-0.000 000 000 742 147 676 585 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 585(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
-0.000 000 000 742 147 676 585(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:

|-0.000 000 000 742 147 676 585| = 0.000 000 000 742 147 676 585


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: 0.000 000 000 742 147 676 585.

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) 0.000 000 000 742 147 676 585 × 2 = 0 + 0.000 000 001 484 295 353 17;
  • 2) 0.000 000 001 484 295 353 17 × 2 = 0 + 0.000 000 002 968 590 706 34;
  • 3) 0.000 000 002 968 590 706 34 × 2 = 0 + 0.000 000 005 937 181 412 68;
  • 4) 0.000 000 005 937 181 412 68 × 2 = 0 + 0.000 000 011 874 362 825 36;
  • 5) 0.000 000 011 874 362 825 36 × 2 = 0 + 0.000 000 023 748 725 650 72;
  • 6) 0.000 000 023 748 725 650 72 × 2 = 0 + 0.000 000 047 497 451 301 44;
  • 7) 0.000 000 047 497 451 301 44 × 2 = 0 + 0.000 000 094 994 902 602 88;
  • 8) 0.000 000 094 994 902 602 88 × 2 = 0 + 0.000 000 189 989 805 205 76;
  • 9) 0.000 000 189 989 805 205 76 × 2 = 0 + 0.000 000 379 979 610 411 52;
  • 10) 0.000 000 379 979 610 411 52 × 2 = 0 + 0.000 000 759 959 220 823 04;
  • 11) 0.000 000 759 959 220 823 04 × 2 = 0 + 0.000 001 519 918 441 646 08;
  • 12) 0.000 001 519 918 441 646 08 × 2 = 0 + 0.000 003 039 836 883 292 16;
  • 13) 0.000 003 039 836 883 292 16 × 2 = 0 + 0.000 006 079 673 766 584 32;
  • 14) 0.000 006 079 673 766 584 32 × 2 = 0 + 0.000 012 159 347 533 168 64;
  • 15) 0.000 012 159 347 533 168 64 × 2 = 0 + 0.000 024 318 695 066 337 28;
  • 16) 0.000 024 318 695 066 337 28 × 2 = 0 + 0.000 048 637 390 132 674 56;
  • 17) 0.000 048 637 390 132 674 56 × 2 = 0 + 0.000 097 274 780 265 349 12;
  • 18) 0.000 097 274 780 265 349 12 × 2 = 0 + 0.000 194 549 560 530 698 24;
  • 19) 0.000 194 549 560 530 698 24 × 2 = 0 + 0.000 389 099 121 061 396 48;
  • 20) 0.000 389 099 121 061 396 48 × 2 = 0 + 0.000 778 198 242 122 792 96;
  • 21) 0.000 778 198 242 122 792 96 × 2 = 0 + 0.001 556 396 484 245 585 92;
  • 22) 0.001 556 396 484 245 585 92 × 2 = 0 + 0.003 112 792 968 491 171 84;
  • 23) 0.003 112 792 968 491 171 84 × 2 = 0 + 0.006 225 585 936 982 343 68;
  • 24) 0.006 225 585 936 982 343 68 × 2 = 0 + 0.012 451 171 873 964 687 36;
  • 25) 0.012 451 171 873 964 687 36 × 2 = 0 + 0.024 902 343 747 929 374 72;
  • 26) 0.024 902 343 747 929 374 72 × 2 = 0 + 0.049 804 687 495 858 749 44;
  • 27) 0.049 804 687 495 858 749 44 × 2 = 0 + 0.099 609 374 991 717 498 88;
  • 28) 0.099 609 374 991 717 498 88 × 2 = 0 + 0.199 218 749 983 434 997 76;
  • 29) 0.199 218 749 983 434 997 76 × 2 = 0 + 0.398 437 499 966 869 995 52;
  • 30) 0.398 437 499 966 869 995 52 × 2 = 0 + 0.796 874 999 933 739 991 04;
  • 31) 0.796 874 999 933 739 991 04 × 2 = 1 + 0.593 749 999 867 479 982 08;
  • 32) 0.593 749 999 867 479 982 08 × 2 = 1 + 0.187 499 999 734 959 964 16;
  • 33) 0.187 499 999 734 959 964 16 × 2 = 0 + 0.374 999 999 469 919 928 32;
  • 34) 0.374 999 999 469 919 928 32 × 2 = 0 + 0.749 999 998 939 839 856 64;
  • 35) 0.749 999 998 939 839 856 64 × 2 = 1 + 0.499 999 997 879 679 713 28;
  • 36) 0.499 999 997 879 679 713 28 × 2 = 0 + 0.999 999 995 759 359 426 56;
  • 37) 0.999 999 995 759 359 426 56 × 2 = 1 + 0.999 999 991 518 718 853 12;
  • 38) 0.999 999 991 518 718 853 12 × 2 = 1 + 0.999 999 983 037 437 706 24;
  • 39) 0.999 999 983 037 437 706 24 × 2 = 1 + 0.999 999 966 074 875 412 48;
  • 40) 0.999 999 966 074 875 412 48 × 2 = 1 + 0.999 999 932 149 750 824 96;
  • 41) 0.999 999 932 149 750 824 96 × 2 = 1 + 0.999 999 864 299 501 649 92;
  • 42) 0.999 999 864 299 501 649 92 × 2 = 1 + 0.999 999 728 599 003 299 84;
  • 43) 0.999 999 728 599 003 299 84 × 2 = 1 + 0.999 999 457 198 006 599 68;
  • 44) 0.999 999 457 198 006 599 68 × 2 = 1 + 0.999 998 914 396 013 199 36;
  • 45) 0.999 998 914 396 013 199 36 × 2 = 1 + 0.999 997 828 792 026 398 72;
  • 46) 0.999 997 828 792 026 398 72 × 2 = 1 + 0.999 995 657 584 052 797 44;
  • 47) 0.999 995 657 584 052 797 44 × 2 = 1 + 0.999 991 315 168 105 594 88;
  • 48) 0.999 991 315 168 105 594 88 × 2 = 1 + 0.999 982 630 336 211 189 76;
  • 49) 0.999 982 630 336 211 189 76 × 2 = 1 + 0.999 965 260 672 422 379 52;
  • 50) 0.999 965 260 672 422 379 52 × 2 = 1 + 0.999 930 521 344 844 759 04;
  • 51) 0.999 930 521 344 844 759 04 × 2 = 1 + 0.999 861 042 689 689 518 08;
  • 52) 0.999 861 042 689 689 518 08 × 2 = 1 + 0.999 722 085 379 379 036 16;
  • 53) 0.999 722 085 379 379 036 16 × 2 = 1 + 0.999 444 170 758 758 072 32;
  • 54) 0.999 444 170 758 758 072 32 × 2 = 1 + 0.998 888 341 517 516 144 64;

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:


0.000 000 000 742 147 676 585(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 585(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(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:


0.000 000 000 742 147 676 585(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) × 20 =

1.1001 0111 1111 1111 1111 111(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 0111 1111 1111 1111 111


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 1011 1111 1111 1111 1111 =

100 1011 1111 1111 1111 1111


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 1011 1111 1111 1111 1111


Decimal number -0.000 000 000 742 147 676 585 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1011 1111 1111 1111 1111

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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