-0.000 005 3 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 005 3(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
-0.000 005 3(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:

|-0.000 005 3| = 0.000 005 3


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: 0.000 005 3.

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) 0.000 005 3 × 2 = 0 + 0.000 010 6;
  • 2) 0.000 010 6 × 2 = 0 + 0.000 021 2;
  • 3) 0.000 021 2 × 2 = 0 + 0.000 042 4;
  • 4) 0.000 042 4 × 2 = 0 + 0.000 084 8;
  • 5) 0.000 084 8 × 2 = 0 + 0.000 169 6;
  • 6) 0.000 169 6 × 2 = 0 + 0.000 339 2;
  • 7) 0.000 339 2 × 2 = 0 + 0.000 678 4;
  • 8) 0.000 678 4 × 2 = 0 + 0.001 356 8;
  • 9) 0.001 356 8 × 2 = 0 + 0.002 713 6;
  • 10) 0.002 713 6 × 2 = 0 + 0.005 427 2;
  • 11) 0.005 427 2 × 2 = 0 + 0.010 854 4;
  • 12) 0.010 854 4 × 2 = 0 + 0.021 708 8;
  • 13) 0.021 708 8 × 2 = 0 + 0.043 417 6;
  • 14) 0.043 417 6 × 2 = 0 + 0.086 835 2;
  • 15) 0.086 835 2 × 2 = 0 + 0.173 670 4;
  • 16) 0.173 670 4 × 2 = 0 + 0.347 340 8;
  • 17) 0.347 340 8 × 2 = 0 + 0.694 681 6;
  • 18) 0.694 681 6 × 2 = 1 + 0.389 363 2;
  • 19) 0.389 363 2 × 2 = 0 + 0.778 726 4;
  • 20) 0.778 726 4 × 2 = 1 + 0.557 452 8;
  • 21) 0.557 452 8 × 2 = 1 + 0.114 905 6;
  • 22) 0.114 905 6 × 2 = 0 + 0.229 811 2;
  • 23) 0.229 811 2 × 2 = 0 + 0.459 622 4;
  • 24) 0.459 622 4 × 2 = 0 + 0.919 244 8;
  • 25) 0.919 244 8 × 2 = 1 + 0.838 489 6;
  • 26) 0.838 489 6 × 2 = 1 + 0.676 979 2;
  • 27) 0.676 979 2 × 2 = 1 + 0.353 958 4;
  • 28) 0.353 958 4 × 2 = 0 + 0.707 916 8;
  • 29) 0.707 916 8 × 2 = 1 + 0.415 833 6;
  • 30) 0.415 833 6 × 2 = 0 + 0.831 667 2;
  • 31) 0.831 667 2 × 2 = 1 + 0.663 334 4;
  • 32) 0.663 334 4 × 2 = 1 + 0.326 668 8;
  • 33) 0.326 668 8 × 2 = 0 + 0.653 337 6;
  • 34) 0.653 337 6 × 2 = 1 + 0.306 675 2;
  • 35) 0.306 675 2 × 2 = 0 + 0.613 350 4;
  • 36) 0.613 350 4 × 2 = 1 + 0.226 700 8;
  • 37) 0.226 700 8 × 2 = 0 + 0.453 401 6;
  • 38) 0.453 401 6 × 2 = 0 + 0.906 803 2;
  • 39) 0.906 803 2 × 2 = 1 + 0.813 606 4;
  • 40) 0.813 606 4 × 2 = 1 + 0.627 212 8;
  • 41) 0.627 212 8 × 2 = 1 + 0.254 425 6;

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:


0.000 005 3(10) =

0.0000 0000 0000 0000 0101 1000 1110 1011 0101 0011 1(2)

6. Positive number before normalization:

0.000 005 3(10) =

0.0000 0000 0000 0000 0101 1000 1110 1011 0101 0011 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 18 positions to the right, so that only one non zero digit remains to the left of it:


0.000 005 3(10) =

0.0000 0000 0000 0000 0101 1000 1110 1011 0101 0011 1(2) =

0.0000 0000 0000 0000 0101 1000 1110 1011 0101 0011 1(2) × 20 =

1.0110 0011 1010 1101 0100 111(2) × 2-18


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): -18

Mantissa (not normalized):
1.0110 0011 1010 1101 0100 111


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-18 + 2(8-1) - 1 =

(-18 + 127)(10) =

109(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;
  • 109 ÷ 2 = 54 + 1;
  • 54 ÷ 2 = 27 + 0;
  • 27 ÷ 2 = 13 + 1;
  • 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) =

109(10) =

0110 1101(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. 011 0001 1101 0110 1010 0111 =

011 0001 1101 0110 1010 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 1101

Mantissa (23 bits) =
011 0001 1101 0110 1010 0111


Decimal number -0.000 005 3 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 1101 - 011 0001 1101 0110 1010 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