-2.211 829 052 383 358 300 119 548 661 694 77 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 694 77(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
-2.211 829 052 383 358 300 119 548 661 694 77(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:

|-2.211 829 052 383 358 300 119 548 661 694 77| = 2.211 829 052 383 358 300 119 548 661 694 77


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: 0.211 829 052 383 358 300 119 548 661 694 77.

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) 0.211 829 052 383 358 300 119 548 661 694 77 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 389 54;
  • 2) 0.423 658 104 766 716 600 239 097 323 389 54 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 779 08;
  • 3) 0.847 316 209 533 433 200 478 194 646 779 08 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 558 16;
  • 4) 0.694 632 419 066 866 400 956 389 293 558 16 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 116 32;
  • 5) 0.389 264 838 133 732 801 912 778 587 116 32 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 232 64;
  • 6) 0.778 529 676 267 465 603 825 557 174 232 64 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 465 28;
  • 7) 0.557 059 352 534 931 207 651 114 348 465 28 × 2 = 1 + 0.114 118 705 069 862 415 302 228 696 930 56;
  • 8) 0.114 118 705 069 862 415 302 228 696 930 56 × 2 = 0 + 0.228 237 410 139 724 830 604 457 393 861 12;
  • 9) 0.228 237 410 139 724 830 604 457 393 861 12 × 2 = 0 + 0.456 474 820 279 449 661 208 914 787 722 24;
  • 10) 0.456 474 820 279 449 661 208 914 787 722 24 × 2 = 0 + 0.912 949 640 558 899 322 417 829 575 444 48;
  • 11) 0.912 949 640 558 899 322 417 829 575 444 48 × 2 = 1 + 0.825 899 281 117 798 644 835 659 150 888 96;
  • 12) 0.825 899 281 117 798 644 835 659 150 888 96 × 2 = 1 + 0.651 798 562 235 597 289 671 318 301 777 92;
  • 13) 0.651 798 562 235 597 289 671 318 301 777 92 × 2 = 1 + 0.303 597 124 471 194 579 342 636 603 555 84;
  • 14) 0.303 597 124 471 194 579 342 636 603 555 84 × 2 = 0 + 0.607 194 248 942 389 158 685 273 207 111 68;
  • 15) 0.607 194 248 942 389 158 685 273 207 111 68 × 2 = 1 + 0.214 388 497 884 778 317 370 546 414 223 36;
  • 16) 0.214 388 497 884 778 317 370 546 414 223 36 × 2 = 0 + 0.428 776 995 769 556 634 741 092 828 446 72;
  • 17) 0.428 776 995 769 556 634 741 092 828 446 72 × 2 = 0 + 0.857 553 991 539 113 269 482 185 656 893 44;
  • 18) 0.857 553 991 539 113 269 482 185 656 893 44 × 2 = 1 + 0.715 107 983 078 226 538 964 371 313 786 88;
  • 19) 0.715 107 983 078 226 538 964 371 313 786 88 × 2 = 1 + 0.430 215 966 156 453 077 928 742 627 573 76;
  • 20) 0.430 215 966 156 453 077 928 742 627 573 76 × 2 = 0 + 0.860 431 932 312 906 155 857 485 255 147 52;
  • 21) 0.860 431 932 312 906 155 857 485 255 147 52 × 2 = 1 + 0.720 863 864 625 812 311 714 970 510 295 04;
  • 22) 0.720 863 864 625 812 311 714 970 510 295 04 × 2 = 1 + 0.441 727 729 251 624 623 429 941 020 590 08;
  • 23) 0.441 727 729 251 624 623 429 941 020 590 08 × 2 = 0 + 0.883 455 458 503 249 246 859 882 041 180 16;
  • 24) 0.883 455 458 503 249 246 859 882 041 180 16 × 2 = 1 + 0.766 910 917 006 498 493 719 764 082 360 32;
  • 25) 0.766 910 917 006 498 493 719 764 082 360 32 × 2 = 1 + 0.533 821 834 012 996 987 439 528 164 720 64;
  • 26) 0.533 821 834 012 996 987 439 528 164 720 64 × 2 = 1 + 0.067 643 668 025 993 974 879 056 329 441 28;
  • 27) 0.067 643 668 025 993 974 879 056 329 441 28 × 2 = 0 + 0.135 287 336 051 987 949 758 112 658 882 56;
  • 28) 0.135 287 336 051 987 949 758 112 658 882 56 × 2 = 0 + 0.270 574 672 103 975 899 516 225 317 765 12;
  • 29) 0.270 574 672 103 975 899 516 225 317 765 12 × 2 = 0 + 0.541 149 344 207 951 799 032 450 635 530 24;
  • 30) 0.541 149 344 207 951 799 032 450 635 530 24 × 2 = 1 + 0.082 298 688 415 903 598 064 901 271 060 48;
  • 31) 0.082 298 688 415 903 598 064 901 271 060 48 × 2 = 0 + 0.164 597 376 831 807 196 129 802 542 120 96;
  • 32) 0.164 597 376 831 807 196 129 802 542 120 96 × 2 = 0 + 0.329 194 753 663 614 392 259 605 084 241 92;
  • 33) 0.329 194 753 663 614 392 259 605 084 241 92 × 2 = 0 + 0.658 389 507 327 228 784 519 210 168 483 84;
  • 34) 0.658 389 507 327 228 784 519 210 168 483 84 × 2 = 1 + 0.316 779 014 654 457 569 038 420 336 967 68;
  • 35) 0.316 779 014 654 457 569 038 420 336 967 68 × 2 = 0 + 0.633 558 029 308 915 138 076 840 673 935 36;
  • 36) 0.633 558 029 308 915 138 076 840 673 935 36 × 2 = 1 + 0.267 116 058 617 830 276 153 681 347 870 72;
  • 37) 0.267 116 058 617 830 276 153 681 347 870 72 × 2 = 0 + 0.534 232 117 235 660 552 307 362 695 741 44;
  • 38) 0.534 232 117 235 660 552 307 362 695 741 44 × 2 = 1 + 0.068 464 234 471 321 104 614 725 391 482 88;
  • 39) 0.068 464 234 471 321 104 614 725 391 482 88 × 2 = 0 + 0.136 928 468 942 642 209 229 450 782 965 76;
  • 40) 0.136 928 468 942 642 209 229 450 782 965 76 × 2 = 0 + 0.273 856 937 885 284 418 458 901 565 931 52;
  • 41) 0.273 856 937 885 284 418 458 901 565 931 52 × 2 = 0 + 0.547 713 875 770 568 836 917 803 131 863 04;
  • 42) 0.547 713 875 770 568 836 917 803 131 863 04 × 2 = 1 + 0.095 427 751 541 137 673 835 606 263 726 08;
  • 43) 0.095 427 751 541 137 673 835 606 263 726 08 × 2 = 0 + 0.190 855 503 082 275 347 671 212 527 452 16;
  • 44) 0.190 855 503 082 275 347 671 212 527 452 16 × 2 = 0 + 0.381 711 006 164 550 695 342 425 054 904 32;
  • 45) 0.381 711 006 164 550 695 342 425 054 904 32 × 2 = 0 + 0.763 422 012 329 101 390 684 850 109 808 64;
  • 46) 0.763 422 012 329 101 390 684 850 109 808 64 × 2 = 1 + 0.526 844 024 658 202 781 369 700 219 617 28;
  • 47) 0.526 844 024 658 202 781 369 700 219 617 28 × 2 = 1 + 0.053 688 049 316 405 562 739 400 439 234 56;
  • 48) 0.053 688 049 316 405 562 739 400 439 234 56 × 2 = 0 + 0.107 376 098 632 811 125 478 800 878 469 12;
  • 49) 0.107 376 098 632 811 125 478 800 878 469 12 × 2 = 0 + 0.214 752 197 265 622 250 957 601 756 938 24;
  • 50) 0.214 752 197 265 622 250 957 601 756 938 24 × 2 = 0 + 0.429 504 394 531 244 501 915 203 513 876 48;
  • 51) 0.429 504 394 531 244 501 915 203 513 876 48 × 2 = 0 + 0.859 008 789 062 489 003 830 407 027 752 96;
  • 52) 0.859 008 789 062 489 003 830 407 027 752 96 × 2 = 1 + 0.718 017 578 124 978 007 660 814 055 505 92;
  • 53) 0.718 017 578 124 978 007 660 814 055 505 92 × 2 = 1 + 0.436 035 156 249 956 015 321 628 111 011 84;

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:


0.211 829 052 383 358 300 119 548 661 694 77(10) =

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

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 694 77(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:


2.211 829 052 383 358 300 119 548 661 694 77(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 -2.211 829 052 383 358 300 119 548 661 694 77 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