-0.000 000 73 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 73(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 73(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. Start with the positive version of the number:

|-0.000 000 73| = 0.000 000 73


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 73.

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 73 × 2 = 0 + 0.000 001 46;
  • 2) 0.000 001 46 × 2 = 0 + 0.000 002 92;
  • 3) 0.000 002 92 × 2 = 0 + 0.000 005 84;
  • 4) 0.000 005 84 × 2 = 0 + 0.000 011 68;
  • 5) 0.000 011 68 × 2 = 0 + 0.000 023 36;
  • 6) 0.000 023 36 × 2 = 0 + 0.000 046 72;
  • 7) 0.000 046 72 × 2 = 0 + 0.000 093 44;
  • 8) 0.000 093 44 × 2 = 0 + 0.000 186 88;
  • 9) 0.000 186 88 × 2 = 0 + 0.000 373 76;
  • 10) 0.000 373 76 × 2 = 0 + 0.000 747 52;
  • 11) 0.000 747 52 × 2 = 0 + 0.001 495 04;
  • 12) 0.001 495 04 × 2 = 0 + 0.002 990 08;
  • 13) 0.002 990 08 × 2 = 0 + 0.005 980 16;
  • 14) 0.005 980 16 × 2 = 0 + 0.011 960 32;
  • 15) 0.011 960 32 × 2 = 0 + 0.023 920 64;
  • 16) 0.023 920 64 × 2 = 0 + 0.047 841 28;
  • 17) 0.047 841 28 × 2 = 0 + 0.095 682 56;
  • 18) 0.095 682 56 × 2 = 0 + 0.191 365 12;
  • 19) 0.191 365 12 × 2 = 0 + 0.382 730 24;
  • 20) 0.382 730 24 × 2 = 0 + 0.765 460 48;
  • 21) 0.765 460 48 × 2 = 1 + 0.530 920 96;
  • 22) 0.530 920 96 × 2 = 1 + 0.061 841 92;
  • 23) 0.061 841 92 × 2 = 0 + 0.123 683 84;
  • 24) 0.123 683 84 × 2 = 0 + 0.247 367 68;
  • 25) 0.247 367 68 × 2 = 0 + 0.494 735 36;
  • 26) 0.494 735 36 × 2 = 0 + 0.989 470 72;
  • 27) 0.989 470 72 × 2 = 1 + 0.978 941 44;
  • 28) 0.978 941 44 × 2 = 1 + 0.957 882 88;
  • 29) 0.957 882 88 × 2 = 1 + 0.915 765 76;
  • 30) 0.915 765 76 × 2 = 1 + 0.831 531 52;
  • 31) 0.831 531 52 × 2 = 1 + 0.663 063 04;
  • 32) 0.663 063 04 × 2 = 1 + 0.326 126 08;
  • 33) 0.326 126 08 × 2 = 0 + 0.652 252 16;
  • 34) 0.652 252 16 × 2 = 1 + 0.304 504 32;
  • 35) 0.304 504 32 × 2 = 0 + 0.609 008 64;
  • 36) 0.609 008 64 × 2 = 1 + 0.218 017 28;
  • 37) 0.218 017 28 × 2 = 0 + 0.436 034 56;
  • 38) 0.436 034 56 × 2 = 0 + 0.872 069 12;
  • 39) 0.872 069 12 × 2 = 1 + 0.744 138 24;
  • 40) 0.744 138 24 × 2 = 1 + 0.488 276 48;
  • 41) 0.488 276 48 × 2 = 0 + 0.976 552 96;
  • 42) 0.976 552 96 × 2 = 1 + 0.953 105 92;
  • 43) 0.953 105 92 × 2 = 1 + 0.906 211 84;
  • 44) 0.906 211 84 × 2 = 1 + 0.812 423 68;

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).


5. 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 73(10) =

0.0000 0000 0000 0000 0000 1100 0011 1111 0101 0011 0111(2)

6. Positive number before normalization:

0.000 000 73(10) =

0.0000 0000 0000 0000 0000 1100 0011 1111 0101 0011 0111(2)

7. Normalize the binary representation of the number.

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


0.000 000 73(10) =

0.0000 0000 0000 0000 0000 1100 0011 1111 0101 0011 0111(2) =

0.0000 0000 0000 0000 0000 1100 0011 1111 0101 0011 0111(2) × 20 =

1.1000 0111 1110 1010 0110 111(2) × 2-21


8. 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 1 (a negative number)

Exponent (unadjusted): -21

Mantissa (not normalized):
1.1000 0111 1110 1010 0110 111


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-21 + 2(8-1) - 1 =

(-21 + 127)(10) =

106(10)


10. 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;
  • 106 ÷ 2 = 53 + 0;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

106(10) =

0110 1010(2)


12. 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 0011 1111 0101 0011 0111 =

100 0011 1111 0101 0011 0111


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

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

Exponent (8 bits) =
0110 1010

Mantissa (23 bits) =
100 0011 1111 0101 0011 0111


Decimal number -0.000 000 73 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 1010 - 100 0011 1111 0101 0011 0111

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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