-0.000 000 030 8 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 030 8(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
-0.000 000 030 8(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:

|-0.000 000 030 8| = 0.000 000 030 8


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: 0.000 000 030 8.

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) 0.000 000 030 8 × 2 = 0 + 0.000 000 061 6;
  • 2) 0.000 000 061 6 × 2 = 0 + 0.000 000 123 2;
  • 3) 0.000 000 123 2 × 2 = 0 + 0.000 000 246 4;
  • 4) 0.000 000 246 4 × 2 = 0 + 0.000 000 492 8;
  • 5) 0.000 000 492 8 × 2 = 0 + 0.000 000 985 6;
  • 6) 0.000 000 985 6 × 2 = 0 + 0.000 001 971 2;
  • 7) 0.000 001 971 2 × 2 = 0 + 0.000 003 942 4;
  • 8) 0.000 003 942 4 × 2 = 0 + 0.000 007 884 8;
  • 9) 0.000 007 884 8 × 2 = 0 + 0.000 015 769 6;
  • 10) 0.000 015 769 6 × 2 = 0 + 0.000 031 539 2;
  • 11) 0.000 031 539 2 × 2 = 0 + 0.000 063 078 4;
  • 12) 0.000 063 078 4 × 2 = 0 + 0.000 126 156 8;
  • 13) 0.000 126 156 8 × 2 = 0 + 0.000 252 313 6;
  • 14) 0.000 252 313 6 × 2 = 0 + 0.000 504 627 2;
  • 15) 0.000 504 627 2 × 2 = 0 + 0.001 009 254 4;
  • 16) 0.001 009 254 4 × 2 = 0 + 0.002 018 508 8;
  • 17) 0.002 018 508 8 × 2 = 0 + 0.004 037 017 6;
  • 18) 0.004 037 017 6 × 2 = 0 + 0.008 074 035 2;
  • 19) 0.008 074 035 2 × 2 = 0 + 0.016 148 070 4;
  • 20) 0.016 148 070 4 × 2 = 0 + 0.032 296 140 8;
  • 21) 0.032 296 140 8 × 2 = 0 + 0.064 592 281 6;
  • 22) 0.064 592 281 6 × 2 = 0 + 0.129 184 563 2;
  • 23) 0.129 184 563 2 × 2 = 0 + 0.258 369 126 4;
  • 24) 0.258 369 126 4 × 2 = 0 + 0.516 738 252 8;
  • 25) 0.516 738 252 8 × 2 = 1 + 0.033 476 505 6;
  • 26) 0.033 476 505 6 × 2 = 0 + 0.066 953 011 2;
  • 27) 0.066 953 011 2 × 2 = 0 + 0.133 906 022 4;
  • 28) 0.133 906 022 4 × 2 = 0 + 0.267 812 044 8;
  • 29) 0.267 812 044 8 × 2 = 0 + 0.535 624 089 6;
  • 30) 0.535 624 089 6 × 2 = 1 + 0.071 248 179 2;
  • 31) 0.071 248 179 2 × 2 = 0 + 0.142 496 358 4;
  • 32) 0.142 496 358 4 × 2 = 0 + 0.284 992 716 8;
  • 33) 0.284 992 716 8 × 2 = 0 + 0.569 985 433 6;
  • 34) 0.569 985 433 6 × 2 = 1 + 0.139 970 867 2;
  • 35) 0.139 970 867 2 × 2 = 0 + 0.279 941 734 4;
  • 36) 0.279 941 734 4 × 2 = 0 + 0.559 883 468 8;
  • 37) 0.559 883 468 8 × 2 = 1 + 0.119 766 937 6;
  • 38) 0.119 766 937 6 × 2 = 0 + 0.239 533 875 2;
  • 39) 0.239 533 875 2 × 2 = 0 + 0.479 067 750 4;
  • 40) 0.479 067 750 4 × 2 = 0 + 0.958 135 500 8;
  • 41) 0.958 135 500 8 × 2 = 1 + 0.916 271 001 6;
  • 42) 0.916 271 001 6 × 2 = 1 + 0.832 542 003 2;
  • 43) 0.832 542 003 2 × 2 = 1 + 0.665 084 006 4;
  • 44) 0.665 084 006 4 × 2 = 1 + 0.330 168 012 8;
  • 45) 0.330 168 012 8 × 2 = 0 + 0.660 336 025 6;
  • 46) 0.660 336 025 6 × 2 = 1 + 0.320 672 051 2;
  • 47) 0.320 672 051 2 × 2 = 0 + 0.641 344 102 4;
  • 48) 0.641 344 102 4 × 2 = 1 + 0.282 688 204 8;

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:


0.000 000 030 8(10) =

0.0000 0000 0000 0000 0000 0000 1000 0100 0100 1000 1111 0101(2)

6. Positive number before normalization:

0.000 000 030 8(10) =

0.0000 0000 0000 0000 0000 0000 1000 0100 0100 1000 1111 0101(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 25 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 030 8(10) =

0.0000 0000 0000 0000 0000 0000 1000 0100 0100 1000 1111 0101(2) =

0.0000 0000 0000 0000 0000 0000 1000 0100 0100 1000 1111 0101(2) × 20 =

1.0000 1000 1001 0001 1110 101(2) × 2-25


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): -25

Mantissa (not normalized):
1.0000 1000 1001 0001 1110 101


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-25 + 2(8-1) - 1 =

(-25 + 127)(10) =

102(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;
  • 102 ÷ 2 = 51 + 0;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 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) =

102(10) =

0110 0110(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. 000 0100 0100 1000 1111 0101 =

000 0100 0100 1000 1111 0101


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 0110

Mantissa (23 bits) =
000 0100 0100 1000 1111 0101


Decimal number -0.000 000 030 8 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0110 - 000 0100 0100 1000 1111 0101

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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