64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 2 666 134 278 319 192 228 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 2 666 134 278 319 192 228(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

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;
  • 2 666 134 278 319 192 228 ÷ 2 = 1 333 067 139 159 596 114 + 0;
  • 1 333 067 139 159 596 114 ÷ 2 = 666 533 569 579 798 057 + 0;
  • 666 533 569 579 798 057 ÷ 2 = 333 266 784 789 899 028 + 1;
  • 333 266 784 789 899 028 ÷ 2 = 166 633 392 394 949 514 + 0;
  • 166 633 392 394 949 514 ÷ 2 = 83 316 696 197 474 757 + 0;
  • 83 316 696 197 474 757 ÷ 2 = 41 658 348 098 737 378 + 1;
  • 41 658 348 098 737 378 ÷ 2 = 20 829 174 049 368 689 + 0;
  • 20 829 174 049 368 689 ÷ 2 = 10 414 587 024 684 344 + 1;
  • 10 414 587 024 684 344 ÷ 2 = 5 207 293 512 342 172 + 0;
  • 5 207 293 512 342 172 ÷ 2 = 2 603 646 756 171 086 + 0;
  • 2 603 646 756 171 086 ÷ 2 = 1 301 823 378 085 543 + 0;
  • 1 301 823 378 085 543 ÷ 2 = 650 911 689 042 771 + 1;
  • 650 911 689 042 771 ÷ 2 = 325 455 844 521 385 + 1;
  • 325 455 844 521 385 ÷ 2 = 162 727 922 260 692 + 1;
  • 162 727 922 260 692 ÷ 2 = 81 363 961 130 346 + 0;
  • 81 363 961 130 346 ÷ 2 = 40 681 980 565 173 + 0;
  • 40 681 980 565 173 ÷ 2 = 20 340 990 282 586 + 1;
  • 20 340 990 282 586 ÷ 2 = 10 170 495 141 293 + 0;
  • 10 170 495 141 293 ÷ 2 = 5 085 247 570 646 + 1;
  • 5 085 247 570 646 ÷ 2 = 2 542 623 785 323 + 0;
  • 2 542 623 785 323 ÷ 2 = 1 271 311 892 661 + 1;
  • 1 271 311 892 661 ÷ 2 = 635 655 946 330 + 1;
  • 635 655 946 330 ÷ 2 = 317 827 973 165 + 0;
  • 317 827 973 165 ÷ 2 = 158 913 986 582 + 1;
  • 158 913 986 582 ÷ 2 = 79 456 993 291 + 0;
  • 79 456 993 291 ÷ 2 = 39 728 496 645 + 1;
  • 39 728 496 645 ÷ 2 = 19 864 248 322 + 1;
  • 19 864 248 322 ÷ 2 = 9 932 124 161 + 0;
  • 9 932 124 161 ÷ 2 = 4 966 062 080 + 1;
  • 4 966 062 080 ÷ 2 = 2 483 031 040 + 0;
  • 2 483 031 040 ÷ 2 = 1 241 515 520 + 0;
  • 1 241 515 520 ÷ 2 = 620 757 760 + 0;
  • 620 757 760 ÷ 2 = 310 378 880 + 0;
  • 310 378 880 ÷ 2 = 155 189 440 + 0;
  • 155 189 440 ÷ 2 = 77 594 720 + 0;
  • 77 594 720 ÷ 2 = 38 797 360 + 0;
  • 38 797 360 ÷ 2 = 19 398 680 + 0;
  • 19 398 680 ÷ 2 = 9 699 340 + 0;
  • 9 699 340 ÷ 2 = 4 849 670 + 0;
  • 4 849 670 ÷ 2 = 2 424 835 + 0;
  • 2 424 835 ÷ 2 = 1 212 417 + 1;
  • 1 212 417 ÷ 2 = 606 208 + 1;
  • 606 208 ÷ 2 = 303 104 + 0;
  • 303 104 ÷ 2 = 151 552 + 0;
  • 151 552 ÷ 2 = 75 776 + 0;
  • 75 776 ÷ 2 = 37 888 + 0;
  • 37 888 ÷ 2 = 18 944 + 0;
  • 18 944 ÷ 2 = 9 472 + 0;
  • 9 472 ÷ 2 = 4 736 + 0;
  • 4 736 ÷ 2 = 2 368 + 0;
  • 2 368 ÷ 2 = 1 184 + 0;
  • 1 184 ÷ 2 = 592 + 0;
  • 592 ÷ 2 = 296 + 0;
  • 296 ÷ 2 = 148 + 0;
  • 148 ÷ 2 = 74 + 0;
  • 74 ÷ 2 = 37 + 0;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


2 666 134 278 319 192 228(10) =


10 0101 0000 0000 0000 0011 0000 0000 0001 0110 1011 0101 0011 1000 1010 0100(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 61 positions to the left, so that only one non zero digit remains to the left of it:


2 666 134 278 319 192 228(10) =


10 0101 0000 0000 0000 0011 0000 0000 0001 0110 1011 0101 0011 1000 1010 0100(2) =


10 0101 0000 0000 0000 0011 0000 0000 0001 0110 1011 0101 0011 1000 1010 0100(2) × 20 =


1.0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100 0101 0010 0(2) × 261


4. 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): 61


Mantissa (not normalized):
1.0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100 0101 0010 0


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


61 + 2(11-1) - 1 =


(61 + 1 023)(10) =


1 084(10)


6. 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;
  • 1 084 ÷ 2 = 542 + 0;
  • 542 ÷ 2 = 271 + 0;
  • 271 ÷ 2 = 135 + 1;
  • 135 ÷ 2 = 67 + 1;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1084(10) =


100 0011 1100(2)


8. 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. 0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100 0 1010 0100 =


0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100


9. 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 0011 1100


Mantissa (52 bits) =
0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100


The base ten decimal number 2 666 134 278 319 192 228 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0011 1100 - 0010 1000 0000 0000 0001 1000 0000 0000 1011 0101 1010 1001 1100

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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