-0.000 000 000 742 147 676 635 9 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 635 9(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
-0.000 000 000 742 147 676 635 9(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:

|-0.000 000 000 742 147 676 635 9| = 0.000 000 000 742 147 676 635 9


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: 0.000 000 000 742 147 676 635 9.

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) 0.000 000 000 742 147 676 635 9 × 2 = 0 + 0.000 000 001 484 295 353 271 8;
  • 2) 0.000 000 001 484 295 353 271 8 × 2 = 0 + 0.000 000 002 968 590 706 543 6;
  • 3) 0.000 000 002 968 590 706 543 6 × 2 = 0 + 0.000 000 005 937 181 413 087 2;
  • 4) 0.000 000 005 937 181 413 087 2 × 2 = 0 + 0.000 000 011 874 362 826 174 4;
  • 5) 0.000 000 011 874 362 826 174 4 × 2 = 0 + 0.000 000 023 748 725 652 348 8;
  • 6) 0.000 000 023 748 725 652 348 8 × 2 = 0 + 0.000 000 047 497 451 304 697 6;
  • 7) 0.000 000 047 497 451 304 697 6 × 2 = 0 + 0.000 000 094 994 902 609 395 2;
  • 8) 0.000 000 094 994 902 609 395 2 × 2 = 0 + 0.000 000 189 989 805 218 790 4;
  • 9) 0.000 000 189 989 805 218 790 4 × 2 = 0 + 0.000 000 379 979 610 437 580 8;
  • 10) 0.000 000 379 979 610 437 580 8 × 2 = 0 + 0.000 000 759 959 220 875 161 6;
  • 11) 0.000 000 759 959 220 875 161 6 × 2 = 0 + 0.000 001 519 918 441 750 323 2;
  • 12) 0.000 001 519 918 441 750 323 2 × 2 = 0 + 0.000 003 039 836 883 500 646 4;
  • 13) 0.000 003 039 836 883 500 646 4 × 2 = 0 + 0.000 006 079 673 767 001 292 8;
  • 14) 0.000 006 079 673 767 001 292 8 × 2 = 0 + 0.000 012 159 347 534 002 585 6;
  • 15) 0.000 012 159 347 534 002 585 6 × 2 = 0 + 0.000 024 318 695 068 005 171 2;
  • 16) 0.000 024 318 695 068 005 171 2 × 2 = 0 + 0.000 048 637 390 136 010 342 4;
  • 17) 0.000 048 637 390 136 010 342 4 × 2 = 0 + 0.000 097 274 780 272 020 684 8;
  • 18) 0.000 097 274 780 272 020 684 8 × 2 = 0 + 0.000 194 549 560 544 041 369 6;
  • 19) 0.000 194 549 560 544 041 369 6 × 2 = 0 + 0.000 389 099 121 088 082 739 2;
  • 20) 0.000 389 099 121 088 082 739 2 × 2 = 0 + 0.000 778 198 242 176 165 478 4;
  • 21) 0.000 778 198 242 176 165 478 4 × 2 = 0 + 0.001 556 396 484 352 330 956 8;
  • 22) 0.001 556 396 484 352 330 956 8 × 2 = 0 + 0.003 112 792 968 704 661 913 6;
  • 23) 0.003 112 792 968 704 661 913 6 × 2 = 0 + 0.006 225 585 937 409 323 827 2;
  • 24) 0.006 225 585 937 409 323 827 2 × 2 = 0 + 0.012 451 171 874 818 647 654 4;
  • 25) 0.012 451 171 874 818 647 654 4 × 2 = 0 + 0.024 902 343 749 637 295 308 8;
  • 26) 0.024 902 343 749 637 295 308 8 × 2 = 0 + 0.049 804 687 499 274 590 617 6;
  • 27) 0.049 804 687 499 274 590 617 6 × 2 = 0 + 0.099 609 374 998 549 181 235 2;
  • 28) 0.099 609 374 998 549 181 235 2 × 2 = 0 + 0.199 218 749 997 098 362 470 4;
  • 29) 0.199 218 749 997 098 362 470 4 × 2 = 0 + 0.398 437 499 994 196 724 940 8;
  • 30) 0.398 437 499 994 196 724 940 8 × 2 = 0 + 0.796 874 999 988 393 449 881 6;
  • 31) 0.796 874 999 988 393 449 881 6 × 2 = 1 + 0.593 749 999 976 786 899 763 2;
  • 32) 0.593 749 999 976 786 899 763 2 × 2 = 1 + 0.187 499 999 953 573 799 526 4;
  • 33) 0.187 499 999 953 573 799 526 4 × 2 = 0 + 0.374 999 999 907 147 599 052 8;
  • 34) 0.374 999 999 907 147 599 052 8 × 2 = 0 + 0.749 999 999 814 295 198 105 6;
  • 35) 0.749 999 999 814 295 198 105 6 × 2 = 1 + 0.499 999 999 628 590 396 211 2;
  • 36) 0.499 999 999 628 590 396 211 2 × 2 = 0 + 0.999 999 999 257 180 792 422 4;
  • 37) 0.999 999 999 257 180 792 422 4 × 2 = 1 + 0.999 999 998 514 361 584 844 8;
  • 38) 0.999 999 998 514 361 584 844 8 × 2 = 1 + 0.999 999 997 028 723 169 689 6;
  • 39) 0.999 999 997 028 723 169 689 6 × 2 = 1 + 0.999 999 994 057 446 339 379 2;
  • 40) 0.999 999 994 057 446 339 379 2 × 2 = 1 + 0.999 999 988 114 892 678 758 4;
  • 41) 0.999 999 988 114 892 678 758 4 × 2 = 1 + 0.999 999 976 229 785 357 516 8;
  • 42) 0.999 999 976 229 785 357 516 8 × 2 = 1 + 0.999 999 952 459 570 715 033 6;
  • 43) 0.999 999 952 459 570 715 033 6 × 2 = 1 + 0.999 999 904 919 141 430 067 2;
  • 44) 0.999 999 904 919 141 430 067 2 × 2 = 1 + 0.999 999 809 838 282 860 134 4;
  • 45) 0.999 999 809 838 282 860 134 4 × 2 = 1 + 0.999 999 619 676 565 720 268 8;
  • 46) 0.999 999 619 676 565 720 268 8 × 2 = 1 + 0.999 999 239 353 131 440 537 6;
  • 47) 0.999 999 239 353 131 440 537 6 × 2 = 1 + 0.999 998 478 706 262 881 075 2;
  • 48) 0.999 998 478 706 262 881 075 2 × 2 = 1 + 0.999 996 957 412 525 762 150 4;
  • 49) 0.999 996 957 412 525 762 150 4 × 2 = 1 + 0.999 993 914 825 051 524 300 8;
  • 50) 0.999 993 914 825 051 524 300 8 × 2 = 1 + 0.999 987 829 650 103 048 601 6;
  • 51) 0.999 987 829 650 103 048 601 6 × 2 = 1 + 0.999 975 659 300 206 097 203 2;
  • 52) 0.999 975 659 300 206 097 203 2 × 2 = 1 + 0.999 951 318 600 412 194 406 4;
  • 53) 0.999 951 318 600 412 194 406 4 × 2 = 1 + 0.999 902 637 200 824 388 812 8;
  • 54) 0.999 902 637 200 824 388 812 8 × 2 = 1 + 0.999 805 274 401 648 777 625 6;

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:


0.000 000 000 742 147 676 635 9(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 635 9(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:


0.000 000 000 742 147 676 635 9(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 -0.000 000 000 742 147 676 635 9 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