0.000 000 009 7 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 009 7(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
0.000 000 009 7(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: 0.000 000 009 7.

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) 0.000 000 009 7 × 2 = 0 + 0.000 000 019 4;
  • 2) 0.000 000 019 4 × 2 = 0 + 0.000 000 038 8;
  • 3) 0.000 000 038 8 × 2 = 0 + 0.000 000 077 6;
  • 4) 0.000 000 077 6 × 2 = 0 + 0.000 000 155 2;
  • 5) 0.000 000 155 2 × 2 = 0 + 0.000 000 310 4;
  • 6) 0.000 000 310 4 × 2 = 0 + 0.000 000 620 8;
  • 7) 0.000 000 620 8 × 2 = 0 + 0.000 001 241 6;
  • 8) 0.000 001 241 6 × 2 = 0 + 0.000 002 483 2;
  • 9) 0.000 002 483 2 × 2 = 0 + 0.000 004 966 4;
  • 10) 0.000 004 966 4 × 2 = 0 + 0.000 009 932 8;
  • 11) 0.000 009 932 8 × 2 = 0 + 0.000 019 865 6;
  • 12) 0.000 019 865 6 × 2 = 0 + 0.000 039 731 2;
  • 13) 0.000 039 731 2 × 2 = 0 + 0.000 079 462 4;
  • 14) 0.000 079 462 4 × 2 = 0 + 0.000 158 924 8;
  • 15) 0.000 158 924 8 × 2 = 0 + 0.000 317 849 6;
  • 16) 0.000 317 849 6 × 2 = 0 + 0.000 635 699 2;
  • 17) 0.000 635 699 2 × 2 = 0 + 0.001 271 398 4;
  • 18) 0.001 271 398 4 × 2 = 0 + 0.002 542 796 8;
  • 19) 0.002 542 796 8 × 2 = 0 + 0.005 085 593 6;
  • 20) 0.005 085 593 6 × 2 = 0 + 0.010 171 187 2;
  • 21) 0.010 171 187 2 × 2 = 0 + 0.020 342 374 4;
  • 22) 0.020 342 374 4 × 2 = 0 + 0.040 684 748 8;
  • 23) 0.040 684 748 8 × 2 = 0 + 0.081 369 497 6;
  • 24) 0.081 369 497 6 × 2 = 0 + 0.162 738 995 2;
  • 25) 0.162 738 995 2 × 2 = 0 + 0.325 477 990 4;
  • 26) 0.325 477 990 4 × 2 = 0 + 0.650 955 980 8;
  • 27) 0.650 955 980 8 × 2 = 1 + 0.301 911 961 6;
  • 28) 0.301 911 961 6 × 2 = 0 + 0.603 823 923 2;
  • 29) 0.603 823 923 2 × 2 = 1 + 0.207 647 846 4;
  • 30) 0.207 647 846 4 × 2 = 0 + 0.415 295 692 8;
  • 31) 0.415 295 692 8 × 2 = 0 + 0.830 591 385 6;
  • 32) 0.830 591 385 6 × 2 = 1 + 0.661 182 771 2;
  • 33) 0.661 182 771 2 × 2 = 1 + 0.322 365 542 4;
  • 34) 0.322 365 542 4 × 2 = 0 + 0.644 731 084 8;
  • 35) 0.644 731 084 8 × 2 = 1 + 0.289 462 169 6;
  • 36) 0.289 462 169 6 × 2 = 0 + 0.578 924 339 2;
  • 37) 0.578 924 339 2 × 2 = 1 + 0.157 848 678 4;
  • 38) 0.157 848 678 4 × 2 = 0 + 0.315 697 356 8;
  • 39) 0.315 697 356 8 × 2 = 0 + 0.631 394 713 6;
  • 40) 0.631 394 713 6 × 2 = 1 + 0.262 789 427 2;
  • 41) 0.262 789 427 2 × 2 = 0 + 0.525 578 854 4;
  • 42) 0.525 578 854 4 × 2 = 1 + 0.051 157 708 8;
  • 43) 0.051 157 708 8 × 2 = 0 + 0.102 315 417 6;
  • 44) 0.102 315 417 6 × 2 = 0 + 0.204 630 835 2;
  • 45) 0.204 630 835 2 × 2 = 0 + 0.409 261 670 4;
  • 46) 0.409 261 670 4 × 2 = 0 + 0.818 523 340 8;
  • 47) 0.818 523 340 8 × 2 = 1 + 0.637 046 681 6;
  • 48) 0.637 046 681 6 × 2 = 1 + 0.274 093 363 2;
  • 49) 0.274 093 363 2 × 2 = 0 + 0.548 186 726 4;
  • 50) 0.548 186 726 4 × 2 = 1 + 0.096 373 452 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).


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:


0.000 000 009 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1001 1010 1001 0100 0011 01(2)

5. Positive number before normalization:

0.000 000 009 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1001 1010 1001 0100 0011 01(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:


0.000 000 009 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1001 1010 1001 0100 0011 01(2) =

0.0000 0000 0000 0000 0000 0000 0010 1001 1010 1001 0100 0011 01(2) × 20 =

1.0100 1101 0100 1010 0001 101(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):
1.0100 1101 0100 1010 0001 101


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. 010 0110 1010 0101 0000 1101 =

010 0110 1010 0101 0000 1101


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) =
010 0110 1010 0101 0000 1101


Decimal number 0.000 000 009 7 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0110 0100 - 010 0110 1010 0101 0000 1101

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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