-2.211 829 052 383 358 300 119 548 661 699 653 280 265 57 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 280 265 57(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
-2.211 829 052 383 358 300 119 548 661 699 653 280 265 57(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:

|-2.211 829 052 383 358 300 119 548 661 699 653 280 265 57| = 2.211 829 052 383 358 300 119 548 661 699 653 280 265 57


2. First, convert to binary (in base 2) the integer part: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

2(10) =

10(2)


4. Convert to binary (base 2) the fractional part: 0.211 829 052 383 358 300 119 548 661 699 653 280 265 57.

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) 0.211 829 052 383 358 300 119 548 661 699 653 280 265 57 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 399 306 560 531 14;
  • 2) 0.423 658 104 766 716 600 239 097 323 399 306 560 531 14 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 798 613 121 062 28;
  • 3) 0.847 316 209 533 433 200 478 194 646 798 613 121 062 28 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 597 226 242 124 56;
  • 4) 0.694 632 419 066 866 400 956 389 293 597 226 242 124 56 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 194 452 484 249 12;
  • 5) 0.389 264 838 133 732 801 912 778 587 194 452 484 249 12 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 388 904 968 498 24;
  • 6) 0.778 529 676 267 465 603 825 557 174 388 904 968 498 24 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 777 809 936 996 48;
  • 7) 0.557 059 352 534 931 207 651 114 348 777 809 936 996 48 × 2 = 1 + 0.114 118 705 069 862 415 302 228 697 555 619 873 992 96;
  • 8) 0.114 118 705 069 862 415 302 228 697 555 619 873 992 96 × 2 = 0 + 0.228 237 410 139 724 830 604 457 395 111 239 747 985 92;
  • 9) 0.228 237 410 139 724 830 604 457 395 111 239 747 985 92 × 2 = 0 + 0.456 474 820 279 449 661 208 914 790 222 479 495 971 84;
  • 10) 0.456 474 820 279 449 661 208 914 790 222 479 495 971 84 × 2 = 0 + 0.912 949 640 558 899 322 417 829 580 444 958 991 943 68;
  • 11) 0.912 949 640 558 899 322 417 829 580 444 958 991 943 68 × 2 = 1 + 0.825 899 281 117 798 644 835 659 160 889 917 983 887 36;
  • 12) 0.825 899 281 117 798 644 835 659 160 889 917 983 887 36 × 2 = 1 + 0.651 798 562 235 597 289 671 318 321 779 835 967 774 72;
  • 13) 0.651 798 562 235 597 289 671 318 321 779 835 967 774 72 × 2 = 1 + 0.303 597 124 471 194 579 342 636 643 559 671 935 549 44;
  • 14) 0.303 597 124 471 194 579 342 636 643 559 671 935 549 44 × 2 = 0 + 0.607 194 248 942 389 158 685 273 287 119 343 871 098 88;
  • 15) 0.607 194 248 942 389 158 685 273 287 119 343 871 098 88 × 2 = 1 + 0.214 388 497 884 778 317 370 546 574 238 687 742 197 76;
  • 16) 0.214 388 497 884 778 317 370 546 574 238 687 742 197 76 × 2 = 0 + 0.428 776 995 769 556 634 741 093 148 477 375 484 395 52;
  • 17) 0.428 776 995 769 556 634 741 093 148 477 375 484 395 52 × 2 = 0 + 0.857 553 991 539 113 269 482 186 296 954 750 968 791 04;
  • 18) 0.857 553 991 539 113 269 482 186 296 954 750 968 791 04 × 2 = 1 + 0.715 107 983 078 226 538 964 372 593 909 501 937 582 08;
  • 19) 0.715 107 983 078 226 538 964 372 593 909 501 937 582 08 × 2 = 1 + 0.430 215 966 156 453 077 928 745 187 819 003 875 164 16;
  • 20) 0.430 215 966 156 453 077 928 745 187 819 003 875 164 16 × 2 = 0 + 0.860 431 932 312 906 155 857 490 375 638 007 750 328 32;
  • 21) 0.860 431 932 312 906 155 857 490 375 638 007 750 328 32 × 2 = 1 + 0.720 863 864 625 812 311 714 980 751 276 015 500 656 64;
  • 22) 0.720 863 864 625 812 311 714 980 751 276 015 500 656 64 × 2 = 1 + 0.441 727 729 251 624 623 429 961 502 552 031 001 313 28;
  • 23) 0.441 727 729 251 624 623 429 961 502 552 031 001 313 28 × 2 = 0 + 0.883 455 458 503 249 246 859 923 005 104 062 002 626 56;
  • 24) 0.883 455 458 503 249 246 859 923 005 104 062 002 626 56 × 2 = 1 + 0.766 910 917 006 498 493 719 846 010 208 124 005 253 12;
  • 25) 0.766 910 917 006 498 493 719 846 010 208 124 005 253 12 × 2 = 1 + 0.533 821 834 012 996 987 439 692 020 416 248 010 506 24;
  • 26) 0.533 821 834 012 996 987 439 692 020 416 248 010 506 24 × 2 = 1 + 0.067 643 668 025 993 974 879 384 040 832 496 021 012 48;
  • 27) 0.067 643 668 025 993 974 879 384 040 832 496 021 012 48 × 2 = 0 + 0.135 287 336 051 987 949 758 768 081 664 992 042 024 96;
  • 28) 0.135 287 336 051 987 949 758 768 081 664 992 042 024 96 × 2 = 0 + 0.270 574 672 103 975 899 517 536 163 329 984 084 049 92;
  • 29) 0.270 574 672 103 975 899 517 536 163 329 984 084 049 92 × 2 = 0 + 0.541 149 344 207 951 799 035 072 326 659 968 168 099 84;
  • 30) 0.541 149 344 207 951 799 035 072 326 659 968 168 099 84 × 2 = 1 + 0.082 298 688 415 903 598 070 144 653 319 936 336 199 68;
  • 31) 0.082 298 688 415 903 598 070 144 653 319 936 336 199 68 × 2 = 0 + 0.164 597 376 831 807 196 140 289 306 639 872 672 399 36;
  • 32) 0.164 597 376 831 807 196 140 289 306 639 872 672 399 36 × 2 = 0 + 0.329 194 753 663 614 392 280 578 613 279 745 344 798 72;
  • 33) 0.329 194 753 663 614 392 280 578 613 279 745 344 798 72 × 2 = 0 + 0.658 389 507 327 228 784 561 157 226 559 490 689 597 44;
  • 34) 0.658 389 507 327 228 784 561 157 226 559 490 689 597 44 × 2 = 1 + 0.316 779 014 654 457 569 122 314 453 118 981 379 194 88;
  • 35) 0.316 779 014 654 457 569 122 314 453 118 981 379 194 88 × 2 = 0 + 0.633 558 029 308 915 138 244 628 906 237 962 758 389 76;
  • 36) 0.633 558 029 308 915 138 244 628 906 237 962 758 389 76 × 2 = 1 + 0.267 116 058 617 830 276 489 257 812 475 925 516 779 52;
  • 37) 0.267 116 058 617 830 276 489 257 812 475 925 516 779 52 × 2 = 0 + 0.534 232 117 235 660 552 978 515 624 951 851 033 559 04;
  • 38) 0.534 232 117 235 660 552 978 515 624 951 851 033 559 04 × 2 = 1 + 0.068 464 234 471 321 105 957 031 249 903 702 067 118 08;
  • 39) 0.068 464 234 471 321 105 957 031 249 903 702 067 118 08 × 2 = 0 + 0.136 928 468 942 642 211 914 062 499 807 404 134 236 16;
  • 40) 0.136 928 468 942 642 211 914 062 499 807 404 134 236 16 × 2 = 0 + 0.273 856 937 885 284 423 828 124 999 614 808 268 472 32;
  • 41) 0.273 856 937 885 284 423 828 124 999 614 808 268 472 32 × 2 = 0 + 0.547 713 875 770 568 847 656 249 999 229 616 536 944 64;
  • 42) 0.547 713 875 770 568 847 656 249 999 229 616 536 944 64 × 2 = 1 + 0.095 427 751 541 137 695 312 499 998 459 233 073 889 28;
  • 43) 0.095 427 751 541 137 695 312 499 998 459 233 073 889 28 × 2 = 0 + 0.190 855 503 082 275 390 624 999 996 918 466 147 778 56;
  • 44) 0.190 855 503 082 275 390 624 999 996 918 466 147 778 56 × 2 = 0 + 0.381 711 006 164 550 781 249 999 993 836 932 295 557 12;
  • 45) 0.381 711 006 164 550 781 249 999 993 836 932 295 557 12 × 2 = 0 + 0.763 422 012 329 101 562 499 999 987 673 864 591 114 24;
  • 46) 0.763 422 012 329 101 562 499 999 987 673 864 591 114 24 × 2 = 1 + 0.526 844 024 658 203 124 999 999 975 347 729 182 228 48;
  • 47) 0.526 844 024 658 203 124 999 999 975 347 729 182 228 48 × 2 = 1 + 0.053 688 049 316 406 249 999 999 950 695 458 364 456 96;
  • 48) 0.053 688 049 316 406 249 999 999 950 695 458 364 456 96 × 2 = 0 + 0.107 376 098 632 812 499 999 999 901 390 916 728 913 92;
  • 49) 0.107 376 098 632 812 499 999 999 901 390 916 728 913 92 × 2 = 0 + 0.214 752 197 265 624 999 999 999 802 781 833 457 827 84;
  • 50) 0.214 752 197 265 624 999 999 999 802 781 833 457 827 84 × 2 = 0 + 0.429 504 394 531 249 999 999 999 605 563 666 915 655 68;
  • 51) 0.429 504 394 531 249 999 999 999 605 563 666 915 655 68 × 2 = 0 + 0.859 008 789 062 499 999 999 999 211 127 333 831 311 36;
  • 52) 0.859 008 789 062 499 999 999 999 211 127 333 831 311 36 × 2 = 1 + 0.718 017 578 124 999 999 999 998 422 254 667 662 622 72;
  • 53) 0.718 017 578 124 999 999 999 998 422 254 667 662 622 72 × 2 = 1 + 0.436 035 156 249 999 999 999 996 844 509 335 325 245 44;

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:


0.211 829 052 383 358 300 119 548 661 699 653 280 265 57(10) =

0.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 699 653 280 265 57(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

7. Normalize the binary representation of the number.

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


2.211 829 052 383 358 300 119 548 661 699 653 280 265 57(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) × 20 =

1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11(2) × 21


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): 1

Mantissa (not normalized):
1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11


9. 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) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 0000(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11 =

0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 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) =
100 0000 0000

Mantissa (52 bits) =
0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


Decimal number -2.211 829 052 383 358 300 119 548 661 699 653 280 265 57 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0000 - 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 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