0.000 000 629 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 629(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
0.000 000 629(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: 0.000 000 629.

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) 0.000 000 629 × 2 = 0 + 0.000 001 258;
  • 2) 0.000 001 258 × 2 = 0 + 0.000 002 516;
  • 3) 0.000 002 516 × 2 = 0 + 0.000 005 032;
  • 4) 0.000 005 032 × 2 = 0 + 0.000 010 064;
  • 5) 0.000 010 064 × 2 = 0 + 0.000 020 128;
  • 6) 0.000 020 128 × 2 = 0 + 0.000 040 256;
  • 7) 0.000 040 256 × 2 = 0 + 0.000 080 512;
  • 8) 0.000 080 512 × 2 = 0 + 0.000 161 024;
  • 9) 0.000 161 024 × 2 = 0 + 0.000 322 048;
  • 10) 0.000 322 048 × 2 = 0 + 0.000 644 096;
  • 11) 0.000 644 096 × 2 = 0 + 0.001 288 192;
  • 12) 0.001 288 192 × 2 = 0 + 0.002 576 384;
  • 13) 0.002 576 384 × 2 = 0 + 0.005 152 768;
  • 14) 0.005 152 768 × 2 = 0 + 0.010 305 536;
  • 15) 0.010 305 536 × 2 = 0 + 0.020 611 072;
  • 16) 0.020 611 072 × 2 = 0 + 0.041 222 144;
  • 17) 0.041 222 144 × 2 = 0 + 0.082 444 288;
  • 18) 0.082 444 288 × 2 = 0 + 0.164 888 576;
  • 19) 0.164 888 576 × 2 = 0 + 0.329 777 152;
  • 20) 0.329 777 152 × 2 = 0 + 0.659 554 304;
  • 21) 0.659 554 304 × 2 = 1 + 0.319 108 608;
  • 22) 0.319 108 608 × 2 = 0 + 0.638 217 216;
  • 23) 0.638 217 216 × 2 = 1 + 0.276 434 432;
  • 24) 0.276 434 432 × 2 = 0 + 0.552 868 864;
  • 25) 0.552 868 864 × 2 = 1 + 0.105 737 728;
  • 26) 0.105 737 728 × 2 = 0 + 0.211 475 456;
  • 27) 0.211 475 456 × 2 = 0 + 0.422 950 912;
  • 28) 0.422 950 912 × 2 = 0 + 0.845 901 824;
  • 29) 0.845 901 824 × 2 = 1 + 0.691 803 648;
  • 30) 0.691 803 648 × 2 = 1 + 0.383 607 296;
  • 31) 0.383 607 296 × 2 = 0 + 0.767 214 592;
  • 32) 0.767 214 592 × 2 = 1 + 0.534 429 184;
  • 33) 0.534 429 184 × 2 = 1 + 0.068 858 368;
  • 34) 0.068 858 368 × 2 = 0 + 0.137 716 736;
  • 35) 0.137 716 736 × 2 = 0 + 0.275 433 472;
  • 36) 0.275 433 472 × 2 = 0 + 0.550 866 944;
  • 37) 0.550 866 944 × 2 = 1 + 0.101 733 888;
  • 38) 0.101 733 888 × 2 = 0 + 0.203 467 776;
  • 39) 0.203 467 776 × 2 = 0 + 0.406 935 552;
  • 40) 0.406 935 552 × 2 = 0 + 0.813 871 104;
  • 41) 0.813 871 104 × 2 = 1 + 0.627 742 208;
  • 42) 0.627 742 208 × 2 = 1 + 0.255 484 416;
  • 43) 0.255 484 416 × 2 = 0 + 0.510 968 832;
  • 44) 0.510 968 832 × 2 = 1 + 0.021 937 664;

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:


0.000 000 629(10) =

0.0000 0000 0000 0000 0000 1010 1000 1101 1000 1000 1101(2)

5. Positive number before normalization:

0.000 000 629(10) =

0.0000 0000 0000 0000 0000 1010 1000 1101 1000 1000 1101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 21 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 629(10) =

0.0000 0000 0000 0000 0000 1010 1000 1101 1000 1000 1101(2) =

0.0000 0000 0000 0000 0000 1010 1000 1101 1000 1000 1101(2) × 20 =

1.0101 0001 1011 0001 0001 101(2) × 2-21


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): -21

Mantissa (not normalized):
1.0101 0001 1011 0001 0001 101


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-21 + 2(8-1) - 1 =

(-21 + 127)(10) =

106(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;
  • 106 ÷ 2 = 53 + 0;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 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) =

106(10) =

0110 1010(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 1000 1101 1000 1000 1101 =

010 1000 1101 1000 1000 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 1010

Mantissa (23 bits) =
010 1000 1101 1000 1000 1101


Decimal number 0.000 000 629 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0110 1010 - 010 1000 1101 1000 1000 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