1.123 412 341 234 123 412 341 328 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.123 412 341 234 123 412 341 328(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
1.123 412 341 234 123 412 341 328(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: 0.123 412 341 234 123 412 341 328.

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) 0.123 412 341 234 123 412 341 328 × 2 = 0 + 0.246 824 682 468 246 824 682 656;
  • 2) 0.246 824 682 468 246 824 682 656 × 2 = 0 + 0.493 649 364 936 493 649 365 312;
  • 3) 0.493 649 364 936 493 649 365 312 × 2 = 0 + 0.987 298 729 872 987 298 730 624;
  • 4) 0.987 298 729 872 987 298 730 624 × 2 = 1 + 0.974 597 459 745 974 597 461 248;
  • 5) 0.974 597 459 745 974 597 461 248 × 2 = 1 + 0.949 194 919 491 949 194 922 496;
  • 6) 0.949 194 919 491 949 194 922 496 × 2 = 1 + 0.898 389 838 983 898 389 844 992;
  • 7) 0.898 389 838 983 898 389 844 992 × 2 = 1 + 0.796 779 677 967 796 779 689 984;
  • 8) 0.796 779 677 967 796 779 689 984 × 2 = 1 + 0.593 559 355 935 593 559 379 968;
  • 9) 0.593 559 355 935 593 559 379 968 × 2 = 1 + 0.187 118 711 871 187 118 759 936;
  • 10) 0.187 118 711 871 187 118 759 936 × 2 = 0 + 0.374 237 423 742 374 237 519 872;
  • 11) 0.374 237 423 742 374 237 519 872 × 2 = 0 + 0.748 474 847 484 748 475 039 744;
  • 12) 0.748 474 847 484 748 475 039 744 × 2 = 1 + 0.496 949 694 969 496 950 079 488;
  • 13) 0.496 949 694 969 496 950 079 488 × 2 = 0 + 0.993 899 389 938 993 900 158 976;
  • 14) 0.993 899 389 938 993 900 158 976 × 2 = 1 + 0.987 798 779 877 987 800 317 952;
  • 15) 0.987 798 779 877 987 800 317 952 × 2 = 1 + 0.975 597 559 755 975 600 635 904;
  • 16) 0.975 597 559 755 975 600 635 904 × 2 = 1 + 0.951 195 119 511 951 201 271 808;
  • 17) 0.951 195 119 511 951 201 271 808 × 2 = 1 + 0.902 390 239 023 902 402 543 616;
  • 18) 0.902 390 239 023 902 402 543 616 × 2 = 1 + 0.804 780 478 047 804 805 087 232;
  • 19) 0.804 780 478 047 804 805 087 232 × 2 = 1 + 0.609 560 956 095 609 610 174 464;
  • 20) 0.609 560 956 095 609 610 174 464 × 2 = 1 + 0.219 121 912 191 219 220 348 928;
  • 21) 0.219 121 912 191 219 220 348 928 × 2 = 0 + 0.438 243 824 382 438 440 697 856;
  • 22) 0.438 243 824 382 438 440 697 856 × 2 = 0 + 0.876 487 648 764 876 881 395 712;
  • 23) 0.876 487 648 764 876 881 395 712 × 2 = 1 + 0.752 975 297 529 753 762 791 424;
  • 24) 0.752 975 297 529 753 762 791 424 × 2 = 1 + 0.505 950 595 059 507 525 582 848;
  • 25) 0.505 950 595 059 507 525 582 848 × 2 = 1 + 0.011 901 190 119 015 051 165 696;
  • 26) 0.011 901 190 119 015 051 165 696 × 2 = 0 + 0.023 802 380 238 030 102 331 392;
  • 27) 0.023 802 380 238 030 102 331 392 × 2 = 0 + 0.047 604 760 476 060 204 662 784;
  • 28) 0.047 604 760 476 060 204 662 784 × 2 = 0 + 0.095 209 520 952 120 409 325 568;
  • 29) 0.095 209 520 952 120 409 325 568 × 2 = 0 + 0.190 419 041 904 240 818 651 136;
  • 30) 0.190 419 041 904 240 818 651 136 × 2 = 0 + 0.380 838 083 808 481 637 302 272;
  • 31) 0.380 838 083 808 481 637 302 272 × 2 = 0 + 0.761 676 167 616 963 274 604 544;
  • 32) 0.761 676 167 616 963 274 604 544 × 2 = 1 + 0.523 352 335 233 926 549 209 088;
  • 33) 0.523 352 335 233 926 549 209 088 × 2 = 1 + 0.046 704 670 467 853 098 418 176;
  • 34) 0.046 704 670 467 853 098 418 176 × 2 = 0 + 0.093 409 340 935 706 196 836 352;
  • 35) 0.093 409 340 935 706 196 836 352 × 2 = 0 + 0.186 818 681 871 412 393 672 704;
  • 36) 0.186 818 681 871 412 393 672 704 × 2 = 0 + 0.373 637 363 742 824 787 345 408;
  • 37) 0.373 637 363 742 824 787 345 408 × 2 = 0 + 0.747 274 727 485 649 574 690 816;
  • 38) 0.747 274 727 485 649 574 690 816 × 2 = 1 + 0.494 549 454 971 299 149 381 632;
  • 39) 0.494 549 454 971 299 149 381 632 × 2 = 0 + 0.989 098 909 942 598 298 763 264;
  • 40) 0.989 098 909 942 598 298 763 264 × 2 = 1 + 0.978 197 819 885 196 597 526 528;
  • 41) 0.978 197 819 885 196 597 526 528 × 2 = 1 + 0.956 395 639 770 393 195 053 056;
  • 42) 0.956 395 639 770 393 195 053 056 × 2 = 1 + 0.912 791 279 540 786 390 106 112;
  • 43) 0.912 791 279 540 786 390 106 112 × 2 = 1 + 0.825 582 559 081 572 780 212 224;
  • 44) 0.825 582 559 081 572 780 212 224 × 2 = 1 + 0.651 165 118 163 145 560 424 448;
  • 45) 0.651 165 118 163 145 560 424 448 × 2 = 1 + 0.302 330 236 326 291 120 848 896;
  • 46) 0.302 330 236 326 291 120 848 896 × 2 = 0 + 0.604 660 472 652 582 241 697 792;
  • 47) 0.604 660 472 652 582 241 697 792 × 2 = 1 + 0.209 320 945 305 164 483 395 584;
  • 48) 0.209 320 945 305 164 483 395 584 × 2 = 0 + 0.418 641 890 610 328 966 791 168;
  • 49) 0.418 641 890 610 328 966 791 168 × 2 = 0 + 0.837 283 781 220 657 933 582 336;
  • 50) 0.837 283 781 220 657 933 582 336 × 2 = 1 + 0.674 567 562 441 315 867 164 672;
  • 51) 0.674 567 562 441 315 867 164 672 × 2 = 1 + 0.349 135 124 882 631 734 329 344;
  • 52) 0.349 135 124 882 631 734 329 344 × 2 = 0 + 0.698 270 249 765 263 468 658 688;
  • 53) 0.698 270 249 765 263 468 658 688 × 2 = 1 + 0.396 540 499 530 526 937 317 376;

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:


0.123 412 341 234 123 412 341 328(10) =

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

5. Positive number before normalization:

1.123 412 341 234 123 412 341 328(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:


1.123 412 341 234 123 412 341 328(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 1.123 412 341 234 123 412 341 328 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