 354 384 248 556 748 8;
  • 39) 0.000 000 002 354 384 248 556 748 8 × 2 = 0 + 0.000 000 004 708 768 497 113 497 6;
  • 40) 0.000 000 004 708 768 497 113 497 6 × 2 = 0 + 0.000 000 009 417 536 994 226 995 2;
  • 41) 0.000 000 009 417 536 994 226 995 2 × 2 = 0 + 0.000 000 018 835 073 988 453 990 4;
  • 42) 0.000 000 018 835 073 988 453 990 4 × 2 = 0 + 0.000 000 037 670 147 976 907 980 8;
  • 43) 0.000 000 037 670 147 976 907 980 8 × 2 = 0 + 0.000 000 075 340 295 953 815 961 6;
  • 44) 0.000 000 075 340 295 953 815 961 6 × 2 = 0 + 0.000 000 150 680 591 907 631 923 2;
  • 45) 0.000 000 150 680 591 907 631 923 2 × 2 = 0 + 0.000 000 301 361 183 815 263 846 4;
  • 46) 0.000 000 301 361 183 815 263 846 4 × 2 = 0 + 0.000 000 602 722 367 630 527 692 8;
  • 47) 0.000 000 602 722 367 630 527 692 8 × 2 = 0 + 0.000 001 205 444 735 261 055 385 6;
  • 48) 0.000 001 205 444 735 261 055 385 6 × 2 = 0 + 0.000 002 410 889 470 522 110 771 2;
  • 49) 0.000 002 410 889 470 522 110 771 2 × 2 = 0 + 0.000 004 821 778 941 044 221 542 4;
  • 50) 0.000 004 821 778 941 044 221 542 4 × 2 = 0 + 0.000 009 643 557 882 088 443 084 8;
  • 51) 0.000 009 643 557 882 088 443 084 8 × 2 = 0 + 0.000 019 287 115 764 176 886 169 6;
  • 52) 0.000 019 287 115 764 176 886 169 6 × 2 = 0 + 0.000 038 574 231 528 353 772 339 2;
  • 53) 0.000 038 574 231 528 353 772 339 2 × 2 = 0 + 0.000 077 148 463 056 707 544 678 4;
  • 54) 0.000 077 148 463 056 707 544 678 4 × 2 = 0 + 0.000 154 296 926 113 415 089 356 8;
  • 55) 0.000 154 296 926 113 415 089 356 8 × 2 = 0 + 0.000 308 593 852 226 830 178 713 6;
  • 56) 0.000 308 593 852 226 830 178 713 6 × 2 = 0 + 0.000 617 187 704 453 660 357 427 2;
  • 57) 0.000 617 187 704 453 660 357 427 2 × 2 = 0 + 0.001 234 375 408 907 320 714 854 4;
  • 58) 0.001 234 375 408 907 320 714 854 4 × 2 = 0 + 0.002 468 750 817 814 641 429 708 8;
  • 59) 0.002 468 750 817 814 641 429 708 8 × 2 = 0 + 0.004 937 501 635 629 282 859 417 6;
  • 60) 0.004 937 501 635 629 282 859 417 6 × 2 = 0 + 0.009 875 003 271 258 565 718 835 2;
  • 61) 0.009 875 003 271 258 565 718 835 2 × 2 = 0 + 0.019 750 006 542 517 131 437 670 4;
  • 62) 0.019 750 006 542 517 131 437 670 4 × 2 = 0 + 0.039 500 013 085 034 262 875 340 8;
  • 63) 0.039 500 013 085 034 262 875 340 8 × 2 = 0 + 0.079 000 026 170 068 525 750 681 6;
  • 64) 0.079 000 026 170 068 525 750 681 6 × 2 = 0 + 0.158 000 052 340 137 051 501 363 2;
  • 65) 0.158 000 052 340 137 051 501 363 2 × 2 = 0 + 0.316 000 104 680 274 103 002 726 4;
  • 66) 0.316 000 104 680 274 103 002 726 4 × 2 = 0 + 0.632 000 209 360 548 206 005 452 8;
  • 67) 0.632 000 209 360 548 206 005 452 8 × 2 = 1 + 0.264 000 418 721 096 412 010 905 6;
  • 68) 0.264 000 418 721 096 412 010 905 6 × 2 = 0 + 0.528 000 837 442 192 824 021 811 2;
  • 69) 0.528 000 837 442 192 824 021 811 2 × 2 = 1 + 0.056 001 674 884 385 648 043 622 4;
  • 70) 0.056 001 674 884 385 648 043 622 4 × 2 = 0 + 0.112 003 349 768 771 296 087 244 8;
  • 71) 0.112 003 349 768 771 296 087 244 8 × 2 = 0 + 0.224 006 699 537 542 592 174 489 6;
  • 72) 0.224 006 699 537 542 592 174 489 6 × 2 = 0 + 0.448 013 399 075 085 184 348 979 2;
  • 73) 0.448 013 399 075 085 184 348 979 2 × 2 = 0 + 0.896 026 798 150 170 368 697 958 4;
  • 74) 0.896 026 798 150 170 368 697 958 4 × 2 = 1 + 0.792 053 596 300 340 737 395 916 8;
  • 75) 0.792 053 596 300 340 737 395 916 8 × 2 = 1 + 0.584 107 192 600 681 474 791 833 6;
  • 76) 0.584 107 192 600 681 474 791 833 6 × 2 = 1 + 0.168 214 385 201 362 949 583 667 2;
  • 77) 0.168 214 385 201 362 949 583 667 2 × 2 = 0 + 0.336 428 770 402 725 899 167 334 4;
  • 78) 0.336 428 770 402 725 899 167 334 4 × 2 = 0 + 0.672 857 540 805 451 798 334 668 8;
  • 79) 0.672 857 540 805 451 798 334 668 8 × 2 = 1 + 0.345 715 081 610 903 596 669 337 6;
  • 80) 0.345 715 081 610 903 596 669 337 6 × 2 = 0 + 0.691 430 163 221 807 193 338 675 2;
  • 81) 0.691 430 163 221 807 193 338 675 2 × 2 = 1 + 0.382 860 326 443 614 386 677 350 4;
  • 82) 0.382 860 326 443 614 386 677 350 4 × 2 = 0 + 0.765 720 652 887 228 773 354 700 8;
  • 83) 0.765 720 652 887 228 773 354 700 8 × 2 = 1 + 0.531 441 305 774 457 546 709 401 6;
  • 84) 0.531 441 305 774 457 546 709 401 6 × 2 = 1 + 0.062 882 611 548 915 093 418 803 2;
  • 85) 0.062 882 611 548 915 093 418 803 2 × 2 = 0 + 0.125 765 223 097 830 186 837 606 4;
  • 86) 0.125 765 223 097 830 186 837 606 4 × 2 = 0 + 0.251 530 446 195 660 373 675 212 8;
  • 87) 0.251 530 446 195 660 373 675 212 8 × 2 = 0 + 0.503 060 892 391 320 747 350 425 6;
  • 88) 0.503 060 892 391 320 747 350 425 6 × 2 = 1 + 0.006 121 784 782 641 494 700 851 2;
  • 89) 0.006 121 784 782 641 494 700 851 2 × 2 = 0 + 0.012 243 569 565 282 989 401 702 4;
  • 90) 0.012 243 569 565 282 989 401 702 4 × 2 = 0 + 0.024 487 139 130 565 978 803 404 8;
  • 91) 0.024 487 139 130 565 978 803 404 8 × 2 = 0 + 0.048 974 278 261 131 957 606 809 6;
  • 92) 0.048 974 278 261 131 957 606 809 6 × 2 = 0 + 0.097 948 556 522 263 915 213 619 2;
  • 93) 0.097 948 556 522 263 915 213 619 2 × 2 = 0 + 0.195 897 113 044 527 830 427 238 4;
  • 94) 0.195 897 113 044 527 830 427 238 4 × 2 = 0 + 0.391 794 226 089 055 660 854 476 8;
  • 95) 0.391 794 226 089 055 660 854 476 8 × 2 = 0 + 0.783 588 452 178 111 321 708 953 6;
  • 96) 0.783 588 452 178 111 321 708 953 6 × 2 = 1 + 0.567 176 904 356 222 643 417 907 2;
  • 97) 0.567 176 904 356 222 643 417 907 2 × 2 = 1 + 0.134 353 808 712 445 286 835 814 4;
  • 98) 0.134 353 808 712 445 286 835 814 4 × 2 = 0 + 0.268 707 617 424 890 573 671 628 8;
  • 99) 0.268 707 617 424 890 573 671 628 8 × 2 = 0 + 0.537 415 234 849 781 147 343 257 6;
  • 100) 0.537 415 234 849 781 147 343 257 6 × 2 = 1 + 0.074 830 469 699 562 294 686 515 2;
  • 101) 0.074 830 469 699 562 294 686 515 2 × 2 = 0 + 0.149 660 939 399 124 589 373 030 4;
  • 102) 0.149 660 939 399 124 589 373 030 4 × 2 = 0 + 0.299 321 878 798 249 178 746 060 8;
  • 103) 0.299 321 878 798 249 178 746 060 8 × 2 = 0 + 0.598 643 757 596 498 357 492 121 6;
  • 104) 0.598 643 757 596 498 357 492 121 6 × 2 = 1 + 0.197 287 515 192 996 714 984 243 2;
  • 105) 0.197 287 515 192 996 714 984 243 2 × 2 = 0 + 0.394 575 030 385 993 429 968 486 4;
  • 106) 0.394 575 030 385 993 429 968 486 4 × 2 = 0 + 0.789 150 060 771 986 859 936 972 8;
  • 107) 0.789 150 060 771 986 859 936 972 8 × 2 = 1 + 0.578 300 121 543 973 719 873 945 6;
  • 108) 0.578 300 121 543 973 719 873 945 6 × 2 = 1 + 0.156 600 243 087 947 439 747 891 2;
  • 109) 0.156 600 243 087 947 439 747 891 2 × 2 = 0 + 0.313 200 486 175 894 879 495 782 4;
  • 110) 0.313 200 486 175 894 879 495 782 4 × 2 = 0 + 0.626 400 972 351 789 758 991 564 8;
  • 111) 0.626 400 972 351 789 758 991 564 8 × 2 = 1 + 0.252 801 944 703 579 517 983 129 6;
  • 112) 0.252 801 944 703 579 517 983 129 6 × 2 = 0 + 0.505 603 889 407 159 035 966 259 2;
  • 113) 0.505 603 889 407 159 035 966 259 2 × 2 = 1 + 0.011 207 778 814 318 071 932 518 4;
  • 114) 0.011 207 778 814 318 071 932 518 4 × 2 = 0 + 0.022 415 557 628 636 143 865 036 8;
  • 115) 0.022 415 557 628 636 143 865 036 8 × 2 = 0 + 0.044 831 115 257 272 287 730 073 6;
  • 116) 0.044 831 115 257 272 287 730 073 6 × 2 = 0 + 0.089 662 230 514 544 575 460 147 2;
  • 117) 0.089 662 230 514 544 575 460 147 2 × 2 = 0 + 0.179 324 461 029 089 150 920 294 4;
  • 118) 0.179 324 461 029 089 150 920 294 4 × 2 = 0 + 0.358 648 922 058 178 301 840 588 8;
  • 119) 0.358 648 922 058 178 301 840 588 8 × 2 = 0 + 0.717 297 844 116 356 603 681 177 6;

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:


0.000 000 000 000 000 000 008 565 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0010 1011 0001 0000 0001 1001 0001 0011 0010 1000 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 565 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0010 1011 0001 0000 0001 1001 0001 0011 0010 1000 000(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:


0.000 000 000 000 000 000 008 565 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0010 1011 0001 0000 0001 1001 0001 0011 0010 1000 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0010 1011 0001 0000 0001 1001 0001 0011 0010 1000 000(2) × 20 =

1.0100 0011 1001 0101 1000 1000 0000 1100 1000 1001 1001 0100 0000(2) × 2-67


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0011 1001 0101 1000 1000 0000 1100 1000 1001 1001 0100 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-67 + 1 023)(10) =

956(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0011 1001 0101 1000 1000 0000 1100 1000 1001 1001 0100 0000 =

0100 0011 1001 0101 1000 1000 0000 1100 1000 1001 1001 0100 0000


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

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


Decimal number 0.000 000 000 000 000 000 008 565 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

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

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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