Convert 1 562 940 256.720 71 to 64 Bit Double Precision IEEE 754 Binary Floating Point Standard, From a Number in Base 10 Decimal System

How to convert the decimal number 1 562 940 256.720 71(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 binary (base 2) the integer part: 1 562 940 256. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 1 562 940 256 ÷ 2 = 781 470 128 + 0;
  • 781 470 128 ÷ 2 = 390 735 064 + 0;
  • 390 735 064 ÷ 2 = 195 367 532 + 0;
  • 195 367 532 ÷ 2 = 97 683 766 + 0;
  • 97 683 766 ÷ 2 = 48 841 883 + 0;
  • 48 841 883 ÷ 2 = 24 420 941 + 1;
  • 24 420 941 ÷ 2 = 12 210 470 + 1;
  • 12 210 470 ÷ 2 = 6 105 235 + 0;
  • 6 105 235 ÷ 2 = 3 052 617 + 1;
  • 3 052 617 ÷ 2 = 1 526 308 + 1;
  • 1 526 308 ÷ 2 = 763 154 + 0;
  • 763 154 ÷ 2 = 381 577 + 0;
  • 381 577 ÷ 2 = 190 788 + 1;
  • 190 788 ÷ 2 = 95 394 + 0;
  • 95 394 ÷ 2 = 47 697 + 0;
  • 47 697 ÷ 2 = 23 848 + 1;
  • 23 848 ÷ 2 = 11 924 + 0;
  • 11 924 ÷ 2 = 5 962 + 0;
  • 5 962 ÷ 2 = 2 981 + 0;
  • 2 981 ÷ 2 = 1 490 + 1;
  • 1 490 ÷ 2 = 745 + 0;
  • 745 ÷ 2 = 372 + 1;
  • 372 ÷ 2 = 186 + 0;
  • 186 ÷ 2 = 93 + 0;
  • 93 ÷ 2 = 46 + 1;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the integer part of the number, by taking all the remainders starting from the bottom of the list constructed above:

1 562 940 256(10) =


101 1101 0010 1000 1001 0011 0110 0000(2)

3. Convert to binary (base 2) the fractional part: 0.720 71. Multiply it 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) 0.720 71 × 2 = 1 + 0.441 42;
  • 2) 0.441 42 × 2 = 0 + 0.882 84;
  • 3) 0.882 84 × 2 = 1 + 0.765 68;
  • 4) 0.765 68 × 2 = 1 + 0.531 36;
  • 5) 0.531 36 × 2 = 1 + 0.062 72;
  • 6) 0.062 72 × 2 = 0 + 0.125 44;
  • 7) 0.125 44 × 2 = 0 + 0.250 88;
  • 8) 0.250 88 × 2 = 0 + 0.501 76;
  • 9) 0.501 76 × 2 = 1 + 0.003 52;
  • 10) 0.003 52 × 2 = 0 + 0.007 04;
  • 11) 0.007 04 × 2 = 0 + 0.014 08;
  • 12) 0.014 08 × 2 = 0 + 0.028 16;
  • 13) 0.028 16 × 2 = 0 + 0.056 32;
  • 14) 0.056 32 × 2 = 0 + 0.112 64;
  • 15) 0.112 64 × 2 = 0 + 0.225 28;
  • 16) 0.225 28 × 2 = 0 + 0.450 56;
  • 17) 0.450 56 × 2 = 0 + 0.901 12;
  • 18) 0.901 12 × 2 = 1 + 0.802 24;
  • 19) 0.802 24 × 2 = 1 + 0.604 48;
  • 20) 0.604 48 × 2 = 1 + 0.208 96;
  • 21) 0.208 96 × 2 = 0 + 0.417 92;
  • 22) 0.417 92 × 2 = 0 + 0.835 84;
  • 23) 0.835 84 × 2 = 1 + 0.671 68;
  • 24) 0.671 68 × 2 = 1 + 0.343 36;
  • 25) 0.343 36 × 2 = 0 + 0.686 72;
  • 26) 0.686 72 × 2 = 1 + 0.373 44;
  • 27) 0.373 44 × 2 = 0 + 0.746 88;
  • 28) 0.746 88 × 2 = 1 + 0.493 76;
  • 29) 0.493 76 × 2 = 0 + 0.987 52;
  • 30) 0.987 52 × 2 = 1 + 0.975 04;
  • 31) 0.975 04 × 2 = 1 + 0.950 08;
  • 32) 0.950 08 × 2 = 1 + 0.900 16;
  • 33) 0.900 16 × 2 = 1 + 0.800 32;
  • 34) 0.800 32 × 2 = 1 + 0.600 64;
  • 35) 0.600 64 × 2 = 1 + 0.201 28;
  • 36) 0.201 28 × 2 = 0 + 0.402 56;
  • 37) 0.402 56 × 2 = 0 + 0.805 12;
  • 38) 0.805 12 × 2 = 1 + 0.610 24;
  • 39) 0.610 24 × 2 = 1 + 0.220 48;
  • 40) 0.220 48 × 2 = 0 + 0.440 96;
  • 41) 0.440 96 × 2 = 0 + 0.881 92;
  • 42) 0.881 92 × 2 = 1 + 0.763 84;
  • 43) 0.763 84 × 2 = 1 + 0.527 68;
  • 44) 0.527 68 × 2 = 1 + 0.055 36;
  • 45) 0.055 36 × 2 = 0 + 0.110 72;
  • 46) 0.110 72 × 2 = 0 + 0.221 44;
  • 47) 0.221 44 × 2 = 0 + 0.442 88;
  • 48) 0.442 88 × 2 = 0 + 0.885 76;
  • 49) 0.885 76 × 2 = 1 + 0.771 52;
  • 50) 0.771 52 × 2 = 1 + 0.543 04;
  • 51) 0.543 04 × 2 = 1 + 0.086 08;
  • 52) 0.086 08 × 2 = 0 + 0.172 16;
  • 53) 0.172 16 × 2 = 0 + 0.344 32;

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, by taking all the integer parts of the multiplying operations, starting from the top of the constructed list above:

0.720 71(10) =


0.1011 1000 1000 0000 0111 0011 0101 0111 1110 0110 0111 0000 1110 0(2)

Positive number before normalization:

1 562 940 256.720 71(10) =


101 1101 0010 1000 1001 0011 0110 0000.1011 1000 1000 0000 0111 0011 0101 0111 1110 0110 0111 0000 1110 0(2)

5. Normalize the binary representation of the number, shifting the decimal mark 30 positions to the left so that only one non zero digit remains to the left of it:

1 562 940 256.720 71(10) =


101 1101 0010 1000 1001 0011 0110 0000.1011 1000 1000 0000 0111 0011 0101 0111 1110 0110 0111 0000 1110 0(2) =


101 1101 0010 1000 1001 0011 0110 0000.1011 1000 1000 0000 0111 0011 0101 0111 1110 0110 0111 0000 1110 0(2) × 20 =


1.0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100 1101 0101 1111 1001 1001 1100 0011 100(2) × 230

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): 30


Mantissa (not normalized): 1.0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100 1101 0101 1111 1001 1001 1100 0011 100

6. 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:

Exponent (adjusted) =


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


30 + 2(11-1) - 1 =


(30 + 1 023)(10) =


1 053(10)


  • division = quotient + remainder;
  • 1 053 ÷ 2 = 526 + 1;
  • 526 ÷ 2 = 263 + 0;
  • 263 ÷ 2 = 131 + 1;
  • 131 ÷ 2 = 65 + 1;
  • 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;

Exponent (adjusted) =


1053(10) =


100 0001 1101(2)

7. Normalize mantissa, remove the leading (the leftmost) bit, since it's allways 1 (and the decimal point, if the case) then 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. 0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100 110 1010 1111 1100 1100 1110 0001 1100 =


0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100

Conclusion:

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 1101


Mantissa (52 bits) =
0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100

Number 1 562 940 256.720 71 converted from decimal system (base 10)
to
64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1101 - 0111 0100 1010 0010 0100 1101 1000 0010 1110 0010 0000 0001 1100

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 0

      59
    • 0

      58
    • 0

      57
    • 1

      56
    • 1

      55
    • 1

      54
    • 0

      53
    • 1

      52
  • Mantissa (52 bits):

    • 0

      51
    • 1

      50
    • 1

      49
    • 1

      48
    • 0

      47
    • 1

      46
    • 0

      45
    • 0

      44
    • 1

      43
    • 0

      42
    • 1

      41
    • 0

      40
    • 0

      39
    • 0

      38
    • 1

      37
    • 0

      36
    • 0

      35
    • 1

      34
    • 0

      33
    • 0

      32
    • 1

      31
    • 1

      30
    • 0

      29
    • 1

      28
    • 1

      27
    • 0

      26
    • 0

      25
    • 0

      24
    • 0

      23
    • 0

      22
    • 1

      21
    • 0

      20
    • 1

      19
    • 1

      18
    • 1

      17
    • 0

      16
    • 0

      15
    • 0

      14
    • 1

      13
    • 0

      12
    • 0

      11
    • 0

      10
    • 0

      9
    • 0

      8
    • 0

      7
    • 0

      6
    • 0

      5
    • 1

      4
    • 1

      3
    • 1

      2
    • 0

      1
    • 0

      0

1 562 940 256.720 7 = ? ... 1 562 940 256.720 72 = ?


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)

Latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

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