0.000 000 013 4 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 013 4(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
0.000 000 013 4(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: 0.000 000 013 4.

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) 0.000 000 013 4 × 2 = 0 + 0.000 000 026 8;
  • 2) 0.000 000 026 8 × 2 = 0 + 0.000 000 053 6;
  • 3) 0.000 000 053 6 × 2 = 0 + 0.000 000 107 2;
  • 4) 0.000 000 107 2 × 2 = 0 + 0.000 000 214 4;
  • 5) 0.000 000 214 4 × 2 = 0 + 0.000 000 428 8;
  • 6) 0.000 000 428 8 × 2 = 0 + 0.000 000 857 6;
  • 7) 0.000 000 857 6 × 2 = 0 + 0.000 001 715 2;
  • 8) 0.000 001 715 2 × 2 = 0 + 0.000 003 430 4;
  • 9) 0.000 003 430 4 × 2 = 0 + 0.000 006 860 8;
  • 10) 0.000 006 860 8 × 2 = 0 + 0.000 013 721 6;
  • 11) 0.000 013 721 6 × 2 = 0 + 0.000 027 443 2;
  • 12) 0.000 027 443 2 × 2 = 0 + 0.000 054 886 4;
  • 13) 0.000 054 886 4 × 2 = 0 + 0.000 109 772 8;
  • 14) 0.000 109 772 8 × 2 = 0 + 0.000 219 545 6;
  • 15) 0.000 219 545 6 × 2 = 0 + 0.000 439 091 2;
  • 16) 0.000 439 091 2 × 2 = 0 + 0.000 878 182 4;
  • 17) 0.000 878 182 4 × 2 = 0 + 0.001 756 364 8;
  • 18) 0.001 756 364 8 × 2 = 0 + 0.003 512 729 6;
  • 19) 0.003 512 729 6 × 2 = 0 + 0.007 025 459 2;
  • 20) 0.007 025 459 2 × 2 = 0 + 0.014 050 918 4;
  • 21) 0.014 050 918 4 × 2 = 0 + 0.028 101 836 8;
  • 22) 0.028 101 836 8 × 2 = 0 + 0.056 203 673 6;
  • 23) 0.056 203 673 6 × 2 = 0 + 0.112 407 347 2;
  • 24) 0.112 407 347 2 × 2 = 0 + 0.224 814 694 4;
  • 25) 0.224 814 694 4 × 2 = 0 + 0.449 629 388 8;
  • 26) 0.449 629 388 8 × 2 = 0 + 0.899 258 777 6;
  • 27) 0.899 258 777 6 × 2 = 1 + 0.798 517 555 2;
  • 28) 0.798 517 555 2 × 2 = 1 + 0.597 035 110 4;
  • 29) 0.597 035 110 4 × 2 = 1 + 0.194 070 220 8;
  • 30) 0.194 070 220 8 × 2 = 0 + 0.388 140 441 6;
  • 31) 0.388 140 441 6 × 2 = 0 + 0.776 280 883 2;
  • 32) 0.776 280 883 2 × 2 = 1 + 0.552 561 766 4;
  • 33) 0.552 561 766 4 × 2 = 1 + 0.105 123 532 8;
  • 34) 0.105 123 532 8 × 2 = 0 + 0.210 247 065 6;
  • 35) 0.210 247 065 6 × 2 = 0 + 0.420 494 131 2;
  • 36) 0.420 494 131 2 × 2 = 0 + 0.840 988 262 4;
  • 37) 0.840 988 262 4 × 2 = 1 + 0.681 976 524 8;
  • 38) 0.681 976 524 8 × 2 = 1 + 0.363 953 049 6;
  • 39) 0.363 953 049 6 × 2 = 0 + 0.727 906 099 2;
  • 40) 0.727 906 099 2 × 2 = 1 + 0.455 812 198 4;
  • 41) 0.455 812 198 4 × 2 = 0 + 0.911 624 396 8;
  • 42) 0.911 624 396 8 × 2 = 1 + 0.823 248 793 6;
  • 43) 0.823 248 793 6 × 2 = 1 + 0.646 497 587 2;
  • 44) 0.646 497 587 2 × 2 = 1 + 0.292 995 174 4;
  • 45) 0.292 995 174 4 × 2 = 0 + 0.585 990 348 8;
  • 46) 0.585 990 348 8 × 2 = 1 + 0.171 980 697 6;
  • 47) 0.171 980 697 6 × 2 = 0 + 0.343 961 395 2;
  • 48) 0.343 961 395 2 × 2 = 0 + 0.687 922 790 4;
  • 49) 0.687 922 790 4 × 2 = 1 + 0.375 845 580 8;
  • 50) 0.375 845 580 8 × 2 = 0 + 0.751 691 161 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).


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:


0.000 000 013 4(10) =

0.0000 0000 0000 0000 0000 0000 0011 1001 1000 1101 0111 0100 10(2)

5. Positive number before normalization:

0.000 000 013 4(10) =

0.0000 0000 0000 0000 0000 0000 0011 1001 1000 1101 0111 0100 10(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 27 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 013 4(10) =

0.0000 0000 0000 0000 0000 0000 0011 1001 1000 1101 0111 0100 10(2) =

0.0000 0000 0000 0000 0000 0000 0011 1001 1000 1101 0111 0100 10(2) × 20 =

1.1100 1100 0110 1011 1010 010(2) × 2-27


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): -27

Mantissa (not normalized):
1.1100 1100 0110 1011 1010 010


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-27 + 2(8-1) - 1 =

(-27 + 127)(10) =

100(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;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 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) =

100(10) =

0110 0100(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. 110 0110 0011 0101 1101 0010 =

110 0110 0011 0101 1101 0010


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 0100

Mantissa (23 bits) =
110 0110 0011 0101 1101 0010


Decimal number 0.000 000 013 4 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0110 0100 - 110 0110 0011 0101 1101 0010

Share

» Convert decimal 0.000 000 008 7 to 32 bit single precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 05, 2026 [May]: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: May - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 minutes

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