0.000 000 38 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 38(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
0.000 000 38(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
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;
  • 0 ÷ 2 = 0 + 0;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 38.

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) 0.000 000 38 × 2 = 0 + 0.000 000 76;
  • 2) 0.000 000 76 × 2 = 0 + 0.000 001 52;
  • 3) 0.000 001 52 × 2 = 0 + 0.000 003 04;
  • 4) 0.000 003 04 × 2 = 0 + 0.000 006 08;
  • 5) 0.000 006 08 × 2 = 0 + 0.000 012 16;
  • 6) 0.000 012 16 × 2 = 0 + 0.000 024 32;
  • 7) 0.000 024 32 × 2 = 0 + 0.000 048 64;
  • 8) 0.000 048 64 × 2 = 0 + 0.000 097 28;
  • 9) 0.000 097 28 × 2 = 0 + 0.000 194 56;
  • 10) 0.000 194 56 × 2 = 0 + 0.000 389 12;
  • 11) 0.000 389 12 × 2 = 0 + 0.000 778 24;
  • 12) 0.000 778 24 × 2 = 0 + 0.001 556 48;
  • 13) 0.001 556 48 × 2 = 0 + 0.003 112 96;
  • 14) 0.003 112 96 × 2 = 0 + 0.006 225 92;
  • 15) 0.006 225 92 × 2 = 0 + 0.012 451 84;
  • 16) 0.012 451 84 × 2 = 0 + 0.024 903 68;
  • 17) 0.024 903 68 × 2 = 0 + 0.049 807 36;
  • 18) 0.049 807 36 × 2 = 0 + 0.099 614 72;
  • 19) 0.099 614 72 × 2 = 0 + 0.199 229 44;
  • 20) 0.199 229 44 × 2 = 0 + 0.398 458 88;
  • 21) 0.398 458 88 × 2 = 0 + 0.796 917 76;
  • 22) 0.796 917 76 × 2 = 1 + 0.593 835 52;
  • 23) 0.593 835 52 × 2 = 1 + 0.187 671 04;
  • 24) 0.187 671 04 × 2 = 0 + 0.375 342 08;
  • 25) 0.375 342 08 × 2 = 0 + 0.750 684 16;
  • 26) 0.750 684 16 × 2 = 1 + 0.501 368 32;
  • 27) 0.501 368 32 × 2 = 1 + 0.002 736 64;
  • 28) 0.002 736 64 × 2 = 0 + 0.005 473 28;
  • 29) 0.005 473 28 × 2 = 0 + 0.010 946 56;
  • 30) 0.010 946 56 × 2 = 0 + 0.021 893 12;
  • 31) 0.021 893 12 × 2 = 0 + 0.043 786 24;
  • 32) 0.043 786 24 × 2 = 0 + 0.087 572 48;
  • 33) 0.087 572 48 × 2 = 0 + 0.175 144 96;
  • 34) 0.175 144 96 × 2 = 0 + 0.350 289 92;
  • 35) 0.350 289 92 × 2 = 0 + 0.700 579 84;
  • 36) 0.700 579 84 × 2 = 1 + 0.401 159 68;
  • 37) 0.401 159 68 × 2 = 0 + 0.802 319 36;
  • 38) 0.802 319 36 × 2 = 1 + 0.604 638 72;
  • 39) 0.604 638 72 × 2 = 1 + 0.209 277 44;
  • 40) 0.209 277 44 × 2 = 0 + 0.418 554 88;
  • 41) 0.418 554 88 × 2 = 0 + 0.837 109 76;
  • 42) 0.837 109 76 × 2 = 1 + 0.674 219 52;
  • 43) 0.674 219 52 × 2 = 1 + 0.348 439 04;
  • 44) 0.348 439 04 × 2 = 0 + 0.696 878 08;
  • 45) 0.696 878 08 × 2 = 1 + 0.393 756 16;

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:


0.000 000 38(10) =

0.0000 0000 0000 0000 0000 0110 0110 0000 0001 0110 0110 1(2)

5. Positive number before normalization:

0.000 000 38(10) =

0.0000 0000 0000 0000 0000 0110 0110 0000 0001 0110 0110 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 22 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 38(10) =

0.0000 0000 0000 0000 0000 0110 0110 0000 0001 0110 0110 1(2) =

0.0000 0000 0000 0000 0000 0110 0110 0000 0001 0110 0110 1(2) × 20 =

1.1001 1000 0000 0101 1001 101(2) × 2-22


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -22

Mantissa (not normalized):
1.1001 1000 0000 0101 1001 101


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-22 + 2(8-1) - 1 =

(-22 + 127)(10) =

105(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 105 ÷ 2 = 52 + 1;
  • 52 ÷ 2 = 26 + 0;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 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) =

105(10) =

0110 1001(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 100 1100 0000 0010 1100 1101 =

100 1100 0000 0010 1100 1101


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (8 bits) =
0110 1001

Mantissa (23 bits) =
100 1100 0000 0010 1100 1101


Decimal number 0.000 000 38 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0110 1001 - 100 1100 0000 0010 1100 1101

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How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =
    1 - 1000 0011 - 100 1010 1100 0110 1010 0111