0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286(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
0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286(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: 0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286.

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) 0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286 × 2 = 0 + 0.004 166 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 572;
  • 2) 0.004 166 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 572 × 2 = 0 + 0.008 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 144;
  • 3) 0.008 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 144 × 2 = 0 + 0.016 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 288;
  • 4) 0.016 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 288 × 2 = 0 + 0.033 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 576;
  • 5) 0.033 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 576 × 2 = 0 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 152;
  • 6) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 152 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 304;
  • 7) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 304 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 608;
  • 8) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 608 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 321 216;
  • 9) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 321 216 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 642 432;
  • 10) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 642 432 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 864;
  • 11) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 864 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 569 728;
  • 12) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 569 728 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 139 456;
  • 13) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 139 456 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 278 912;
  • 14) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 278 912 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 557 824;
  • 15) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 557 824 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 115 648;
  • 16) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 115 648 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 231 296;
  • 17) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 231 296 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 462 592;
  • 18) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 462 592 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 925 184;
  • 19) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 925 184 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 850 368;
  • 20) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 850 368 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 283 700 736;
  • 21) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 283 700 736 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 567 401 472;
  • 22) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 567 401 472 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 134 802 944;
  • 23) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 134 802 944 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 269 605 888;
  • 24) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 269 605 888 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 539 211 776;
  • 25) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 539 211 776 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 078 423 552;
  • 26) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 078 423 552 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 156 847 104;
  • 27) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 156 847 104 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 313 694 208;
  • 28) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 313 694 208 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 627 388 416;
  • 29) 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 627 388 416 × 2 = 1 + 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 254 776 832;
  • 30) 0.066 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 254 776 832 × 2 = 0 + 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 282 509 553 664;
  • 31) 0.133 333 333 333 333 333 333 333 333 333 333 333 333 333 333 282 509 553 664 × 2 = 0 + 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 565 019 107 328;
  • 32) 0.266 666 666 666 666 666 666 666 666 666 666 666 666 666 666 565 019 107 328 × 2 = 0 + 0.533 333 333 333 333 333 333 333 333 333 333 333 333 333 333 130 038 214 656;

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:


0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286(10) =

0.0000 0000 1000 1000 1000 1000 1000 1000(2)

5. Positive number before normalization:

0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286(10) =

0.0000 0000 1000 1000 1000 1000 1000 1000(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 9 positions to the right, so that only one non zero digit remains to the left of it:


0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286(10) =

0.0000 0000 1000 1000 1000 1000 1000 1000(2) =

0.0000 0000 1000 1000 1000 1000 1000 1000(2) × 20 =

1.0001 0001 0001 0001 0001 000(2) × 2-9


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): -9

Mantissa (not normalized):
1.0001 0001 0001 0001 0001 000


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-9 + 2(8-1) - 1 =

(-9 + 127)(10) =

118(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;
  • 118 ÷ 2 = 59 + 0;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

118(10) =

0111 0110(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. 000 1000 1000 1000 1000 1000 =

000 1000 1000 1000 1000 1000


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 0110

Mantissa (23 bits) =
000 1000 1000 1000 1000 1000


Decimal number 0.002 083 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 286 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0111 0110 - 000 1000 1000 1000 1000 1000

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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