Convert 0.000 065 24 to 32 Bit Single Precision IEEE 754 Binary Floating Point Standard, From a Base 10 Decimal Number

0.000 065 24(10) to 32 bit single precision IEEE 754 binary floating point (1 bit for sign, 8 bits for exponent, 23 bits for mantissa) = ?

1. First, convert to the binary (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 the binary (base 2) the fractional part: 0.000 065 24.

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) 0.000 065 24 × 2 = 0 + 0.000 130 48;
  • 2) 0.000 130 48 × 2 = 0 + 0.000 260 96;
  • 3) 0.000 260 96 × 2 = 0 + 0.000 521 92;
  • 4) 0.000 521 92 × 2 = 0 + 0.001 043 84;
  • 5) 0.001 043 84 × 2 = 0 + 0.002 087 68;
  • 6) 0.002 087 68 × 2 = 0 + 0.004 175 36;
  • 7) 0.004 175 36 × 2 = 0 + 0.008 350 72;
  • 8) 0.008 350 72 × 2 = 0 + 0.016 701 44;
  • 9) 0.016 701 44 × 2 = 0 + 0.033 402 88;
  • 10) 0.033 402 88 × 2 = 0 + 0.066 805 76;
  • 11) 0.066 805 76 × 2 = 0 + 0.133 611 52;
  • 12) 0.133 611 52 × 2 = 0 + 0.267 223 04;
  • 13) 0.267 223 04 × 2 = 0 + 0.534 446 08;
  • 14) 0.534 446 08 × 2 = 1 + 0.068 892 16;
  • 15) 0.068 892 16 × 2 = 0 + 0.137 784 32;
  • 16) 0.137 784 32 × 2 = 0 + 0.275 568 64;
  • 17) 0.275 568 64 × 2 = 0 + 0.551 137 28;
  • 18) 0.551 137 28 × 2 = 1 + 0.102 274 56;
  • 19) 0.102 274 56 × 2 = 0 + 0.204 549 12;
  • 20) 0.204 549 12 × 2 = 0 + 0.409 098 24;
  • 21) 0.409 098 24 × 2 = 0 + 0.818 196 48;
  • 22) 0.818 196 48 × 2 = 1 + 0.636 392 96;
  • 23) 0.636 392 96 × 2 = 1 + 0.272 785 92;
  • 24) 0.272 785 92 × 2 = 0 + 0.545 571 84;
  • 25) 0.545 571 84 × 2 = 1 + 0.091 143 68;
  • 26) 0.091 143 68 × 2 = 0 + 0.182 287 36;
  • 27) 0.182 287 36 × 2 = 0 + 0.364 574 72;
  • 28) 0.364 574 72 × 2 = 0 + 0.729 149 44;
  • 29) 0.729 149 44 × 2 = 1 + 0.458 298 88;
  • 30) 0.458 298 88 × 2 = 0 + 0.916 597 76;
  • 31) 0.916 597 76 × 2 = 1 + 0.833 195 52;
  • 32) 0.833 195 52 × 2 = 1 + 0.666 391 04;
  • 33) 0.666 391 04 × 2 = 1 + 0.332 782 08;
  • 34) 0.332 782 08 × 2 = 0 + 0.665 564 16;
  • 35) 0.665 564 16 × 2 = 1 + 0.331 128 32;
  • 36) 0.331 128 32 × 2 = 0 + 0.662 256 64;
  • 37) 0.662 256 64 × 2 = 1 + 0.324 513 28;

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:


0.000 065 24(10) =

0.0000 0000 0000 0100 0100 0110 1000 1011 1010 1(2)


5. Positive number before normalization:

0.000 065 24(10) =

0.0000 0000 0000 0100 0100 0110 1000 1011 1010 1(2)


6. Normalize the binary representation of the number.

Shift the decimal mark 14 positions to the right so that only one non zero digit remains to the left of it:


0.000 065 24(10) =

0.0000 0000 0000 0100 0100 0110 1000 1011 1010 1(2) =

0.0000 0000 0000 0100 0100 0110 1000 1011 1010 1(2) × 20 =

1.0001 0001 1010 0010 1110 101(2) × 2-14


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): -14

Mantissa (not normalized):
1.0001 0001 1010 0010 1110 101


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-14 + 2(8-1) - 1 =

(-14 + 127)(10) =

113(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;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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) =

113(10) =

0111 0001(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. 000 1000 1101 0001 0111 0101 =

000 1000 1101 0001 0111 0101


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) =
0111 0001

Mantissa (23 bits) =
000 1000 1101 0001 0111 0101


Number 0.000 065 24 converted from decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point:
0 - 0111 0001 - 000 1000 1101 0001 0111 0101

(32 bits IEEE 754)

More operations of this kind:

0.000 065 23 = ? ... 0.000 065 25 = ?


Convert to 32 bit single precision IEEE 754 binary floating point standard

A number in 32 bit single precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (8 bits) and mantissa (23 bits)

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All base ten decimal numbers converted to 32 bit single precision IEEE 754 binary floating point

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