-2.211 829 052 383 358 300 119 548 661 699 653 277 92 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 277 92(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
-2.211 829 052 383 358 300 119 548 661 699 653 277 92(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:

|-2.211 829 052 383 358 300 119 548 661 699 653 277 92| = 2.211 829 052 383 358 300 119 548 661 699 653 277 92


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: 0.211 829 052 383 358 300 119 548 661 699 653 277 92.

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) 0.211 829 052 383 358 300 119 548 661 699 653 277 92 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 399 306 555 84;
  • 2) 0.423 658 104 766 716 600 239 097 323 399 306 555 84 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 798 613 111 68;
  • 3) 0.847 316 209 533 433 200 478 194 646 798 613 111 68 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 597 226 223 36;
  • 4) 0.694 632 419 066 866 400 956 389 293 597 226 223 36 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 194 452 446 72;
  • 5) 0.389 264 838 133 732 801 912 778 587 194 452 446 72 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 388 904 893 44;
  • 6) 0.778 529 676 267 465 603 825 557 174 388 904 893 44 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 777 809 786 88;
  • 7) 0.557 059 352 534 931 207 651 114 348 777 809 786 88 × 2 = 1 + 0.114 118 705 069 862 415 302 228 697 555 619 573 76;
  • 8) 0.114 118 705 069 862 415 302 228 697 555 619 573 76 × 2 = 0 + 0.228 237 410 139 724 830 604 457 395 111 239 147 52;
  • 9) 0.228 237 410 139 724 830 604 457 395 111 239 147 52 × 2 = 0 + 0.456 474 820 279 449 661 208 914 790 222 478 295 04;
  • 10) 0.456 474 820 279 449 661 208 914 790 222 478 295 04 × 2 = 0 + 0.912 949 640 558 899 322 417 829 580 444 956 590 08;
  • 11) 0.912 949 640 558 899 322 417 829 580 444 956 590 08 × 2 = 1 + 0.825 899 281 117 798 644 835 659 160 889 913 180 16;
  • 12) 0.825 899 281 117 798 644 835 659 160 889 913 180 16 × 2 = 1 + 0.651 798 562 235 597 289 671 318 321 779 826 360 32;
  • 13) 0.651 798 562 235 597 289 671 318 321 779 826 360 32 × 2 = 1 + 0.303 597 124 471 194 579 342 636 643 559 652 720 64;
  • 14) 0.303 597 124 471 194 579 342 636 643 559 652 720 64 × 2 = 0 + 0.607 194 248 942 389 158 685 273 287 119 305 441 28;
  • 15) 0.607 194 248 942 389 158 685 273 287 119 305 441 28 × 2 = 1 + 0.214 388 497 884 778 317 370 546 574 238 610 882 56;
  • 16) 0.214 388 497 884 778 317 370 546 574 238 610 882 56 × 2 = 0 + 0.428 776 995 769 556 634 741 093 148 477 221 765 12;
  • 17) 0.428 776 995 769 556 634 741 093 148 477 221 765 12 × 2 = 0 + 0.857 553 991 539 113 269 482 186 296 954 443 530 24;
  • 18) 0.857 553 991 539 113 269 482 186 296 954 443 530 24 × 2 = 1 + 0.715 107 983 078 226 538 964 372 593 908 887 060 48;
  • 19) 0.715 107 983 078 226 538 964 372 593 908 887 060 48 × 2 = 1 + 0.430 215 966 156 453 077 928 745 187 817 774 120 96;
  • 20) 0.430 215 966 156 453 077 928 745 187 817 774 120 96 × 2 = 0 + 0.860 431 932 312 906 155 857 490 375 635 548 241 92;
  • 21) 0.860 431 932 312 906 155 857 490 375 635 548 241 92 × 2 = 1 + 0.720 863 864 625 812 311 714 980 751 271 096 483 84;
  • 22) 0.720 863 864 625 812 311 714 980 751 271 096 483 84 × 2 = 1 + 0.441 727 729 251 624 623 429 961 502 542 192 967 68;
  • 23) 0.441 727 729 251 624 623 429 961 502 542 192 967 68 × 2 = 0 + 0.883 455 458 503 249 246 859 923 005 084 385 935 36;
  • 24) 0.883 455 458 503 249 246 859 923 005 084 385 935 36 × 2 = 1 + 0.766 910 917 006 498 493 719 846 010 168 771 870 72;
  • 25) 0.766 910 917 006 498 493 719 846 010 168 771 870 72 × 2 = 1 + 0.533 821 834 012 996 987 439 692 020 337 543 741 44;
  • 26) 0.533 821 834 012 996 987 439 692 020 337 543 741 44 × 2 = 1 + 0.067 643 668 025 993 974 879 384 040 675 087 482 88;
  • 27) 0.067 643 668 025 993 974 879 384 040 675 087 482 88 × 2 = 0 + 0.135 287 336 051 987 949 758 768 081 350 174 965 76;
  • 28) 0.135 287 336 051 987 949 758 768 081 350 174 965 76 × 2 = 0 + 0.270 574 672 103 975 899 517 536 162 700 349 931 52;
  • 29) 0.270 574 672 103 975 899 517 536 162 700 349 931 52 × 2 = 0 + 0.541 149 344 207 951 799 035 072 325 400 699 863 04;
  • 30) 0.541 149 344 207 951 799 035 072 325 400 699 863 04 × 2 = 1 + 0.082 298 688 415 903 598 070 144 650 801 399 726 08;
  • 31) 0.082 298 688 415 903 598 070 144 650 801 399 726 08 × 2 = 0 + 0.164 597 376 831 807 196 140 289 301 602 799 452 16;
  • 32) 0.164 597 376 831 807 196 140 289 301 602 799 452 16 × 2 = 0 + 0.329 194 753 663 614 392 280 578 603 205 598 904 32;
  • 33) 0.329 194 753 663 614 392 280 578 603 205 598 904 32 × 2 = 0 + 0.658 389 507 327 228 784 561 157 206 411 197 808 64;
  • 34) 0.658 389 507 327 228 784 561 157 206 411 197 808 64 × 2 = 1 + 0.316 779 014 654 457 569 122 314 412 822 395 617 28;
  • 35) 0.316 779 014 654 457 569 122 314 412 822 395 617 28 × 2 = 0 + 0.633 558 029 308 915 138 244 628 825 644 791 234 56;
  • 36) 0.633 558 029 308 915 138 244 628 825 644 791 234 56 × 2 = 1 + 0.267 116 058 617 830 276 489 257 651 289 582 469 12;
  • 37) 0.267 116 058 617 830 276 489 257 651 289 582 469 12 × 2 = 0 + 0.534 232 117 235 660 552 978 515 302 579 164 938 24;
  • 38) 0.534 232 117 235 660 552 978 515 302 579 164 938 24 × 2 = 1 + 0.068 464 234 471 321 105 957 030 605 158 329 876 48;
  • 39) 0.068 464 234 471 321 105 957 030 605 158 329 876 48 × 2 = 0 + 0.136 928 468 942 642 211 914 061 210 316 659 752 96;
  • 40) 0.136 928 468 942 642 211 914 061 210 316 659 752 96 × 2 = 0 + 0.273 856 937 885 284 423 828 122 420 633 319 505 92;
  • 41) 0.273 856 937 885 284 423 828 122 420 633 319 505 92 × 2 = 0 + 0.547 713 875 770 568 847 656 244 841 266 639 011 84;
  • 42) 0.547 713 875 770 568 847 656 244 841 266 639 011 84 × 2 = 1 + 0.095 427 751 541 137 695 312 489 682 533 278 023 68;
  • 43) 0.095 427 751 541 137 695 312 489 682 533 278 023 68 × 2 = 0 + 0.190 855 503 082 275 390 624 979 365 066 556 047 36;
  • 44) 0.190 855 503 082 275 390 624 979 365 066 556 047 36 × 2 = 0 + 0.381 711 006 164 550 781 249 958 730 133 112 094 72;
  • 45) 0.381 711 006 164 550 781 249 958 730 133 112 094 72 × 2 = 0 + 0.763 422 012 329 101 562 499 917 460 266 224 189 44;
  • 46) 0.763 422 012 329 101 562 499 917 460 266 224 189 44 × 2 = 1 + 0.526 844 024 658 203 124 999 834 920 532 448 378 88;
  • 47) 0.526 844 024 658 203 124 999 834 920 532 448 378 88 × 2 = 1 + 0.053 688 049 316 406 249 999 669 841 064 896 757 76;
  • 48) 0.053 688 049 316 406 249 999 669 841 064 896 757 76 × 2 = 0 + 0.107 376 098 632 812 499 999 339 682 129 793 515 52;
  • 49) 0.107 376 098 632 812 499 999 339 682 129 793 515 52 × 2 = 0 + 0.214 752 197 265 624 999 998 679 364 259 587 031 04;
  • 50) 0.214 752 197 265 624 999 998 679 364 259 587 031 04 × 2 = 0 + 0.429 504 394 531 249 999 997 358 728 519 174 062 08;
  • 51) 0.429 504 394 531 249 999 997 358 728 519 174 062 08 × 2 = 0 + 0.859 008 789 062 499 999 994 717 457 038 348 124 16;
  • 52) 0.859 008 789 062 499 999 994 717 457 038 348 124 16 × 2 = 1 + 0.718 017 578 124 999 999 989 434 914 076 696 248 32;
  • 53) 0.718 017 578 124 999 999 989 434 914 076 696 248 32 × 2 = 1 + 0.436 035 156 249 999 999 978 869 828 153 392 496 64;

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:


0.211 829 052 383 358 300 119 548 661 699 653 277 92(10) =

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

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 699 653 277 92(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:


2.211 829 052 383 358 300 119 548 661 699 653 277 92(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 -2.211 829 052 383 358 300 119 548 661 699 653 277 92 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