-0.000 000 349 613 904 800 000 238 907 549 083 000 006 04 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 006 04(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
-0.000 000 349 613 904 800 000 238 907 549 083 000 006 04(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. Start with the positive version of the number:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 006 04| = 0.000 000 349 613 904 800 000 238 907 549 083 000 006 04


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 349 613 904 800 000 238 907 549 083 000 006 04.

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) 0.000 000 349 613 904 800 000 238 907 549 083 000 006 04 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 012 08;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 012 08 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 024 16;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 024 16 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 048 32;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 048 32 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 096 64;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 096 64 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 193 28;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 193 28 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 000 386 56;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 000 386 56 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 000 773 12;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 000 773 12 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 001 546 24;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 001 546 24 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 003 092 48;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 003 092 48 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 006 184 96;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 006 184 96 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 012 369 92;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 012 369 92 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 024 739 84;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 024 739 84 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 049 479 68;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 049 479 68 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 098 959 36;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 098 959 36 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 197 918 72;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 197 918 72 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 488 395 837 44;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 488 395 837 44 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 976 791 674 88;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 976 791 674 88 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 953 583 349 76;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 953 583 349 76 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 907 166 699 52;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 907 166 699 52 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 814 333 399 04;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 814 333 399 04 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 628 666 798 08;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 628 666 798 08 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 257 333 596 16;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 257 333 596 16 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 514 667 192 32;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 514 667 192 32 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 029 334 384 64;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 029 334 384 64 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 058 668 769 28;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 058 668 769 28 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 117 337 538 56;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 117 337 538 56 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 744 234 675 077 12;
  • 28) 0.924 383 979 464 326 465 628 439 968 744 234 675 077 12 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 488 469 350 154 24;
  • 29) 0.848 767 958 928 652 931 256 879 937 488 469 350 154 24 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 976 938 700 308 48;
  • 30) 0.697 535 917 857 305 862 513 759 874 976 938 700 308 48 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 953 877 400 616 96;
  • 31) 0.395 071 835 714 611 725 027 519 749 953 877 400 616 96 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 907 754 801 233 92;
  • 32) 0.790 143 671 429 223 450 055 039 499 907 754 801 233 92 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 815 509 602 467 84;
  • 33) 0.580 287 342 858 446 900 110 078 999 815 509 602 467 84 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 631 019 204 935 68;
  • 34) 0.160 574 685 716 893 800 220 157 999 631 019 204 935 68 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 262 038 409 871 36;
  • 35) 0.321 149 371 433 787 600 440 315 999 262 038 409 871 36 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 524 076 819 742 72;
  • 36) 0.642 298 742 867 575 200 880 631 998 524 076 819 742 72 × 2 = 1 + 0.284 597 485 735 150 401 761 263 997 048 153 639 485 44;
  • 37) 0.284 597 485 735 150 401 761 263 997 048 153 639 485 44 × 2 = 0 + 0.569 194 971 470 300 803 522 527 994 096 307 278 970 88;
  • 38) 0.569 194 971 470 300 803 522 527 994 096 307 278 970 88 × 2 = 1 + 0.138 389 942 940 601 607 045 055 988 192 614 557 941 76;
  • 39) 0.138 389 942 940 601 607 045 055 988 192 614 557 941 76 × 2 = 0 + 0.276 779 885 881 203 214 090 111 976 385 229 115 883 52;
  • 40) 0.276 779 885 881 203 214 090 111 976 385 229 115 883 52 × 2 = 0 + 0.553 559 771 762 406 428 180 223 952 770 458 231 767 04;
  • 41) 0.553 559 771 762 406 428 180 223 952 770 458 231 767 04 × 2 = 1 + 0.107 119 543 524 812 856 360 447 905 540 916 463 534 08;
  • 42) 0.107 119 543 524 812 856 360 447 905 540 916 463 534 08 × 2 = 0 + 0.214 239 087 049 625 712 720 895 811 081 832 927 068 16;
  • 43) 0.214 239 087 049 625 712 720 895 811 081 832 927 068 16 × 2 = 0 + 0.428 478 174 099 251 425 441 791 622 163 665 854 136 32;
  • 44) 0.428 478 174 099 251 425 441 791 622 163 665 854 136 32 × 2 = 0 + 0.856 956 348 198 502 850 883 583 244 327 331 708 272 64;
  • 45) 0.856 956 348 198 502 850 883 583 244 327 331 708 272 64 × 2 = 1 + 0.713 912 696 397 005 701 767 166 488 654 663 416 545 28;
  • 46) 0.713 912 696 397 005 701 767 166 488 654 663 416 545 28 × 2 = 1 + 0.427 825 392 794 011 403 534 332 977 309 326 833 090 56;
  • 47) 0.427 825 392 794 011 403 534 332 977 309 326 833 090 56 × 2 = 0 + 0.855 650 785 588 022 807 068 665 954 618 653 666 181 12;
  • 48) 0.855 650 785 588 022 807 068 665 954 618 653 666 181 12 × 2 = 1 + 0.711 301 571 176 045 614 137 331 909 237 307 332 362 24;
  • 49) 0.711 301 571 176 045 614 137 331 909 237 307 332 362 24 × 2 = 1 + 0.422 603 142 352 091 228 274 663 818 474 614 664 724 48;
  • 50) 0.422 603 142 352 091 228 274 663 818 474 614 664 724 48 × 2 = 0 + 0.845 206 284 704 182 456 549 327 636 949 229 329 448 96;
  • 51) 0.845 206 284 704 182 456 549 327 636 949 229 329 448 96 × 2 = 1 + 0.690 412 569 408 364 913 098 655 273 898 458 658 897 92;
  • 52) 0.690 412 569 408 364 913 098 655 273 898 458 658 897 92 × 2 = 1 + 0.380 825 138 816 729 826 197 310 547 796 917 317 795 84;
  • 53) 0.380 825 138 816 729 826 197 310 547 796 917 317 795 84 × 2 = 0 + 0.761 650 277 633 459 652 394 621 095 593 834 635 591 68;
  • 54) 0.761 650 277 633 459 652 394 621 095 593 834 635 591 68 × 2 = 1 + 0.523 300 555 266 919 304 789 242 191 187 669 271 183 36;
  • 55) 0.523 300 555 266 919 304 789 242 191 187 669 271 183 36 × 2 = 1 + 0.046 601 110 533 838 609 578 484 382 375 338 542 366 72;
  • 56) 0.046 601 110 533 838 609 578 484 382 375 338 542 366 72 × 2 = 0 + 0.093 202 221 067 677 219 156 968 764 750 677 084 733 44;
  • 57) 0.093 202 221 067 677 219 156 968 764 750 677 084 733 44 × 2 = 0 + 0.186 404 442 135 354 438 313 937 529 501 354 169 466 88;
  • 58) 0.186 404 442 135 354 438 313 937 529 501 354 169 466 88 × 2 = 0 + 0.372 808 884 270 708 876 627 875 059 002 708 338 933 76;
  • 59) 0.372 808 884 270 708 876 627 875 059 002 708 338 933 76 × 2 = 0 + 0.745 617 768 541 417 753 255 750 118 005 416 677 867 52;
  • 60) 0.745 617 768 541 417 753 255 750 118 005 416 677 867 52 × 2 = 1 + 0.491 235 537 082 835 506 511 500 236 010 833 355 735 04;
  • 61) 0.491 235 537 082 835 506 511 500 236 010 833 355 735 04 × 2 = 0 + 0.982 471 074 165 671 013 023 000 472 021 666 711 470 08;
  • 62) 0.982 471 074 165 671 013 023 000 472 021 666 711 470 08 × 2 = 1 + 0.964 942 148 331 342 026 046 000 944 043 333 422 940 16;
  • 63) 0.964 942 148 331 342 026 046 000 944 043 333 422 940 16 × 2 = 1 + 0.929 884 296 662 684 052 092 001 888 086 666 845 880 32;
  • 64) 0.929 884 296 662 684 052 092 001 888 086 666 845 880 32 × 2 = 1 + 0.859 768 593 325 368 104 184 003 776 173 333 691 760 64;
  • 65) 0.859 768 593 325 368 104 184 003 776 173 333 691 760 64 × 2 = 1 + 0.719 537 186 650 736 208 368 007 552 346 667 383 521 28;
  • 66) 0.719 537 186 650 736 208 368 007 552 346 667 383 521 28 × 2 = 1 + 0.439 074 373 301 472 416 736 015 104 693 334 767 042 56;
  • 67) 0.439 074 373 301 472 416 736 015 104 693 334 767 042 56 × 2 = 0 + 0.878 148 746 602 944 833 472 030 209 386 669 534 085 12;
  • 68) 0.878 148 746 602 944 833 472 030 209 386 669 534 085 12 × 2 = 1 + 0.756 297 493 205 889 666 944 060 418 773 339 068 170 24;
  • 69) 0.756 297 493 205 889 666 944 060 418 773 339 068 170 24 × 2 = 1 + 0.512 594 986 411 779 333 888 120 837 546 678 136 340 48;
  • 70) 0.512 594 986 411 779 333 888 120 837 546 678 136 340 48 × 2 = 1 + 0.025 189 972 823 558 667 776 241 675 093 356 272 680 96;
  • 71) 0.025 189 972 823 558 667 776 241 675 093 356 272 680 96 × 2 = 0 + 0.050 379 945 647 117 335 552 483 350 186 712 545 361 92;
  • 72) 0.050 379 945 647 117 335 552 483 350 186 712 545 361 92 × 2 = 0 + 0.100 759 891 294 234 671 104 966 700 373 425 090 723 84;
  • 73) 0.100 759 891 294 234 671 104 966 700 373 425 090 723 84 × 2 = 0 + 0.201 519 782 588 469 342 209 933 400 746 850 181 447 68;
  • 74) 0.201 519 782 588 469 342 209 933 400 746 850 181 447 68 × 2 = 0 + 0.403 039 565 176 938 684 419 866 801 493 700 362 895 36;

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).


5. 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:


0.000 000 349 613 904 800 000 238 907 549 083 000 006 04(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

6. Positive number before normalization:

0.000 000 349 613 904 800 000 238 907 549 083 000 006 04(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 22 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 349 613 904 800 000 238 907 549 083 000 006 04(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) × 20 =

1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000(2) × 2-22


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): -22

Mantissa (not normalized):
1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-22 + 2(11-1) - 1 =

(-22 + 1 023)(10) =

1 001(10)


10. 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;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

1001(10) =

011 1110 1001(2)


12. 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. 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000 =

0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


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

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1110 1001

Mantissa (52 bits) =
0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


Decimal number -0.000 000 349 613 904 800 000 238 907 549 083 000 006 04 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1110 1001 - 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000

» Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 006 3 to 64 bit double precision IEEE 754 binary floating point representation standard

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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