1.333 333 333 333 333 333 333 333 333 333 333 333 279 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.333 333 333 333 333 333 333 333 333 333 333 333 279(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
1.333 333 333 333 333 333 333 333 333 333 333 333 279(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.333 333 333 333 333 333 333 333 333 333 333 333 279.

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) 0.333 333 333 333 333 333 333 333 333 333 333 333 279 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 558;
  • 2) 0.666 666 666 666 666 666 666 666 666 666 666 666 558 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 116;
  • 3) 0.333 333 333 333 333 333 333 333 333 333 333 333 116 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 232;
  • 4) 0.666 666 666 666 666 666 666 666 666 666 666 666 232 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 332 464;
  • 5) 0.333 333 333 333 333 333 333 333 333 333 333 332 464 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 664 928;
  • 6) 0.666 666 666 666 666 666 666 666 666 666 666 664 928 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 329 856;
  • 7) 0.333 333 333 333 333 333 333 333 333 333 333 329 856 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 659 712;
  • 8) 0.666 666 666 666 666 666 666 666 666 666 666 659 712 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 319 424;
  • 9) 0.333 333 333 333 333 333 333 333 333 333 333 319 424 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 638 848;
  • 10) 0.666 666 666 666 666 666 666 666 666 666 666 638 848 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 277 696;
  • 11) 0.333 333 333 333 333 333 333 333 333 333 333 277 696 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 555 392;
  • 12) 0.666 666 666 666 666 666 666 666 666 666 666 555 392 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 110 784;
  • 13) 0.333 333 333 333 333 333 333 333 333 333 333 110 784 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 221 568;
  • 14) 0.666 666 666 666 666 666 666 666 666 666 666 221 568 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 332 443 136;
  • 15) 0.333 333 333 333 333 333 333 333 333 333 332 443 136 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 664 886 272;
  • 16) 0.666 666 666 666 666 666 666 666 666 666 664 886 272 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 329 772 544;
  • 17) 0.333 333 333 333 333 333 333 333 333 333 329 772 544 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 659 545 088;
  • 18) 0.666 666 666 666 666 666 666 666 666 666 659 545 088 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 319 090 176;
  • 19) 0.333 333 333 333 333 333 333 333 333 333 319 090 176 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 638 180 352;
  • 20) 0.666 666 666 666 666 666 666 666 666 666 638 180 352 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 276 360 704;
  • 21) 0.333 333 333 333 333 333 333 333 333 333 276 360 704 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 552 721 408;
  • 22) 0.666 666 666 666 666 666 666 666 666 666 552 721 408 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 105 442 816;
  • 23) 0.333 333 333 333 333 333 333 333 333 333 105 442 816 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 210 885 632;
  • 24) 0.666 666 666 666 666 666 666 666 666 666 210 885 632 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 332 421 771 264;

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:


0.333 333 333 333 333 333 333 333 333 333 333 333 279(10) =

0.0101 0101 0101 0101 0101 0101(2)

5. Positive number before normalization:

1.333 333 333 333 333 333 333 333 333 333 333 333 279(10) =

1.0101 0101 0101 0101 0101 0101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.333 333 333 333 333 333 333 333 333 333 333 333 279(10) =

1.0101 0101 0101 0101 0101 0101(2) =

1.0101 0101 0101 0101 0101 0101(2) × 20


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): 0

Mantissa (not normalized):
1.0101 0101 0101 0101 0101 0101


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(8-1) - 1 =

(0 + 127)(10) =

127(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;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 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) =

127(10) =

0111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 010 1010 1010 1010 1010 1010 1 =

010 1010 1010 1010 1010 1010


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) =
0111 1111

Mantissa (23 bits) =
010 1010 1010 1010 1010 1010


Decimal number 1.333 333 333 333 333 333 333 333 333 333 333 333 279 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0111 1111 - 010 1010 1010 1010 1010 1010

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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