-0.000 000 349 613 904 800 000 238 907 549 083 000 013 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 013(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
-0.000 000 349 613 904 800 000 238 907 549 083 000 013(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:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 013| = 0.000 000 349 613 904 800 000 238 907 549 083 000 013


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: 0.000 000 349 613 904 800 000 238 907 549 083 000 013.

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) 0.000 000 349 613 904 800 000 238 907 549 083 000 013 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 026;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 026 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 052;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 052 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 104;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 104 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 208;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 208 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 416;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 416 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 000 832;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 000 832 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 001 664;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 001 664 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 003 328;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 003 328 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 006 656;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 006 656 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 013 312;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 013 312 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 026 624;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 026 624 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 053 248;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 053 248 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 106 496;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 106 496 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 212 992;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 212 992 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 425 984;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 425 984 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 488 851 968;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 488 851 968 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 977 703 936;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 977 703 936 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 955 407 872;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 955 407 872 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 910 815 744;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 910 815 744 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 821 631 488;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 821 631 488 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 643 262 976;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 643 262 976 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 286 525 952;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 286 525 952 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 573 051 904;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 573 051 904 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 146 103 808;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 146 103 808 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 292 207 616;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 292 207 616 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 584 415 232;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 584 415 232 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 745 168 830 464;
  • 28) 0.924 383 979 464 326 465 628 439 968 745 168 830 464 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 490 337 660 928;
  • 29) 0.848 767 958 928 652 931 256 879 937 490 337 660 928 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 980 675 321 856;
  • 30) 0.697 535 917 857 305 862 513 759 874 980 675 321 856 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 961 350 643 712;
  • 31) 0.395 071 835 714 611 725 027 519 749 961 350 643 712 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 922 701 287 424;
  • 32) 0.790 143 671 429 223 450 055 039 499 922 701 287 424 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 845 402 574 848;
  • 33) 0.580 287 342 858 446 900 110 078 999 845 402 574 848 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 690 805 149 696;
  • 34) 0.160 574 685 716 893 800 220 157 999 690 805 149 696 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 381 610 299 392;
  • 35) 0.321 149 371 433 787 600 440 315 999 381 610 299 392 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 763 220 598 784;
  • 36) 0.642 298 742 867 575 200 880 631 998 763 220 598 784 × 2 = 1 + 0.284 597 485 735 150 401 761 263 997 526 441 197 568;
  • 37) 0.284 597 485 735 150 401 761 263 997 526 441 197 568 × 2 = 0 + 0.569 194 971 470 300 803 522 527 995 052 882 395 136;
  • 38) 0.569 194 971 470 300 803 522 527 995 052 882 395 136 × 2 = 1 + 0.138 389 942 940 601 607 045 055 990 105 764 790 272;
  • 39) 0.138 389 942 940 601 607 045 055 990 105 764 790 272 × 2 = 0 + 0.276 779 885 881 203 214 090 111 980 211 529 580 544;
  • 40) 0.276 779 885 881 203 214 090 111 980 211 529 580 544 × 2 = 0 + 0.553 559 771 762 406 428 180 223 960 423 059 161 088;
  • 41) 0.553 559 771 762 406 428 180 223 960 423 059 161 088 × 2 = 1 + 0.107 119 543 524 812 856 360 447 920 846 118 322 176;
  • 42) 0.107 119 543 524 812 856 360 447 920 846 118 322 176 × 2 = 0 + 0.214 239 087 049 625 712 720 895 841 692 236 644 352;
  • 43) 0.214 239 087 049 625 712 720 895 841 692 236 644 352 × 2 = 0 + 0.428 478 174 099 251 425 441 791 683 384 473 288 704;
  • 44) 0.428 478 174 099 251 425 441 791 683 384 473 288 704 × 2 = 0 + 0.856 956 348 198 502 850 883 583 366 768 946 577 408;
  • 45) 0.856 956 348 198 502 850 883 583 366 768 946 577 408 × 2 = 1 + 0.713 912 696 397 005 701 767 166 733 537 893 154 816;
  • 46) 0.713 912 696 397 005 701 767 166 733 537 893 154 816 × 2 = 1 + 0.427 825 392 794 011 403 534 333 467 075 786 309 632;
  • 47) 0.427 825 392 794 011 403 534 333 467 075 786 309 632 × 2 = 0 + 0.855 650 785 588 022 807 068 666 934 151 572 619 264;
  • 48) 0.855 650 785 588 022 807 068 666 934 151 572 619 264 × 2 = 1 + 0.711 301 571 176 045 614 137 333 868 303 145 238 528;
  • 49) 0.711 301 571 176 045 614 137 333 868 303 145 238 528 × 2 = 1 + 0.422 603 142 352 091 228 274 667 736 606 290 477 056;
  • 50) 0.422 603 142 352 091 228 274 667 736 606 290 477 056 × 2 = 0 + 0.845 206 284 704 182 456 549 335 473 212 580 954 112;
  • 51) 0.845 206 284 704 182 456 549 335 473 212 580 954 112 × 2 = 1 + 0.690 412 569 408 364 913 098 670 946 425 161 908 224;
  • 52) 0.690 412 569 408 364 913 098 670 946 425 161 908 224 × 2 = 1 + 0.380 825 138 816 729 826 197 341 892 850 323 816 448;
  • 53) 0.380 825 138 816 729 826 197 341 892 850 323 816 448 × 2 = 0 + 0.761 650 277 633 459 652 394 683 785 700 647 632 896;
  • 54) 0.761 650 277 633 459 652 394 683 785 700 647 632 896 × 2 = 1 + 0.523 300 555 266 919 304 789 367 571 401 295 265 792;
  • 55) 0.523 300 555 266 919 304 789 367 571 401 295 265 792 × 2 = 1 + 0.046 601 110 533 838 609 578 735 142 802 590 531 584;
  • 56) 0.046 601 110 533 838 609 578 735 142 802 590 531 584 × 2 = 0 + 0.093 202 221 067 677 219 157 470 285 605 181 063 168;
  • 57) 0.093 202 221 067 677 219 157 470 285 605 181 063 168 × 2 = 0 + 0.186 404 442 135 354 438 314 940 571 210 362 126 336;
  • 58) 0.186 404 442 135 354 438 314 940 571 210 362 126 336 × 2 = 0 + 0.372 808 884 270 708 876 629 881 142 420 724 252 672;
  • 59) 0.372 808 884 270 708 876 629 881 142 420 724 252 672 × 2 = 0 + 0.745 617 768 541 417 753 259 762 284 841 448 505 344;
  • 60) 0.745 617 768 541 417 753 259 762 284 841 448 505 344 × 2 = 1 + 0.491 235 537 082 835 506 519 524 569 682 897 010 688;
  • 61) 0.491 235 537 082 835 506 519 524 569 682 897 010 688 × 2 = 0 + 0.982 471 074 165 671 013 039 049 139 365 794 021 376;
  • 62) 0.982 471 074 165 671 013 039 049 139 365 794 021 376 × 2 = 1 + 0.964 942 148 331 342 026 078 098 278 731 588 042 752;
  • 63) 0.964 942 148 331 342 026 078 098 278 731 588 042 752 × 2 = 1 + 0.929 884 296 662 684 052 156 196 557 463 176 085 504;
  • 64) 0.929 884 296 662 684 052 156 196 557 463 176 085 504 × 2 = 1 + 0.859 768 593 325 368 104 312 393 114 926 352 171 008;
  • 65) 0.859 768 593 325 368 104 312 393 114 926 352 171 008 × 2 = 1 + 0.719 537 186 650 736 208 624 786 229 852 704 342 016;
  • 66) 0.719 537 186 650 736 208 624 786 229 852 704 342 016 × 2 = 1 + 0.439 074 373 301 472 417 249 572 459 705 408 684 032;
  • 67) 0.439 074 373 301 472 417 249 572 459 705 408 684 032 × 2 = 0 + 0.878 148 746 602 944 834 499 144 919 410 817 368 064;
  • 68) 0.878 148 746 602 944 834 499 144 919 410 817 368 064 × 2 = 1 + 0.756 297 493 205 889 668 998 289 838 821 634 736 128;
  • 69) 0.756 297 493 205 889 668 998 289 838 821 634 736 128 × 2 = 1 + 0.512 594 986 411 779 337 996 579 677 643 269 472 256;
  • 70) 0.512 594 986 411 779 337 996 579 677 643 269 472 256 × 2 = 1 + 0.025 189 972 823 558 675 993 159 355 286 538 944 512;
  • 71) 0.025 189 972 823 558 675 993 159 355 286 538 944 512 × 2 = 0 + 0.050 379 945 647 117 351 986 318 710 573 077 889 024;
  • 72) 0.050 379 945 647 117 351 986 318 710 573 077 889 024 × 2 = 0 + 0.100 759 891 294 234 703 972 637 421 146 155 778 048;
  • 73) 0.100 759 891 294 234 703 972 637 421 146 155 778 048 × 2 = 0 + 0.201 519 782 588 469 407 945 274 842 292 311 556 096;
  • 74) 0.201 519 782 588 469 407 945 274 842 292 311 556 096 × 2 = 0 + 0.403 039 565 176 938 815 890 549 684 584 623 112 192;

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:


0.000 000 349 613 904 800 000 238 907 549 083 000 013(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:

0.000 000 349 613 904 800 000 238 907 549 083 000 013(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:


0.000 000 349 613 904 800 000 238 907 549 083 000 013(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 -0.000 000 349 613 904 800 000 238 907 549 083 000 013 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 -0.000 000 349 613 904 800 000 238 907 549 083 000 027 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