-0.000 000 035 6 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 035 6(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
-0.000 000 035 6(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:

|-0.000 000 035 6| = 0.000 000 035 6


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: 0.000 000 035 6.

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) 0.000 000 035 6 × 2 = 0 + 0.000 000 071 2;
  • 2) 0.000 000 071 2 × 2 = 0 + 0.000 000 142 4;
  • 3) 0.000 000 142 4 × 2 = 0 + 0.000 000 284 8;
  • 4) 0.000 000 284 8 × 2 = 0 + 0.000 000 569 6;
  • 5) 0.000 000 569 6 × 2 = 0 + 0.000 001 139 2;
  • 6) 0.000 001 139 2 × 2 = 0 + 0.000 002 278 4;
  • 7) 0.000 002 278 4 × 2 = 0 + 0.000 004 556 8;
  • 8) 0.000 004 556 8 × 2 = 0 + 0.000 009 113 6;
  • 9) 0.000 009 113 6 × 2 = 0 + 0.000 018 227 2;
  • 10) 0.000 018 227 2 × 2 = 0 + 0.000 036 454 4;
  • 11) 0.000 036 454 4 × 2 = 0 + 0.000 072 908 8;
  • 12) 0.000 072 908 8 × 2 = 0 + 0.000 145 817 6;
  • 13) 0.000 145 817 6 × 2 = 0 + 0.000 291 635 2;
  • 14) 0.000 291 635 2 × 2 = 0 + 0.000 583 270 4;
  • 15) 0.000 583 270 4 × 2 = 0 + 0.001 166 540 8;
  • 16) 0.001 166 540 8 × 2 = 0 + 0.002 333 081 6;
  • 17) 0.002 333 081 6 × 2 = 0 + 0.004 666 163 2;
  • 18) 0.004 666 163 2 × 2 = 0 + 0.009 332 326 4;
  • 19) 0.009 332 326 4 × 2 = 0 + 0.018 664 652 8;
  • 20) 0.018 664 652 8 × 2 = 0 + 0.037 329 305 6;
  • 21) 0.037 329 305 6 × 2 = 0 + 0.074 658 611 2;
  • 22) 0.074 658 611 2 × 2 = 0 + 0.149 317 222 4;
  • 23) 0.149 317 222 4 × 2 = 0 + 0.298 634 444 8;
  • 24) 0.298 634 444 8 × 2 = 0 + 0.597 268 889 6;
  • 25) 0.597 268 889 6 × 2 = 1 + 0.194 537 779 2;
  • 26) 0.194 537 779 2 × 2 = 0 + 0.389 075 558 4;
  • 27) 0.389 075 558 4 × 2 = 0 + 0.778 151 116 8;
  • 28) 0.778 151 116 8 × 2 = 1 + 0.556 302 233 6;
  • 29) 0.556 302 233 6 × 2 = 1 + 0.112 604 467 2;
  • 30) 0.112 604 467 2 × 2 = 0 + 0.225 208 934 4;
  • 31) 0.225 208 934 4 × 2 = 0 + 0.450 417 868 8;
  • 32) 0.450 417 868 8 × 2 = 0 + 0.900 835 737 6;
  • 33) 0.900 835 737 6 × 2 = 1 + 0.801 671 475 2;
  • 34) 0.801 671 475 2 × 2 = 1 + 0.603 342 950 4;
  • 35) 0.603 342 950 4 × 2 = 1 + 0.206 685 900 8;
  • 36) 0.206 685 900 8 × 2 = 0 + 0.413 371 801 6;
  • 37) 0.413 371 801 6 × 2 = 0 + 0.826 743 603 2;
  • 38) 0.826 743 603 2 × 2 = 1 + 0.653 487 206 4;
  • 39) 0.653 487 206 4 × 2 = 1 + 0.306 974 412 8;
  • 40) 0.306 974 412 8 × 2 = 0 + 0.613 948 825 6;
  • 41) 0.613 948 825 6 × 2 = 1 + 0.227 897 651 2;
  • 42) 0.227 897 651 2 × 2 = 0 + 0.455 795 302 4;
  • 43) 0.455 795 302 4 × 2 = 0 + 0.911 590 604 8;
  • 44) 0.911 590 604 8 × 2 = 1 + 0.823 181 209 6;
  • 45) 0.823 181 209 6 × 2 = 1 + 0.646 362 419 2;
  • 46) 0.646 362 419 2 × 2 = 1 + 0.292 724 838 4;
  • 47) 0.292 724 838 4 × 2 = 0 + 0.585 449 676 8;
  • 48) 0.585 449 676 8 × 2 = 1 + 0.170 899 353 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:


0.000 000 035 6(10) =

0.0000 0000 0000 0000 0000 0000 1001 1000 1110 0110 1001 1101(2)

6. Positive number before normalization:

0.000 000 035 6(10) =

0.0000 0000 0000 0000 0000 0000 1001 1000 1110 0110 1001 1101(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:


0.000 000 035 6(10) =

0.0000 0000 0000 0000 0000 0000 1001 1000 1110 0110 1001 1101(2) =

0.0000 0000 0000 0000 0000 0000 1001 1000 1110 0110 1001 1101(2) × 20 =

1.0011 0001 1100 1101 0011 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):
1.0011 0001 1100 1101 0011 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. 001 1000 1110 0110 1001 1101 =

001 1000 1110 0110 1001 1101


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) =
001 1000 1110 0110 1001 1101


Decimal number -0.000 000 035 6 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0110 - 001 1000 1110 0110 1001 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