0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266(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
0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266(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: 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266.

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) 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266 × 2 = 1 + 0.570 860 049 649 700 895 268 369 897 652 419 700 540 091 025 730 532;
  • 2) 0.570 860 049 649 700 895 268 369 897 652 419 700 540 091 025 730 532 × 2 = 1 + 0.141 720 099 299 401 790 536 739 795 304 839 401 080 182 051 461 064;
  • 3) 0.141 720 099 299 401 790 536 739 795 304 839 401 080 182 051 461 064 × 2 = 0 + 0.283 440 198 598 803 581 073 479 590 609 678 802 160 364 102 922 128;
  • 4) 0.283 440 198 598 803 581 073 479 590 609 678 802 160 364 102 922 128 × 2 = 0 + 0.566 880 397 197 607 162 146 959 181 219 357 604 320 728 205 844 256;
  • 5) 0.566 880 397 197 607 162 146 959 181 219 357 604 320 728 205 844 256 × 2 = 1 + 0.133 760 794 395 214 324 293 918 362 438 715 208 641 456 411 688 512;
  • 6) 0.133 760 794 395 214 324 293 918 362 438 715 208 641 456 411 688 512 × 2 = 0 + 0.267 521 588 790 428 648 587 836 724 877 430 417 282 912 823 377 024;
  • 7) 0.267 521 588 790 428 648 587 836 724 877 430 417 282 912 823 377 024 × 2 = 0 + 0.535 043 177 580 857 297 175 673 449 754 860 834 565 825 646 754 048;
  • 8) 0.535 043 177 580 857 297 175 673 449 754 860 834 565 825 646 754 048 × 2 = 1 + 0.070 086 355 161 714 594 351 346 899 509 721 669 131 651 293 508 096;
  • 9) 0.070 086 355 161 714 594 351 346 899 509 721 669 131 651 293 508 096 × 2 = 0 + 0.140 172 710 323 429 188 702 693 799 019 443 338 263 302 587 016 192;
  • 10) 0.140 172 710 323 429 188 702 693 799 019 443 338 263 302 587 016 192 × 2 = 0 + 0.280 345 420 646 858 377 405 387 598 038 886 676 526 605 174 032 384;
  • 11) 0.280 345 420 646 858 377 405 387 598 038 886 676 526 605 174 032 384 × 2 = 0 + 0.560 690 841 293 716 754 810 775 196 077 773 353 053 210 348 064 768;
  • 12) 0.560 690 841 293 716 754 810 775 196 077 773 353 053 210 348 064 768 × 2 = 1 + 0.121 381 682 587 433 509 621 550 392 155 546 706 106 420 696 129 536;
  • 13) 0.121 381 682 587 433 509 621 550 392 155 546 706 106 420 696 129 536 × 2 = 0 + 0.242 763 365 174 867 019 243 100 784 311 093 412 212 841 392 259 072;
  • 14) 0.242 763 365 174 867 019 243 100 784 311 093 412 212 841 392 259 072 × 2 = 0 + 0.485 526 730 349 734 038 486 201 568 622 186 824 425 682 784 518 144;
  • 15) 0.485 526 730 349 734 038 486 201 568 622 186 824 425 682 784 518 144 × 2 = 0 + 0.971 053 460 699 468 076 972 403 137 244 373 648 851 365 569 036 288;
  • 16) 0.971 053 460 699 468 076 972 403 137 244 373 648 851 365 569 036 288 × 2 = 1 + 0.942 106 921 398 936 153 944 806 274 488 747 297 702 731 138 072 576;
  • 17) 0.942 106 921 398 936 153 944 806 274 488 747 297 702 731 138 072 576 × 2 = 1 + 0.884 213 842 797 872 307 889 612 548 977 494 595 405 462 276 145 152;
  • 18) 0.884 213 842 797 872 307 889 612 548 977 494 595 405 462 276 145 152 × 2 = 1 + 0.768 427 685 595 744 615 779 225 097 954 989 190 810 924 552 290 304;
  • 19) 0.768 427 685 595 744 615 779 225 097 954 989 190 810 924 552 290 304 × 2 = 1 + 0.536 855 371 191 489 231 558 450 195 909 978 381 621 849 104 580 608;
  • 20) 0.536 855 371 191 489 231 558 450 195 909 978 381 621 849 104 580 608 × 2 = 1 + 0.073 710 742 382 978 463 116 900 391 819 956 763 243 698 209 161 216;
  • 21) 0.073 710 742 382 978 463 116 900 391 819 956 763 243 698 209 161 216 × 2 = 0 + 0.147 421 484 765 956 926 233 800 783 639 913 526 487 396 418 322 432;
  • 22) 0.147 421 484 765 956 926 233 800 783 639 913 526 487 396 418 322 432 × 2 = 0 + 0.294 842 969 531 913 852 467 601 567 279 827 052 974 792 836 644 864;
  • 23) 0.294 842 969 531 913 852 467 601 567 279 827 052 974 792 836 644 864 × 2 = 0 + 0.589 685 939 063 827 704 935 203 134 559 654 105 949 585 673 289 728;
  • 24) 0.589 685 939 063 827 704 935 203 134 559 654 105 949 585 673 289 728 × 2 = 1 + 0.179 371 878 127 655 409 870 406 269 119 308 211 899 171 346 579 456;
  • 25) 0.179 371 878 127 655 409 870 406 269 119 308 211 899 171 346 579 456 × 2 = 0 + 0.358 743 756 255 310 819 740 812 538 238 616 423 798 342 693 158 912;
  • 26) 0.358 743 756 255 310 819 740 812 538 238 616 423 798 342 693 158 912 × 2 = 0 + 0.717 487 512 510 621 639 481 625 076 477 232 847 596 685 386 317 824;
  • 27) 0.717 487 512 510 621 639 481 625 076 477 232 847 596 685 386 317 824 × 2 = 1 + 0.434 975 025 021 243 278 963 250 152 954 465 695 193 370 772 635 648;
  • 28) 0.434 975 025 021 243 278 963 250 152 954 465 695 193 370 772 635 648 × 2 = 0 + 0.869 950 050 042 486 557 926 500 305 908 931 390 386 741 545 271 296;
  • 29) 0.869 950 050 042 486 557 926 500 305 908 931 390 386 741 545 271 296 × 2 = 1 + 0.739 900 100 084 973 115 853 000 611 817 862 780 773 483 090 542 592;
  • 30) 0.739 900 100 084 973 115 853 000 611 817 862 780 773 483 090 542 592 × 2 = 1 + 0.479 800 200 169 946 231 706 001 223 635 725 561 546 966 181 085 184;
  • 31) 0.479 800 200 169 946 231 706 001 223 635 725 561 546 966 181 085 184 × 2 = 0 + 0.959 600 400 339 892 463 412 002 447 271 451 123 093 932 362 170 368;
  • 32) 0.959 600 400 339 892 463 412 002 447 271 451 123 093 932 362 170 368 × 2 = 1 + 0.919 200 800 679 784 926 824 004 894 542 902 246 187 864 724 340 736;
  • 33) 0.919 200 800 679 784 926 824 004 894 542 902 246 187 864 724 340 736 × 2 = 1 + 0.838 401 601 359 569 853 648 009 789 085 804 492 375 729 448 681 472;
  • 34) 0.838 401 601 359 569 853 648 009 789 085 804 492 375 729 448 681 472 × 2 = 1 + 0.676 803 202 719 139 707 296 019 578 171 608 984 751 458 897 362 944;
  • 35) 0.676 803 202 719 139 707 296 019 578 171 608 984 751 458 897 362 944 × 2 = 1 + 0.353 606 405 438 279 414 592 039 156 343 217 969 502 917 794 725 888;
  • 36) 0.353 606 405 438 279 414 592 039 156 343 217 969 502 917 794 725 888 × 2 = 0 + 0.707 212 810 876 558 829 184 078 312 686 435 939 005 835 589 451 776;
  • 37) 0.707 212 810 876 558 829 184 078 312 686 435 939 005 835 589 451 776 × 2 = 1 + 0.414 425 621 753 117 658 368 156 625 372 871 878 011 671 178 903 552;
  • 38) 0.414 425 621 753 117 658 368 156 625 372 871 878 011 671 178 903 552 × 2 = 0 + 0.828 851 243 506 235 316 736 313 250 745 743 756 023 342 357 807 104;
  • 39) 0.828 851 243 506 235 316 736 313 250 745 743 756 023 342 357 807 104 × 2 = 1 + 0.657 702 487 012 470 633 472 626 501 491 487 512 046 684 715 614 208;
  • 40) 0.657 702 487 012 470 633 472 626 501 491 487 512 046 684 715 614 208 × 2 = 1 + 0.315 404 974 024 941 266 945 253 002 982 975 024 093 369 431 228 416;
  • 41) 0.315 404 974 024 941 266 945 253 002 982 975 024 093 369 431 228 416 × 2 = 0 + 0.630 809 948 049 882 533 890 506 005 965 950 048 186 738 862 456 832;
  • 42) 0.630 809 948 049 882 533 890 506 005 965 950 048 186 738 862 456 832 × 2 = 1 + 0.261 619 896 099 765 067 781 012 011 931 900 096 373 477 724 913 664;
  • 43) 0.261 619 896 099 765 067 781 012 011 931 900 096 373 477 724 913 664 × 2 = 0 + 0.523 239 792 199 530 135 562 024 023 863 800 192 746 955 449 827 328;
  • 44) 0.523 239 792 199 530 135 562 024 023 863 800 192 746 955 449 827 328 × 2 = 1 + 0.046 479 584 399 060 271 124 048 047 727 600 385 493 910 899 654 656;
  • 45) 0.046 479 584 399 060 271 124 048 047 727 600 385 493 910 899 654 656 × 2 = 0 + 0.092 959 168 798 120 542 248 096 095 455 200 770 987 821 799 309 312;
  • 46) 0.092 959 168 798 120 542 248 096 095 455 200 770 987 821 799 309 312 × 2 = 0 + 0.185 918 337 596 241 084 496 192 190 910 401 541 975 643 598 618 624;
  • 47) 0.185 918 337 596 241 084 496 192 190 910 401 541 975 643 598 618 624 × 2 = 0 + 0.371 836 675 192 482 168 992 384 381 820 803 083 951 287 197 237 248;
  • 48) 0.371 836 675 192 482 168 992 384 381 820 803 083 951 287 197 237 248 × 2 = 0 + 0.743 673 350 384 964 337 984 768 763 641 606 167 902 574 394 474 496;
  • 49) 0.743 673 350 384 964 337 984 768 763 641 606 167 902 574 394 474 496 × 2 = 1 + 0.487 346 700 769 928 675 969 537 527 283 212 335 805 148 788 948 992;
  • 50) 0.487 346 700 769 928 675 969 537 527 283 212 335 805 148 788 948 992 × 2 = 0 + 0.974 693 401 539 857 351 939 075 054 566 424 671 610 297 577 897 984;
  • 51) 0.974 693 401 539 857 351 939 075 054 566 424 671 610 297 577 897 984 × 2 = 1 + 0.949 386 803 079 714 703 878 150 109 132 849 343 220 595 155 795 968;
  • 52) 0.949 386 803 079 714 703 878 150 109 132 849 343 220 595 155 795 968 × 2 = 1 + 0.898 773 606 159 429 407 756 300 218 265 698 686 441 190 311 591 936;
  • 53) 0.898 773 606 159 429 407 756 300 218 265 698 686 441 190 311 591 936 × 2 = 1 + 0.797 547 212 318 858 815 512 600 436 531 397 372 882 380 623 183 872;

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:


0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

5. Positive number before normalization:

0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) × 20 =

1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111(2) × 2-1


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): -1

Mantissa (not normalized):
1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-1 + 1 023)(10) =

1 022(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;
  • 1 022 ÷ 2 = 511 + 0;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

1022(10) =

011 1111 1110(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. 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111 =

1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


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 1111 1110

Mantissa (52 bits) =
1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


Decimal number 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 266 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1110 - 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111

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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