27.257 299 999 999 997 197 619 450 162 164 866 924 284 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 27.257 299 999 999 997 197 619 450 162 164 866 924 284 9(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
27.257 299 999 999 997 197 619 450 162 164 866 924 284 9(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: 0.257 299 999 999 997 197 619 450 162 164 866 924 284 9.

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) 0.257 299 999 999 997 197 619 450 162 164 866 924 284 9 × 2 = 0 + 0.514 599 999 999 994 395 238 900 324 329 733 848 569 8;
  • 2) 0.514 599 999 999 994 395 238 900 324 329 733 848 569 8 × 2 = 1 + 0.029 199 999 999 988 790 477 800 648 659 467 697 139 6;
  • 3) 0.029 199 999 999 988 790 477 800 648 659 467 697 139 6 × 2 = 0 + 0.058 399 999 999 977 580 955 601 297 318 935 394 279 2;
  • 4) 0.058 399 999 999 977 580 955 601 297 318 935 394 279 2 × 2 = 0 + 0.116 799 999 999 955 161 911 202 594 637 870 788 558 4;
  • 5) 0.116 799 999 999 955 161 911 202 594 637 870 788 558 4 × 2 = 0 + 0.233 599 999 999 910 323 822 405 189 275 741 577 116 8;
  • 6) 0.233 599 999 999 910 323 822 405 189 275 741 577 116 8 × 2 = 0 + 0.467 199 999 999 820 647 644 810 378 551 483 154 233 6;
  • 7) 0.467 199 999 999 820 647 644 810 378 551 483 154 233 6 × 2 = 0 + 0.934 399 999 999 641 295 289 620 757 102 966 308 467 2;
  • 8) 0.934 399 999 999 641 295 289 620 757 102 966 308 467 2 × 2 = 1 + 0.868 799 999 999 282 590 579 241 514 205 932 616 934 4;
  • 9) 0.868 799 999 999 282 590 579 241 514 205 932 616 934 4 × 2 = 1 + 0.737 599 999 998 565 181 158 483 028 411 865 233 868 8;
  • 10) 0.737 599 999 998 565 181 158 483 028 411 865 233 868 8 × 2 = 1 + 0.475 199 999 997 130 362 316 966 056 823 730 467 737 6;
  • 11) 0.475 199 999 997 130 362 316 966 056 823 730 467 737 6 × 2 = 0 + 0.950 399 999 994 260 724 633 932 113 647 460 935 475 2;
  • 12) 0.950 399 999 994 260 724 633 932 113 647 460 935 475 2 × 2 = 1 + 0.900 799 999 988 521 449 267 864 227 294 921 870 950 4;
  • 13) 0.900 799 999 988 521 449 267 864 227 294 921 870 950 4 × 2 = 1 + 0.801 599 999 977 042 898 535 728 454 589 843 741 900 8;
  • 14) 0.801 599 999 977 042 898 535 728 454 589 843 741 900 8 × 2 = 1 + 0.603 199 999 954 085 797 071 456 909 179 687 483 801 6;
  • 15) 0.603 199 999 954 085 797 071 456 909 179 687 483 801 6 × 2 = 1 + 0.206 399 999 908 171 594 142 913 818 359 374 967 603 2;
  • 16) 0.206 399 999 908 171 594 142 913 818 359 374 967 603 2 × 2 = 0 + 0.412 799 999 816 343 188 285 827 636 718 749 935 206 4;
  • 17) 0.412 799 999 816 343 188 285 827 636 718 749 935 206 4 × 2 = 0 + 0.825 599 999 632 686 376 571 655 273 437 499 870 412 8;
  • 18) 0.825 599 999 632 686 376 571 655 273 437 499 870 412 8 × 2 = 1 + 0.651 199 999 265 372 753 143 310 546 874 999 740 825 6;
  • 19) 0.651 199 999 265 372 753 143 310 546 874 999 740 825 6 × 2 = 1 + 0.302 399 998 530 745 506 286 621 093 749 999 481 651 2;
  • 20) 0.302 399 998 530 745 506 286 621 093 749 999 481 651 2 × 2 = 0 + 0.604 799 997 061 491 012 573 242 187 499 998 963 302 4;
  • 21) 0.604 799 997 061 491 012 573 242 187 499 998 963 302 4 × 2 = 1 + 0.209 599 994 122 982 025 146 484 374 999 997 926 604 8;
  • 22) 0.209 599 994 122 982 025 146 484 374 999 997 926 604 8 × 2 = 0 + 0.419 199 988 245 964 050 292 968 749 999 995 853 209 6;
  • 23) 0.419 199 988 245 964 050 292 968 749 999 995 853 209 6 × 2 = 0 + 0.838 399 976 491 928 100 585 937 499 999 991 706 419 2;
  • 24) 0.838 399 976 491 928 100 585 937 499 999 991 706 419 2 × 2 = 1 + 0.676 799 952 983 856 201 171 874 999 999 983 412 838 4;
  • 25) 0.676 799 952 983 856 201 171 874 999 999 983 412 838 4 × 2 = 1 + 0.353 599 905 967 712 402 343 749 999 999 966 825 676 8;
  • 26) 0.353 599 905 967 712 402 343 749 999 999 966 825 676 8 × 2 = 0 + 0.707 199 811 935 424 804 687 499 999 999 933 651 353 6;
  • 27) 0.707 199 811 935 424 804 687 499 999 999 933 651 353 6 × 2 = 1 + 0.414 399 623 870 849 609 374 999 999 999 867 302 707 2;
  • 28) 0.414 399 623 870 849 609 374 999 999 999 867 302 707 2 × 2 = 0 + 0.828 799 247 741 699 218 749 999 999 999 734 605 414 4;
  • 29) 0.828 799 247 741 699 218 749 999 999 999 734 605 414 4 × 2 = 1 + 0.657 598 495 483 398 437 499 999 999 999 469 210 828 8;
  • 30) 0.657 598 495 483 398 437 499 999 999 999 469 210 828 8 × 2 = 1 + 0.315 196 990 966 796 874 999 999 999 998 938 421 657 6;
  • 31) 0.315 196 990 966 796 874 999 999 999 998 938 421 657 6 × 2 = 0 + 0.630 393 981 933 593 749 999 999 999 997 876 843 315 2;
  • 32) 0.630 393 981 933 593 749 999 999 999 997 876 843 315 2 × 2 = 1 + 0.260 787 963 867 187 499 999 999 999 995 753 686 630 4;
  • 33) 0.260 787 963 867 187 499 999 999 999 995 753 686 630 4 × 2 = 0 + 0.521 575 927 734 374 999 999 999 999 991 507 373 260 8;
  • 34) 0.521 575 927 734 374 999 999 999 999 991 507 373 260 8 × 2 = 1 + 0.043 151 855 468 749 999 999 999 999 983 014 746 521 6;
  • 35) 0.043 151 855 468 749 999 999 999 999 983 014 746 521 6 × 2 = 0 + 0.086 303 710 937 499 999 999 999 999 966 029 493 043 2;
  • 36) 0.086 303 710 937 499 999 999 999 999 966 029 493 043 2 × 2 = 0 + 0.172 607 421 874 999 999 999 999 999 932 058 986 086 4;
  • 37) 0.172 607 421 874 999 999 999 999 999 932 058 986 086 4 × 2 = 0 + 0.345 214 843 749 999 999 999 999 999 864 117 972 172 8;
  • 38) 0.345 214 843 749 999 999 999 999 999 864 117 972 172 8 × 2 = 0 + 0.690 429 687 499 999 999 999 999 999 728 235 944 345 6;
  • 39) 0.690 429 687 499 999 999 999 999 999 728 235 944 345 6 × 2 = 1 + 0.380 859 374 999 999 999 999 999 999 456 471 888 691 2;
  • 40) 0.380 859 374 999 999 999 999 999 999 456 471 888 691 2 × 2 = 0 + 0.761 718 749 999 999 999 999 999 998 912 943 777 382 4;
  • 41) 0.761 718 749 999 999 999 999 999 998 912 943 777 382 4 × 2 = 1 + 0.523 437 499 999 999 999 999 999 997 825 887 554 764 8;
  • 42) 0.523 437 499 999 999 999 999 999 997 825 887 554 764 8 × 2 = 1 + 0.046 874 999 999 999 999 999 999 995 651 775 109 529 6;
  • 43) 0.046 874 999 999 999 999 999 999 995 651 775 109 529 6 × 2 = 0 + 0.093 749 999 999 999 999 999 999 991 303 550 219 059 2;
  • 44) 0.093 749 999 999 999 999 999 999 991 303 550 219 059 2 × 2 = 0 + 0.187 499 999 999 999 999 999 999 982 607 100 438 118 4;
  • 45) 0.187 499 999 999 999 999 999 999 982 607 100 438 118 4 × 2 = 0 + 0.374 999 999 999 999 999 999 999 965 214 200 876 236 8;
  • 46) 0.374 999 999 999 999 999 999 999 965 214 200 876 236 8 × 2 = 0 + 0.749 999 999 999 999 999 999 999 930 428 401 752 473 6;
  • 47) 0.749 999 999 999 999 999 999 999 930 428 401 752 473 6 × 2 = 1 + 0.499 999 999 999 999 999 999 999 860 856 803 504 947 2;
  • 48) 0.499 999 999 999 999 999 999 999 860 856 803 504 947 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 721 713 607 009 894 4;
  • 49) 0.999 999 999 999 999 999 999 999 721 713 607 009 894 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 443 427 214 019 788 8;
  • 50) 0.999 999 999 999 999 999 999 999 443 427 214 019 788 8 × 2 = 1 + 0.999 999 999 999 999 999 999 998 886 854 428 039 577 6;
  • 51) 0.999 999 999 999 999 999 999 998 886 854 428 039 577 6 × 2 = 1 + 0.999 999 999 999 999 999 999 997 773 708 856 079 155 2;
  • 52) 0.999 999 999 999 999 999 999 997 773 708 856 079 155 2 × 2 = 1 + 0.999 999 999 999 999 999 999 995 547 417 712 158 310 4;
  • 53) 0.999 999 999 999 999 999 999 995 547 417 712 158 310 4 × 2 = 1 + 0.999 999 999 999 999 999 999 991 094 835 424 316 620 8;

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:


0.257 299 999 999 997 197 619 450 162 164 866 924 284 9(10) =

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

5. Positive number before normalization:

27.257 299 999 999 997 197 619 450 162 164 866 924 284 9(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:


27.257 299 999 999 997 197 619 450 162 164 866 924 284 9(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 27.257 299 999 999 997 197 619 450 162 164 866 924 284 9 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