-0.000 000 000 742 147 676 66 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 66(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
-0.000 000 000 742 147 676 66(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:

|-0.000 000 000 742 147 676 66| = 0.000 000 000 742 147 676 66


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: 0.000 000 000 742 147 676 66.

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) 0.000 000 000 742 147 676 66 × 2 = 0 + 0.000 000 001 484 295 353 32;
  • 2) 0.000 000 001 484 295 353 32 × 2 = 0 + 0.000 000 002 968 590 706 64;
  • 3) 0.000 000 002 968 590 706 64 × 2 = 0 + 0.000 000 005 937 181 413 28;
  • 4) 0.000 000 005 937 181 413 28 × 2 = 0 + 0.000 000 011 874 362 826 56;
  • 5) 0.000 000 011 874 362 826 56 × 2 = 0 + 0.000 000 023 748 725 653 12;
  • 6) 0.000 000 023 748 725 653 12 × 2 = 0 + 0.000 000 047 497 451 306 24;
  • 7) 0.000 000 047 497 451 306 24 × 2 = 0 + 0.000 000 094 994 902 612 48;
  • 8) 0.000 000 094 994 902 612 48 × 2 = 0 + 0.000 000 189 989 805 224 96;
  • 9) 0.000 000 189 989 805 224 96 × 2 = 0 + 0.000 000 379 979 610 449 92;
  • 10) 0.000 000 379 979 610 449 92 × 2 = 0 + 0.000 000 759 959 220 899 84;
  • 11) 0.000 000 759 959 220 899 84 × 2 = 0 + 0.000 001 519 918 441 799 68;
  • 12) 0.000 001 519 918 441 799 68 × 2 = 0 + 0.000 003 039 836 883 599 36;
  • 13) 0.000 003 039 836 883 599 36 × 2 = 0 + 0.000 006 079 673 767 198 72;
  • 14) 0.000 006 079 673 767 198 72 × 2 = 0 + 0.000 012 159 347 534 397 44;
  • 15) 0.000 012 159 347 534 397 44 × 2 = 0 + 0.000 024 318 695 068 794 88;
  • 16) 0.000 024 318 695 068 794 88 × 2 = 0 + 0.000 048 637 390 137 589 76;
  • 17) 0.000 048 637 390 137 589 76 × 2 = 0 + 0.000 097 274 780 275 179 52;
  • 18) 0.000 097 274 780 275 179 52 × 2 = 0 + 0.000 194 549 560 550 359 04;
  • 19) 0.000 194 549 560 550 359 04 × 2 = 0 + 0.000 389 099 121 100 718 08;
  • 20) 0.000 389 099 121 100 718 08 × 2 = 0 + 0.000 778 198 242 201 436 16;
  • 21) 0.000 778 198 242 201 436 16 × 2 = 0 + 0.001 556 396 484 402 872 32;
  • 22) 0.001 556 396 484 402 872 32 × 2 = 0 + 0.003 112 792 968 805 744 64;
  • 23) 0.003 112 792 968 805 744 64 × 2 = 0 + 0.006 225 585 937 611 489 28;
  • 24) 0.006 225 585 937 611 489 28 × 2 = 0 + 0.012 451 171 875 222 978 56;
  • 25) 0.012 451 171 875 222 978 56 × 2 = 0 + 0.024 902 343 750 445 957 12;
  • 26) 0.024 902 343 750 445 957 12 × 2 = 0 + 0.049 804 687 500 891 914 24;
  • 27) 0.049 804 687 500 891 914 24 × 2 = 0 + 0.099 609 375 001 783 828 48;
  • 28) 0.099 609 375 001 783 828 48 × 2 = 0 + 0.199 218 750 003 567 656 96;
  • 29) 0.199 218 750 003 567 656 96 × 2 = 0 + 0.398 437 500 007 135 313 92;
  • 30) 0.398 437 500 007 135 313 92 × 2 = 0 + 0.796 875 000 014 270 627 84;
  • 31) 0.796 875 000 014 270 627 84 × 2 = 1 + 0.593 750 000 028 541 255 68;
  • 32) 0.593 750 000 028 541 255 68 × 2 = 1 + 0.187 500 000 057 082 511 36;
  • 33) 0.187 500 000 057 082 511 36 × 2 = 0 + 0.375 000 000 114 165 022 72;
  • 34) 0.375 000 000 114 165 022 72 × 2 = 0 + 0.750 000 000 228 330 045 44;
  • 35) 0.750 000 000 228 330 045 44 × 2 = 1 + 0.500 000 000 456 660 090 88;
  • 36) 0.500 000 000 456 660 090 88 × 2 = 1 + 0.000 000 000 913 320 181 76;
  • 37) 0.000 000 000 913 320 181 76 × 2 = 0 + 0.000 000 001 826 640 363 52;
  • 38) 0.000 000 001 826 640 363 52 × 2 = 0 + 0.000 000 003 653 280 727 04;
  • 39) 0.000 000 003 653 280 727 04 × 2 = 0 + 0.000 000 007 306 561 454 08;
  • 40) 0.000 000 007 306 561 454 08 × 2 = 0 + 0.000 000 014 613 122 908 16;
  • 41) 0.000 000 014 613 122 908 16 × 2 = 0 + 0.000 000 029 226 245 816 32;
  • 42) 0.000 000 029 226 245 816 32 × 2 = 0 + 0.000 000 058 452 491 632 64;
  • 43) 0.000 000 058 452 491 632 64 × 2 = 0 + 0.000 000 116 904 983 265 28;
  • 44) 0.000 000 116 904 983 265 28 × 2 = 0 + 0.000 000 233 809 966 530 56;
  • 45) 0.000 000 233 809 966 530 56 × 2 = 0 + 0.000 000 467 619 933 061 12;
  • 46) 0.000 000 467 619 933 061 12 × 2 = 0 + 0.000 000 935 239 866 122 24;
  • 47) 0.000 000 935 239 866 122 24 × 2 = 0 + 0.000 001 870 479 732 244 48;
  • 48) 0.000 001 870 479 732 244 48 × 2 = 0 + 0.000 003 740 959 464 488 96;
  • 49) 0.000 003 740 959 464 488 96 × 2 = 0 + 0.000 007 481 918 928 977 92;
  • 50) 0.000 007 481 918 928 977 92 × 2 = 0 + 0.000 014 963 837 857 955 84;
  • 51) 0.000 014 963 837 857 955 84 × 2 = 0 + 0.000 029 927 675 715 911 68;
  • 52) 0.000 029 927 675 715 911 68 × 2 = 0 + 0.000 059 855 351 431 823 36;
  • 53) 0.000 059 855 351 431 823 36 × 2 = 0 + 0.000 119 710 702 863 646 72;
  • 54) 0.000 119 710 702 863 646 72 × 2 = 0 + 0.000 239 421 405 727 293 44;

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:


0.000 000 000 742 147 676 66(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 66(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 742 147 676 66(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2) × 20 =

1.1001 1000 0000 0000 0000 000(2) × 2-31


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): -31

Mantissa (not normalized):
1.1001 1000 0000 0000 0000 000


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(8-1) - 1 =

(-31 + 127)(10) =

96(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;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 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) =

96(10) =

0110 0000(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. 100 1100 0000 0000 0000 0000 =

100 1100 0000 0000 0000 0000


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 0000

Mantissa (23 bits) =
100 1100 0000 0000 0000 0000


Decimal number -0.000 000 000 742 147 676 66 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1100 0000 0000 0000 0000

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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