1.123 412 341 234 123 412 341 234 123 412 341 234 123 02 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.123 412 341 234 123 412 341 234 123 412 341 234 123 02(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
1.123 412 341 234 123 412 341 234 123 412 341 234 123 02(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: 1.
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;
  • 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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.123 412 341 234 123 412 341 234 123 412 341 234 123 02.

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) 0.123 412 341 234 123 412 341 234 123 412 341 234 123 02 × 2 = 0 + 0.246 824 682 468 246 824 682 468 246 824 682 468 246 04;
  • 2) 0.246 824 682 468 246 824 682 468 246 824 682 468 246 04 × 2 = 0 + 0.493 649 364 936 493 649 364 936 493 649 364 936 492 08;
  • 3) 0.493 649 364 936 493 649 364 936 493 649 364 936 492 08 × 2 = 0 + 0.987 298 729 872 987 298 729 872 987 298 729 872 984 16;
  • 4) 0.987 298 729 872 987 298 729 872 987 298 729 872 984 16 × 2 = 1 + 0.974 597 459 745 974 597 459 745 974 597 459 745 968 32;
  • 5) 0.974 597 459 745 974 597 459 745 974 597 459 745 968 32 × 2 = 1 + 0.949 194 919 491 949 194 919 491 949 194 919 491 936 64;
  • 6) 0.949 194 919 491 949 194 919 491 949 194 919 491 936 64 × 2 = 1 + 0.898 389 838 983 898 389 838 983 898 389 838 983 873 28;
  • 7) 0.898 389 838 983 898 389 838 983 898 389 838 983 873 28 × 2 = 1 + 0.796 779 677 967 796 779 677 967 796 779 677 967 746 56;
  • 8) 0.796 779 677 967 796 779 677 967 796 779 677 967 746 56 × 2 = 1 + 0.593 559 355 935 593 559 355 935 593 559 355 935 493 12;
  • 9) 0.593 559 355 935 593 559 355 935 593 559 355 935 493 12 × 2 = 1 + 0.187 118 711 871 187 118 711 871 187 118 711 870 986 24;
  • 10) 0.187 118 711 871 187 118 711 871 187 118 711 870 986 24 × 2 = 0 + 0.374 237 423 742 374 237 423 742 374 237 423 741 972 48;
  • 11) 0.374 237 423 742 374 237 423 742 374 237 423 741 972 48 × 2 = 0 + 0.748 474 847 484 748 474 847 484 748 474 847 483 944 96;
  • 12) 0.748 474 847 484 748 474 847 484 748 474 847 483 944 96 × 2 = 1 + 0.496 949 694 969 496 949 694 969 496 949 694 967 889 92;
  • 13) 0.496 949 694 969 496 949 694 969 496 949 694 967 889 92 × 2 = 0 + 0.993 899 389 938 993 899 389 938 993 899 389 935 779 84;
  • 14) 0.993 899 389 938 993 899 389 938 993 899 389 935 779 84 × 2 = 1 + 0.987 798 779 877 987 798 779 877 987 798 779 871 559 68;
  • 15) 0.987 798 779 877 987 798 779 877 987 798 779 871 559 68 × 2 = 1 + 0.975 597 559 755 975 597 559 755 975 597 559 743 119 36;
  • 16) 0.975 597 559 755 975 597 559 755 975 597 559 743 119 36 × 2 = 1 + 0.951 195 119 511 951 195 119 511 951 195 119 486 238 72;
  • 17) 0.951 195 119 511 951 195 119 511 951 195 119 486 238 72 × 2 = 1 + 0.902 390 239 023 902 390 239 023 902 390 238 972 477 44;
  • 18) 0.902 390 239 023 902 390 239 023 902 390 238 972 477 44 × 2 = 1 + 0.804 780 478 047 804 780 478 047 804 780 477 944 954 88;
  • 19) 0.804 780 478 047 804 780 478 047 804 780 477 944 954 88 × 2 = 1 + 0.609 560 956 095 609 560 956 095 609 560 955 889 909 76;
  • 20) 0.609 560 956 095 609 560 956 095 609 560 955 889 909 76 × 2 = 1 + 0.219 121 912 191 219 121 912 191 219 121 911 779 819 52;
  • 21) 0.219 121 912 191 219 121 912 191 219 121 911 779 819 52 × 2 = 0 + 0.438 243 824 382 438 243 824 382 438 243 823 559 639 04;
  • 22) 0.438 243 824 382 438 243 824 382 438 243 823 559 639 04 × 2 = 0 + 0.876 487 648 764 876 487 648 764 876 487 647 119 278 08;
  • 23) 0.876 487 648 764 876 487 648 764 876 487 647 119 278 08 × 2 = 1 + 0.752 975 297 529 752 975 297 529 752 975 294 238 556 16;
  • 24) 0.752 975 297 529 752 975 297 529 752 975 294 238 556 16 × 2 = 1 + 0.505 950 595 059 505 950 595 059 505 950 588 477 112 32;
  • 25) 0.505 950 595 059 505 950 595 059 505 950 588 477 112 32 × 2 = 1 + 0.011 901 190 119 011 901 190 119 011 901 176 954 224 64;
  • 26) 0.011 901 190 119 011 901 190 119 011 901 176 954 224 64 × 2 = 0 + 0.023 802 380 238 023 802 380 238 023 802 353 908 449 28;
  • 27) 0.023 802 380 238 023 802 380 238 023 802 353 908 449 28 × 2 = 0 + 0.047 604 760 476 047 604 760 476 047 604 707 816 898 56;
  • 28) 0.047 604 760 476 047 604 760 476 047 604 707 816 898 56 × 2 = 0 + 0.095 209 520 952 095 209 520 952 095 209 415 633 797 12;
  • 29) 0.095 209 520 952 095 209 520 952 095 209 415 633 797 12 × 2 = 0 + 0.190 419 041 904 190 419 041 904 190 418 831 267 594 24;
  • 30) 0.190 419 041 904 190 419 041 904 190 418 831 267 594 24 × 2 = 0 + 0.380 838 083 808 380 838 083 808 380 837 662 535 188 48;
  • 31) 0.380 838 083 808 380 838 083 808 380 837 662 535 188 48 × 2 = 0 + 0.761 676 167 616 761 676 167 616 761 675 325 070 376 96;
  • 32) 0.761 676 167 616 761 676 167 616 761 675 325 070 376 96 × 2 = 1 + 0.523 352 335 233 523 352 335 233 523 350 650 140 753 92;
  • 33) 0.523 352 335 233 523 352 335 233 523 350 650 140 753 92 × 2 = 1 + 0.046 704 670 467 046 704 670 467 046 701 300 281 507 84;
  • 34) 0.046 704 670 467 046 704 670 467 046 701 300 281 507 84 × 2 = 0 + 0.093 409 340 934 093 409 340 934 093 402 600 563 015 68;
  • 35) 0.093 409 340 934 093 409 340 934 093 402 600 563 015 68 × 2 = 0 + 0.186 818 681 868 186 818 681 868 186 805 201 126 031 36;
  • 36) 0.186 818 681 868 186 818 681 868 186 805 201 126 031 36 × 2 = 0 + 0.373 637 363 736 373 637 363 736 373 610 402 252 062 72;
  • 37) 0.373 637 363 736 373 637 363 736 373 610 402 252 062 72 × 2 = 0 + 0.747 274 727 472 747 274 727 472 747 220 804 504 125 44;
  • 38) 0.747 274 727 472 747 274 727 472 747 220 804 504 125 44 × 2 = 1 + 0.494 549 454 945 494 549 454 945 494 441 609 008 250 88;
  • 39) 0.494 549 454 945 494 549 454 945 494 441 609 008 250 88 × 2 = 0 + 0.989 098 909 890 989 098 909 890 988 883 218 016 501 76;
  • 40) 0.989 098 909 890 989 098 909 890 988 883 218 016 501 76 × 2 = 1 + 0.978 197 819 781 978 197 819 781 977 766 436 033 003 52;
  • 41) 0.978 197 819 781 978 197 819 781 977 766 436 033 003 52 × 2 = 1 + 0.956 395 639 563 956 395 639 563 955 532 872 066 007 04;
  • 42) 0.956 395 639 563 956 395 639 563 955 532 872 066 007 04 × 2 = 1 + 0.912 791 279 127 912 791 279 127 911 065 744 132 014 08;
  • 43) 0.912 791 279 127 912 791 279 127 911 065 744 132 014 08 × 2 = 1 + 0.825 582 558 255 825 582 558 255 822 131 488 264 028 16;
  • 44) 0.825 582 558 255 825 582 558 255 822 131 488 264 028 16 × 2 = 1 + 0.651 165 116 511 651 165 116 511 644 262 976 528 056 32;
  • 45) 0.651 165 116 511 651 165 116 511 644 262 976 528 056 32 × 2 = 1 + 0.302 330 233 023 302 330 233 023 288 525 953 056 112 64;
  • 46) 0.302 330 233 023 302 330 233 023 288 525 953 056 112 64 × 2 = 0 + 0.604 660 466 046 604 660 466 046 577 051 906 112 225 28;
  • 47) 0.604 660 466 046 604 660 466 046 577 051 906 112 225 28 × 2 = 1 + 0.209 320 932 093 209 320 932 093 154 103 812 224 450 56;
  • 48) 0.209 320 932 093 209 320 932 093 154 103 812 224 450 56 × 2 = 0 + 0.418 641 864 186 418 641 864 186 308 207 624 448 901 12;
  • 49) 0.418 641 864 186 418 641 864 186 308 207 624 448 901 12 × 2 = 0 + 0.837 283 728 372 837 283 728 372 616 415 248 897 802 24;
  • 50) 0.837 283 728 372 837 283 728 372 616 415 248 897 802 24 × 2 = 1 + 0.674 567 456 745 674 567 456 745 232 830 497 795 604 48;
  • 51) 0.674 567 456 745 674 567 456 745 232 830 497 795 604 48 × 2 = 1 + 0.349 134 913 491 349 134 913 490 465 660 995 591 208 96;
  • 52) 0.349 134 913 491 349 134 913 490 465 660 995 591 208 96 × 2 = 0 + 0.698 269 826 982 698 269 826 980 931 321 991 182 417 92;
  • 53) 0.698 269 826 982 698 269 826 980 931 321 991 182 417 92 × 2 = 1 + 0.396 539 653 965 396 539 653 961 862 643 982 364 835 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).


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:


0.123 412 341 234 123 412 341 234 123 412 341 234 123 02(10) =

0.0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1(2)

5. Positive number before normalization:

1.123 412 341 234 123 412 341 234 123 412 341 234 123 02(10) =

1.0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.123 412 341 234 123 412 341 234 123 412 341 234 123 02(10) =

1.0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1(2) =

1.0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1(2) × 20


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): 0

Mantissa (not normalized):
1.0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1023(10) =

011 1111 1111(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. 0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110 1 =

0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110


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) =
011 1111 1111

Mantissa (52 bits) =
0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110


Decimal number 1.123 412 341 234 123 412 341 234 123 412 341 234 123 02 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0001 1111 1001 0111 1111 0011 1000 0001 1000 0101 1111 1010 0110

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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