27.257 299 999 999 997 197 619 450 162 164 866 924 283 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 27.257 299 999 999 997 197 619 450 162 164 866 924 283 2(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
27.257 299 999 999 997 197 619 450 162 164 866 924 283 2(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: 27.
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;
  • 27 ÷ 2 = 13 + 1;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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.

27(10) =

1 1011(2)


3. Convert to binary (base 2) the fractional part: 0.257 299 999 999 997 197 619 450 162 164 866 924 283 2.

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) 0.257 299 999 999 997 197 619 450 162 164 866 924 283 2 × 2 = 0 + 0.514 599 999 999 994 395 238 900 324 329 733 848 566 4;
  • 2) 0.514 599 999 999 994 395 238 900 324 329 733 848 566 4 × 2 = 1 + 0.029 199 999 999 988 790 477 800 648 659 467 697 132 8;
  • 3) 0.029 199 999 999 988 790 477 800 648 659 467 697 132 8 × 2 = 0 + 0.058 399 999 999 977 580 955 601 297 318 935 394 265 6;
  • 4) 0.058 399 999 999 977 580 955 601 297 318 935 394 265 6 × 2 = 0 + 0.116 799 999 999 955 161 911 202 594 637 870 788 531 2;
  • 5) 0.116 799 999 999 955 161 911 202 594 637 870 788 531 2 × 2 = 0 + 0.233 599 999 999 910 323 822 405 189 275 741 577 062 4;
  • 6) 0.233 599 999 999 910 323 822 405 189 275 741 577 062 4 × 2 = 0 + 0.467 199 999 999 820 647 644 810 378 551 483 154 124 8;
  • 7) 0.467 199 999 999 820 647 644 810 378 551 483 154 124 8 × 2 = 0 + 0.934 399 999 999 641 295 289 620 757 102 966 308 249 6;
  • 8) 0.934 399 999 999 641 295 289 620 757 102 966 308 249 6 × 2 = 1 + 0.868 799 999 999 282 590 579 241 514 205 932 616 499 2;
  • 9) 0.868 799 999 999 282 590 579 241 514 205 932 616 499 2 × 2 = 1 + 0.737 599 999 998 565 181 158 483 028 411 865 232 998 4;
  • 10) 0.737 599 999 998 565 181 158 483 028 411 865 232 998 4 × 2 = 1 + 0.475 199 999 997 130 362 316 966 056 823 730 465 996 8;
  • 11) 0.475 199 999 997 130 362 316 966 056 823 730 465 996 8 × 2 = 0 + 0.950 399 999 994 260 724 633 932 113 647 460 931 993 6;
  • 12) 0.950 399 999 994 260 724 633 932 113 647 460 931 993 6 × 2 = 1 + 0.900 799 999 988 521 449 267 864 227 294 921 863 987 2;
  • 13) 0.900 799 999 988 521 449 267 864 227 294 921 863 987 2 × 2 = 1 + 0.801 599 999 977 042 898 535 728 454 589 843 727 974 4;
  • 14) 0.801 599 999 977 042 898 535 728 454 589 843 727 974 4 × 2 = 1 + 0.603 199 999 954 085 797 071 456 909 179 687 455 948 8;
  • 15) 0.603 199 999 954 085 797 071 456 909 179 687 455 948 8 × 2 = 1 + 0.206 399 999 908 171 594 142 913 818 359 374 911 897 6;
  • 16) 0.206 399 999 908 171 594 142 913 818 359 374 911 897 6 × 2 = 0 + 0.412 799 999 816 343 188 285 827 636 718 749 823 795 2;
  • 17) 0.412 799 999 816 343 188 285 827 636 718 749 823 795 2 × 2 = 0 + 0.825 599 999 632 686 376 571 655 273 437 499 647 590 4;
  • 18) 0.825 599 999 632 686 376 571 655 273 437 499 647 590 4 × 2 = 1 + 0.651 199 999 265 372 753 143 310 546 874 999 295 180 8;
  • 19) 0.651 199 999 265 372 753 143 310 546 874 999 295 180 8 × 2 = 1 + 0.302 399 998 530 745 506 286 621 093 749 998 590 361 6;
  • 20) 0.302 399 998 530 745 506 286 621 093 749 998 590 361 6 × 2 = 0 + 0.604 799 997 061 491 012 573 242 187 499 997 180 723 2;
  • 21) 0.604 799 997 061 491 012 573 242 187 499 997 180 723 2 × 2 = 1 + 0.209 599 994 122 982 025 146 484 374 999 994 361 446 4;
  • 22) 0.209 599 994 122 982 025 146 484 374 999 994 361 446 4 × 2 = 0 + 0.419 199 988 245 964 050 292 968 749 999 988 722 892 8;
  • 23) 0.419 199 988 245 964 050 292 968 749 999 988 722 892 8 × 2 = 0 + 0.838 399 976 491 928 100 585 937 499 999 977 445 785 6;
  • 24) 0.838 399 976 491 928 100 585 937 499 999 977 445 785 6 × 2 = 1 + 0.676 799 952 983 856 201 171 874 999 999 954 891 571 2;
  • 25) 0.676 799 952 983 856 201 171 874 999 999 954 891 571 2 × 2 = 1 + 0.353 599 905 967 712 402 343 749 999 999 909 783 142 4;
  • 26) 0.353 599 905 967 712 402 343 749 999 999 909 783 142 4 × 2 = 0 + 0.707 199 811 935 424 804 687 499 999 999 819 566 284 8;
  • 27) 0.707 199 811 935 424 804 687 499 999 999 819 566 284 8 × 2 = 1 + 0.414 399 623 870 849 609 374 999 999 999 639 132 569 6;
  • 28) 0.414 399 623 870 849 609 374 999 999 999 639 132 569 6 × 2 = 0 + 0.828 799 247 741 699 218 749 999 999 999 278 265 139 2;
  • 29) 0.828 799 247 741 699 218 749 999 999 999 278 265 139 2 × 2 = 1 + 0.657 598 495 483 398 437 499 999 999 998 556 530 278 4;
  • 30) 0.657 598 495 483 398 437 499 999 999 998 556 530 278 4 × 2 = 1 + 0.315 196 990 966 796 874 999 999 999 997 113 060 556 8;
  • 31) 0.315 196 990 966 796 874 999 999 999 997 113 060 556 8 × 2 = 0 + 0.630 393 981 933 593 749 999 999 999 994 226 121 113 6;
  • 32) 0.630 393 981 933 593 749 999 999 999 994 226 121 113 6 × 2 = 1 + 0.260 787 963 867 187 499 999 999 999 988 452 242 227 2;
  • 33) 0.260 787 963 867 187 499 999 999 999 988 452 242 227 2 × 2 = 0 + 0.521 575 927 734 374 999 999 999 999 976 904 484 454 4;
  • 34) 0.521 575 927 734 374 999 999 999 999 976 904 484 454 4 × 2 = 1 + 0.043 151 855 468 749 999 999 999 999 953 808 968 908 8;
  • 35) 0.043 151 855 468 749 999 999 999 999 953 808 968 908 8 × 2 = 0 + 0.086 303 710 937 499 999 999 999 999 907 617 937 817 6;
  • 36) 0.086 303 710 937 499 999 999 999 999 907 617 937 817 6 × 2 = 0 + 0.172 607 421 874 999 999 999 999 999 815 235 875 635 2;
  • 37) 0.172 607 421 874 999 999 999 999 999 815 235 875 635 2 × 2 = 0 + 0.345 214 843 749 999 999 999 999 999 630 471 751 270 4;
  • 38) 0.345 214 843 749 999 999 999 999 999 630 471 751 270 4 × 2 = 0 + 0.690 429 687 499 999 999 999 999 999 260 943 502 540 8;
  • 39) 0.690 429 687 499 999 999 999 999 999 260 943 502 540 8 × 2 = 1 + 0.380 859 374 999 999 999 999 999 998 521 887 005 081 6;
  • 40) 0.380 859 374 999 999 999 999 999 998 521 887 005 081 6 × 2 = 0 + 0.761 718 749 999 999 999 999 999 997 043 774 010 163 2;
  • 41) 0.761 718 749 999 999 999 999 999 997 043 774 010 163 2 × 2 = 1 + 0.523 437 499 999 999 999 999 999 994 087 548 020 326 4;
  • 42) 0.523 437 499 999 999 999 999 999 994 087 548 020 326 4 × 2 = 1 + 0.046 874 999 999 999 999 999 999 988 175 096 040 652 8;
  • 43) 0.046 874 999 999 999 999 999 999 988 175 096 040 652 8 × 2 = 0 + 0.093 749 999 999 999 999 999 999 976 350 192 081 305 6;
  • 44) 0.093 749 999 999 999 999 999 999 976 350 192 081 305 6 × 2 = 0 + 0.187 499 999 999 999 999 999 999 952 700 384 162 611 2;
  • 45) 0.187 499 999 999 999 999 999 999 952 700 384 162 611 2 × 2 = 0 + 0.374 999 999 999 999 999 999 999 905 400 768 325 222 4;
  • 46) 0.374 999 999 999 999 999 999 999 905 400 768 325 222 4 × 2 = 0 + 0.749 999 999 999 999 999 999 999 810 801 536 650 444 8;
  • 47) 0.749 999 999 999 999 999 999 999 810 801 536 650 444 8 × 2 = 1 + 0.499 999 999 999 999 999 999 999 621 603 073 300 889 6;
  • 48) 0.499 999 999 999 999 999 999 999 621 603 073 300 889 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 243 206 146 601 779 2;
  • 49) 0.999 999 999 999 999 999 999 999 243 206 146 601 779 2 × 2 = 1 + 0.999 999 999 999 999 999 999 998 486 412 293 203 558 4;
  • 50) 0.999 999 999 999 999 999 999 998 486 412 293 203 558 4 × 2 = 1 + 0.999 999 999 999 999 999 999 996 972 824 586 407 116 8;
  • 51) 0.999 999 999 999 999 999 999 996 972 824 586 407 116 8 × 2 = 1 + 0.999 999 999 999 999 999 999 993 945 649 172 814 233 6;
  • 52) 0.999 999 999 999 999 999 999 993 945 649 172 814 233 6 × 2 = 1 + 0.999 999 999 999 999 999 999 987 891 298 345 628 467 2;
  • 53) 0.999 999 999 999 999 999 999 987 891 298 345 628 467 2 × 2 = 1 + 0.999 999 999 999 999 999 999 975 782 596 691 256 934 4;

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:


0.257 299 999 999 997 197 619 450 162 164 866 924 283 2(10) =

0.0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1(2)

5. Positive number before normalization:

27.257 299 999 999 997 197 619 450 162 164 866 924 283 2(10) =

1 1011.0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


27.257 299 999 999 997 197 619 450 162 164 866 924 283 2(10) =

1 1011.0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1(2) =

1 1011.0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1(2) × 20 =

1.1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1(2) × 24


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): 4

Mantissa (not normalized):
1.1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1111 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 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;

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) =

1027(10) =

100 0000 0011(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, 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. 1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010 1 1111 =

1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010


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) =
100 0000 0011

Mantissa (52 bits) =
1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010


Decimal number 27.257 299 999 999 997 197 619 450 162 164 866 924 283 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 1011 0100 0001 1101 1110 0110 1001 1010 1101 0100 0010 1100 0010

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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