Convert 3 677 080.252 847 to 64 Bit Double Precision IEEE 754 Binary Floating Point Standard, From a Number in Base 10 Decimal System

3 677 080.252 847(10) to 64 bit double precision IEEE 754 binary floating point (1 bit for sign, 11 bits for exponent, 52 bits for mantissa) = ?

1. First, convert to the binary (base 2) the integer part: 3 677 080.
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;
  • 3 677 080 ÷ 2 = 1 838 540 + 0;
  • 1 838 540 ÷ 2 = 919 270 + 0;
  • 919 270 ÷ 2 = 459 635 + 0;
  • 459 635 ÷ 2 = 229 817 + 1;
  • 229 817 ÷ 2 = 114 908 + 1;
  • 114 908 ÷ 2 = 57 454 + 0;
  • 57 454 ÷ 2 = 28 727 + 0;
  • 28 727 ÷ 2 = 14 363 + 1;
  • 14 363 ÷ 2 = 7 181 + 1;
  • 7 181 ÷ 2 = 3 590 + 1;
  • 3 590 ÷ 2 = 1 795 + 0;
  • 1 795 ÷ 2 = 897 + 1;
  • 897 ÷ 2 = 448 + 1;
  • 448 ÷ 2 = 224 + 0;
  • 224 ÷ 2 = 112 + 0;
  • 112 ÷ 2 = 56 + 0;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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.


3 677 080(10) =


11 1000 0001 1011 1001 1000(2)


3. Convert to the binary (base 2) the fractional part: 0.252 847.

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) 0.252 847 × 2 = 0 + 0.505 694;
  • 2) 0.505 694 × 2 = 1 + 0.011 388;
  • 3) 0.011 388 × 2 = 0 + 0.022 776;
  • 4) 0.022 776 × 2 = 0 + 0.045 552;
  • 5) 0.045 552 × 2 = 0 + 0.091 104;
  • 6) 0.091 104 × 2 = 0 + 0.182 208;
  • 7) 0.182 208 × 2 = 0 + 0.364 416;
  • 8) 0.364 416 × 2 = 0 + 0.728 832;
  • 9) 0.728 832 × 2 = 1 + 0.457 664;
  • 10) 0.457 664 × 2 = 0 + 0.915 328;
  • 11) 0.915 328 × 2 = 1 + 0.830 656;
  • 12) 0.830 656 × 2 = 1 + 0.661 312;
  • 13) 0.661 312 × 2 = 1 + 0.322 624;
  • 14) 0.322 624 × 2 = 0 + 0.645 248;
  • 15) 0.645 248 × 2 = 1 + 0.290 496;
  • 16) 0.290 496 × 2 = 0 + 0.580 992;
  • 17) 0.580 992 × 2 = 1 + 0.161 984;
  • 18) 0.161 984 × 2 = 0 + 0.323 968;
  • 19) 0.323 968 × 2 = 0 + 0.647 936;
  • 20) 0.647 936 × 2 = 1 + 0.295 872;
  • 21) 0.295 872 × 2 = 0 + 0.591 744;
  • 22) 0.591 744 × 2 = 1 + 0.183 488;
  • 23) 0.183 488 × 2 = 0 + 0.366 976;
  • 24) 0.366 976 × 2 = 0 + 0.733 952;
  • 25) 0.733 952 × 2 = 1 + 0.467 904;
  • 26) 0.467 904 × 2 = 0 + 0.935 808;
  • 27) 0.935 808 × 2 = 1 + 0.871 616;
  • 28) 0.871 616 × 2 = 1 + 0.743 232;
  • 29) 0.743 232 × 2 = 1 + 0.486 464;
  • 30) 0.486 464 × 2 = 0 + 0.972 928;
  • 31) 0.972 928 × 2 = 1 + 0.945 856;
  • 32) 0.945 856 × 2 = 1 + 0.891 712;
  • 33) 0.891 712 × 2 = 1 + 0.783 424;
  • 34) 0.783 424 × 2 = 1 + 0.566 848;
  • 35) 0.566 848 × 2 = 1 + 0.133 696;
  • 36) 0.133 696 × 2 = 0 + 0.267 392;
  • 37) 0.267 392 × 2 = 0 + 0.534 784;
  • 38) 0.534 784 × 2 = 1 + 0.069 568;
  • 39) 0.069 568 × 2 = 0 + 0.139 136;
  • 40) 0.139 136 × 2 = 0 + 0.278 272;
  • 41) 0.278 272 × 2 = 0 + 0.556 544;
  • 42) 0.556 544 × 2 = 1 + 0.113 088;
  • 43) 0.113 088 × 2 = 0 + 0.226 176;
  • 44) 0.226 176 × 2 = 0 + 0.452 352;
  • 45) 0.452 352 × 2 = 0 + 0.904 704;
  • 46) 0.904 704 × 2 = 1 + 0.809 408;
  • 47) 0.809 408 × 2 = 1 + 0.618 816;
  • 48) 0.618 816 × 2 = 1 + 0.237 632;
  • 49) 0.237 632 × 2 = 0 + 0.475 264;
  • 50) 0.475 264 × 2 = 0 + 0.950 528;
  • 51) 0.950 528 × 2 = 1 + 0.901 056;
  • 52) 0.901 056 × 2 = 1 + 0.802 112;
  • 53) 0.802 112 × 2 = 1 + 0.604 224;

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...)


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:


0.252 847(10) =


0.0100 0000 1011 1010 1001 0100 1011 1011 1110 0100 0100 0111 0011 1(2)


5. Positive number before normalization:

3 677 080.252 847(10) =


11 1000 0001 1011 1001 1000.0100 0000 1011 1010 1001 0100 1011 1011 1110 0100 0100 0111 0011 1(2)


6. Normalize the binary representation of the number.

Shift the decimal mark 21 positions to the left so that only one non zero digit remains to the left of it:


3 677 080.252 847(10) =


11 1000 0001 1011 1001 1000.0100 0000 1011 1010 1001 0100 1011 1011 1110 0100 0100 0111 0011 1(2) =


11 1000 0001 1011 1001 1000.0100 0000 1011 1010 1001 0100 1011 1011 1110 0100 0100 0111 0011 1(2) × 20 =


1.1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101 1111 0010 0010 0011 1001 11(2) × 221


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): 21


Mantissa (not normalized):
1.1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101 1111 0010 0010 0011 1001 11


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


21 + 2(11-1) - 1 =


(21 + 1 023)(10) =


1 044(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;
  • 1 044 ÷ 2 = 522 + 0;
  • 522 ÷ 2 = 261 + 0;
  • 261 ÷ 2 = 130 + 1;
  • 130 ÷ 2 = 65 + 0;
  • 65 ÷ 2 = 32 + 1;
  • 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) =


1044(10) =


100 0001 0100(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. 1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101 11 1100 1000 1000 1110 0111 =


1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101


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 0001 0100


Mantissa (52 bits) =
1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101


Number 3 677 080.252 847 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 0100 - 1100 0000 1101 1100 1100 0010 0000 0101 1101 0100 1010 0101 1101

(64 bits IEEE 754)

More operations of this kind:

3 677 080.252 846 = ? ... 3 677 080.252 848 = ?


Convert to 64 bit double precision IEEE 754 binary floating point standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes one bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

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All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

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