Convert 40.521 988 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 40.521 988(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: 40.
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;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 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.

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

40(10) =


10 1000(2)


3. Convert to the binary (base 2) the fractional part: 0.521 988.

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) 0.521 988 × 2 = 1 + 0.043 976;
  • 2) 0.043 976 × 2 = 0 + 0.087 952;
  • 3) 0.087 952 × 2 = 0 + 0.175 904;
  • 4) 0.175 904 × 2 = 0 + 0.351 808;
  • 5) 0.351 808 × 2 = 0 + 0.703 616;
  • 6) 0.703 616 × 2 = 1 + 0.407 232;
  • 7) 0.407 232 × 2 = 0 + 0.814 464;
  • 8) 0.814 464 × 2 = 1 + 0.628 928;
  • 9) 0.628 928 × 2 = 1 + 0.257 856;
  • 10) 0.257 856 × 2 = 0 + 0.515 712;
  • 11) 0.515 712 × 2 = 1 + 0.031 424;
  • 12) 0.031 424 × 2 = 0 + 0.062 848;
  • 13) 0.062 848 × 2 = 0 + 0.125 696;
  • 14) 0.125 696 × 2 = 0 + 0.251 392;
  • 15) 0.251 392 × 2 = 0 + 0.502 784;
  • 16) 0.502 784 × 2 = 1 + 0.005 568;
  • 17) 0.005 568 × 2 = 0 + 0.011 136;
  • 18) 0.011 136 × 2 = 0 + 0.022 272;
  • 19) 0.022 272 × 2 = 0 + 0.044 544;
  • 20) 0.044 544 × 2 = 0 + 0.089 088;
  • 21) 0.089 088 × 2 = 0 + 0.178 176;
  • 22) 0.178 176 × 2 = 0 + 0.356 352;
  • 23) 0.356 352 × 2 = 0 + 0.712 704;
  • 24) 0.712 704 × 2 = 1 + 0.425 408;
  • 25) 0.425 408 × 2 = 0 + 0.850 816;
  • 26) 0.850 816 × 2 = 1 + 0.701 632;
  • 27) 0.701 632 × 2 = 1 + 0.403 264;
  • 28) 0.403 264 × 2 = 0 + 0.806 528;
  • 29) 0.806 528 × 2 = 1 + 0.613 056;
  • 30) 0.613 056 × 2 = 1 + 0.226 112;
  • 31) 0.226 112 × 2 = 0 + 0.452 224;
  • 32) 0.452 224 × 2 = 0 + 0.904 448;
  • 33) 0.904 448 × 2 = 1 + 0.808 896;
  • 34) 0.808 896 × 2 = 1 + 0.617 792;
  • 35) 0.617 792 × 2 = 1 + 0.235 584;
  • 36) 0.235 584 × 2 = 0 + 0.471 168;
  • 37) 0.471 168 × 2 = 0 + 0.942 336;
  • 38) 0.942 336 × 2 = 1 + 0.884 672;
  • 39) 0.884 672 × 2 = 1 + 0.769 344;
  • 40) 0.769 344 × 2 = 1 + 0.538 688;
  • 41) 0.538 688 × 2 = 1 + 0.077 376;
  • 42) 0.077 376 × 2 = 0 + 0.154 752;
  • 43) 0.154 752 × 2 = 0 + 0.309 504;
  • 44) 0.309 504 × 2 = 0 + 0.619 008;
  • 45) 0.619 008 × 2 = 1 + 0.238 016;
  • 46) 0.238 016 × 2 = 0 + 0.476 032;
  • 47) 0.476 032 × 2 = 0 + 0.952 064;
  • 48) 0.952 064 × 2 = 1 + 0.904 128;
  • 49) 0.904 128 × 2 = 1 + 0.808 256;
  • 50) 0.808 256 × 2 = 1 + 0.616 512;
  • 51) 0.616 512 × 2 = 1 + 0.233 024;
  • 52) 0.233 024 × 2 = 0 + 0.466 048;
  • 53) 0.466 048 × 2 = 0 + 0.932 096;

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:

0.521 988(10) =


0.1000 0101 1010 0001 0000 0001 0110 1100 1110 0111 1000 1001 1110 0(2)


5. Positive number before normalization:

40.521 988(10) =


10 1000.1000 0101 1010 0001 0000 0001 0110 1100 1110 0111 1000 1001 1110 0(2)


6. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the left so that only one non zero digit remains to the left of it:

40.521 988(10) =


10 1000.1000 0101 1010 0001 0000 0001 0110 1100 1110 0111 1000 1001 1110 0(2) =


10 1000.1000 0101 1010 0001 0000 0001 0110 1100 1110 0111 1000 1001 1110 0(2) × 20 =


1.0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100 1111 00(2) × 25


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): 5


Mantissa (not normalized):
1.0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100 1111 00


8. Adjust the exponent.

Use the 11 bit excess/bias notation:

Exponent (adjusted) =


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


5 + 2(11-1) - 1 =


(5 + 1 023)(10) =


1 028(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;
  • 1 028 ÷ 2 = 514 + 0;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 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) =


1028(10) =


100 0000 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. 0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100 11 1100 =


0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100


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 0000 0100


Mantissa (52 bits) =
0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100


Conclusion:

Number 40.521 988 converted from decimal system (base 10)
to
64 bit double precision IEEE 754 binary floating point:
0 - 100 0000 0100 - 0100 0100 0010 1101 0000 1000 0000 1011 0110 0111 0011 1100 0100

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 0

      59
    • 0

      58
    • 0

      57
    • 0

      56
    • 0

      55
    • 1

      54
    • 0

      53
    • 0

      52
  • Mantissa (52 bits):

    • 0

      51
    • 1

      50
    • 0

      49
    • 0

      48
    • 0

      47
    • 1

      46
    • 0

      45
    • 0

      44
    • 0

      43
    • 0

      42
    • 1

      41
    • 0

      40
    • 1

      39
    • 1

      38
    • 0

      37
    • 1

      36
    • 0

      35
    • 0

      34
    • 0

      33
    • 0

      32
    • 1

      31
    • 0

      30
    • 0

      29
    • 0

      28
    • 0

      27
    • 0

      26
    • 0

      25
    • 0

      24
    • 1

      23
    • 0

      22
    • 1

      21
    • 1

      20
    • 0

      19
    • 1

      18
    • 1

      17
    • 0

      16
    • 0

      15
    • 1

      14
    • 1

      13
    • 1

      12
    • 0

      11
    • 0

      10
    • 1

      9
    • 1

      8
    • 1

      7
    • 1

      6
    • 0

      5
    • 0

      4
    • 0

      3
    • 1

      2
    • 0

      1
    • 0

      0

More operations of this kind:

40.521 987 = ? ... 40.521 989 = ?


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