-0.000 000 000 742 147 673 91 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 673 91(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
-0.000 000 000 742 147 673 91(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:

|-0.000 000 000 742 147 673 91| = 0.000 000 000 742 147 673 91


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: 0.000 000 000 742 147 673 91.

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) 0.000 000 000 742 147 673 91 × 2 = 0 + 0.000 000 001 484 295 347 82;
  • 2) 0.000 000 001 484 295 347 82 × 2 = 0 + 0.000 000 002 968 590 695 64;
  • 3) 0.000 000 002 968 590 695 64 × 2 = 0 + 0.000 000 005 937 181 391 28;
  • 4) 0.000 000 005 937 181 391 28 × 2 = 0 + 0.000 000 011 874 362 782 56;
  • 5) 0.000 000 011 874 362 782 56 × 2 = 0 + 0.000 000 023 748 725 565 12;
  • 6) 0.000 000 023 748 725 565 12 × 2 = 0 + 0.000 000 047 497 451 130 24;
  • 7) 0.000 000 047 497 451 130 24 × 2 = 0 + 0.000 000 094 994 902 260 48;
  • 8) 0.000 000 094 994 902 260 48 × 2 = 0 + 0.000 000 189 989 804 520 96;
  • 9) 0.000 000 189 989 804 520 96 × 2 = 0 + 0.000 000 379 979 609 041 92;
  • 10) 0.000 000 379 979 609 041 92 × 2 = 0 + 0.000 000 759 959 218 083 84;
  • 11) 0.000 000 759 959 218 083 84 × 2 = 0 + 0.000 001 519 918 436 167 68;
  • 12) 0.000 001 519 918 436 167 68 × 2 = 0 + 0.000 003 039 836 872 335 36;
  • 13) 0.000 003 039 836 872 335 36 × 2 = 0 + 0.000 006 079 673 744 670 72;
  • 14) 0.000 006 079 673 744 670 72 × 2 = 0 + 0.000 012 159 347 489 341 44;
  • 15) 0.000 012 159 347 489 341 44 × 2 = 0 + 0.000 024 318 694 978 682 88;
  • 16) 0.000 024 318 694 978 682 88 × 2 = 0 + 0.000 048 637 389 957 365 76;
  • 17) 0.000 048 637 389 957 365 76 × 2 = 0 + 0.000 097 274 779 914 731 52;
  • 18) 0.000 097 274 779 914 731 52 × 2 = 0 + 0.000 194 549 559 829 463 04;
  • 19) 0.000 194 549 559 829 463 04 × 2 = 0 + 0.000 389 099 119 658 926 08;
  • 20) 0.000 389 099 119 658 926 08 × 2 = 0 + 0.000 778 198 239 317 852 16;
  • 21) 0.000 778 198 239 317 852 16 × 2 = 0 + 0.001 556 396 478 635 704 32;
  • 22) 0.001 556 396 478 635 704 32 × 2 = 0 + 0.003 112 792 957 271 408 64;
  • 23) 0.003 112 792 957 271 408 64 × 2 = 0 + 0.006 225 585 914 542 817 28;
  • 24) 0.006 225 585 914 542 817 28 × 2 = 0 + 0.012 451 171 829 085 634 56;
  • 25) 0.012 451 171 829 085 634 56 × 2 = 0 + 0.024 902 343 658 171 269 12;
  • 26) 0.024 902 343 658 171 269 12 × 2 = 0 + 0.049 804 687 316 342 538 24;
  • 27) 0.049 804 687 316 342 538 24 × 2 = 0 + 0.099 609 374 632 685 076 48;
  • 28) 0.099 609 374 632 685 076 48 × 2 = 0 + 0.199 218 749 265 370 152 96;
  • 29) 0.199 218 749 265 370 152 96 × 2 = 0 + 0.398 437 498 530 740 305 92;
  • 30) 0.398 437 498 530 740 305 92 × 2 = 0 + 0.796 874 997 061 480 611 84;
  • 31) 0.796 874 997 061 480 611 84 × 2 = 1 + 0.593 749 994 122 961 223 68;
  • 32) 0.593 749 994 122 961 223 68 × 2 = 1 + 0.187 499 988 245 922 447 36;
  • 33) 0.187 499 988 245 922 447 36 × 2 = 0 + 0.374 999 976 491 844 894 72;
  • 34) 0.374 999 976 491 844 894 72 × 2 = 0 + 0.749 999 952 983 689 789 44;
  • 35) 0.749 999 952 983 689 789 44 × 2 = 1 + 0.499 999 905 967 379 578 88;
  • 36) 0.499 999 905 967 379 578 88 × 2 = 0 + 0.999 999 811 934 759 157 76;
  • 37) 0.999 999 811 934 759 157 76 × 2 = 1 + 0.999 999 623 869 518 315 52;
  • 38) 0.999 999 623 869 518 315 52 × 2 = 1 + 0.999 999 247 739 036 631 04;
  • 39) 0.999 999 247 739 036 631 04 × 2 = 1 + 0.999 998 495 478 073 262 08;
  • 40) 0.999 998 495 478 073 262 08 × 2 = 1 + 0.999 996 990 956 146 524 16;
  • 41) 0.999 996 990 956 146 524 16 × 2 = 1 + 0.999 993 981 912 293 048 32;
  • 42) 0.999 993 981 912 293 048 32 × 2 = 1 + 0.999 987 963 824 586 096 64;
  • 43) 0.999 987 963 824 586 096 64 × 2 = 1 + 0.999 975 927 649 172 193 28;
  • 44) 0.999 975 927 649 172 193 28 × 2 = 1 + 0.999 951 855 298 344 386 56;
  • 45) 0.999 951 855 298 344 386 56 × 2 = 1 + 0.999 903 710 596 688 773 12;
  • 46) 0.999 903 710 596 688 773 12 × 2 = 1 + 0.999 807 421 193 377 546 24;
  • 47) 0.999 807 421 193 377 546 24 × 2 = 1 + 0.999 614 842 386 755 092 48;
  • 48) 0.999 614 842 386 755 092 48 × 2 = 1 + 0.999 229 684 773 510 184 96;
  • 49) 0.999 229 684 773 510 184 96 × 2 = 1 + 0.998 459 369 547 020 369 92;
  • 50) 0.998 459 369 547 020 369 92 × 2 = 1 + 0.996 918 739 094 040 739 84;
  • 51) 0.996 918 739 094 040 739 84 × 2 = 1 + 0.993 837 478 188 081 479 68;
  • 52) 0.993 837 478 188 081 479 68 × 2 = 1 + 0.987 674 956 376 162 959 36;
  • 53) 0.987 674 956 376 162 959 36 × 2 = 1 + 0.975 349 912 752 325 918 72;
  • 54) 0.975 349 912 752 325 918 72 × 2 = 1 + 0.950 699 825 504 651 837 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:


0.000 000 000 742 147 673 91(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 673 91(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:


0.000 000 000 742 147 673 91(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 -0.000 000 000 742 147 673 91 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