-0.000 000 000 742 147 676 646 713 9 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 713 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
-0.000 000 000 742 147 676 646 713 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:

|-0.000 000 000 742 147 676 646 713 9| = 0.000 000 000 742 147 676 646 713 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: 0.000 000 000 742 147 676 646 713 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) 0.000 000 000 742 147 676 646 713 9 × 2 = 0 + 0.000 000 001 484 295 353 293 427 8;
  • 2) 0.000 000 001 484 295 353 293 427 8 × 2 = 0 + 0.000 000 002 968 590 706 586 855 6;
  • 3) 0.000 000 002 968 590 706 586 855 6 × 2 = 0 + 0.000 000 005 937 181 413 173 711 2;
  • 4) 0.000 000 005 937 181 413 173 711 2 × 2 = 0 + 0.000 000 011 874 362 826 347 422 4;
  • 5) 0.000 000 011 874 362 826 347 422 4 × 2 = 0 + 0.000 000 023 748 725 652 694 844 8;
  • 6) 0.000 000 023 748 725 652 694 844 8 × 2 = 0 + 0.000 000 047 497 451 305 389 689 6;
  • 7) 0.000 000 047 497 451 305 389 689 6 × 2 = 0 + 0.000 000 094 994 902 610 779 379 2;
  • 8) 0.000 000 094 994 902 610 779 379 2 × 2 = 0 + 0.000 000 189 989 805 221 558 758 4;
  • 9) 0.000 000 189 989 805 221 558 758 4 × 2 = 0 + 0.000 000 379 979 610 443 117 516 8;
  • 10) 0.000 000 379 979 610 443 117 516 8 × 2 = 0 + 0.000 000 759 959 220 886 235 033 6;
  • 11) 0.000 000 759 959 220 886 235 033 6 × 2 = 0 + 0.000 001 519 918 441 772 470 067 2;
  • 12) 0.000 001 519 918 441 772 470 067 2 × 2 = 0 + 0.000 003 039 836 883 544 940 134 4;
  • 13) 0.000 003 039 836 883 544 940 134 4 × 2 = 0 + 0.000 006 079 673 767 089 880 268 8;
  • 14) 0.000 006 079 673 767 089 880 268 8 × 2 = 0 + 0.000 012 159 347 534 179 760 537 6;
  • 15) 0.000 012 159 347 534 179 760 537 6 × 2 = 0 + 0.000 024 318 695 068 359 521 075 2;
  • 16) 0.000 024 318 695 068 359 521 075 2 × 2 = 0 + 0.000 048 637 390 136 719 042 150 4;
  • 17) 0.000 048 637 390 136 719 042 150 4 × 2 = 0 + 0.000 097 274 780 273 438 084 300 8;
  • 18) 0.000 097 274 780 273 438 084 300 8 × 2 = 0 + 0.000 194 549 560 546 876 168 601 6;
  • 19) 0.000 194 549 560 546 876 168 601 6 × 2 = 0 + 0.000 389 099 121 093 752 337 203 2;
  • 20) 0.000 389 099 121 093 752 337 203 2 × 2 = 0 + 0.000 778 198 242 187 504 674 406 4;
  • 21) 0.000 778 198 242 187 504 674 406 4 × 2 = 0 + 0.001 556 396 484 375 009 348 812 8;
  • 22) 0.001 556 396 484 375 009 348 812 8 × 2 = 0 + 0.003 112 792 968 750 018 697 625 6;
  • 23) 0.003 112 792 968 750 018 697 625 6 × 2 = 0 + 0.006 225 585 937 500 037 395 251 2;
  • 24) 0.006 225 585 937 500 037 395 251 2 × 2 = 0 + 0.012 451 171 875 000 074 790 502 4;
  • 25) 0.012 451 171 875 000 074 790 502 4 × 2 = 0 + 0.024 902 343 750 000 149 581 004 8;
  • 26) 0.024 902 343 750 000 149 581 004 8 × 2 = 0 + 0.049 804 687 500 000 299 162 009 6;
  • 27) 0.049 804 687 500 000 299 162 009 6 × 2 = 0 + 0.099 609 375 000 000 598 324 019 2;
  • 28) 0.099 609 375 000 000 598 324 019 2 × 2 = 0 + 0.199 218 750 000 001 196 648 038 4;
  • 29) 0.199 218 750 000 001 196 648 038 4 × 2 = 0 + 0.398 437 500 000 002 393 296 076 8;
  • 30) 0.398 437 500 000 002 393 296 076 8 × 2 = 0 + 0.796 875 000 000 004 786 592 153 6;
  • 31) 0.796 875 000 000 004 786 592 153 6 × 2 = 1 + 0.593 750 000 000 009 573 184 307 2;
  • 32) 0.593 750 000 000 009 573 184 307 2 × 2 = 1 + 0.187 500 000 000 019 146 368 614 4;
  • 33) 0.187 500 000 000 019 146 368 614 4 × 2 = 0 + 0.375 000 000 000 038 292 737 228 8;
  • 34) 0.375 000 000 000 038 292 737 228 8 × 2 = 0 + 0.750 000 000 000 076 585 474 457 6;
  • 35) 0.750 000 000 000 076 585 474 457 6 × 2 = 1 + 0.500 000 000 000 153 170 948 915 2;
  • 36) 0.500 000 000 000 153 170 948 915 2 × 2 = 1 + 0.000 000 000 000 306 341 897 830 4;
  • 37) 0.000 000 000 000 306 341 897 830 4 × 2 = 0 + 0.000 000 000 000 612 683 795 660 8;
  • 38) 0.000 000 000 000 612 683 795 660 8 × 2 = 0 + 0.000 000 000 001 225 367 591 321 6;
  • 39) 0.000 000 000 001 225 367 591 321 6 × 2 = 0 + 0.000 000 000 002 450 735 182 643 2;
  • 40) 0.000 000 000 002 450 735 182 643 2 × 2 = 0 + 0.000 000 000 004 901 470 365 286 4;
  • 41) 0.000 000 000 004 901 470 365 286 4 × 2 = 0 + 0.000 000 000 009 802 940 730 572 8;
  • 42) 0.000 000 000 009 802 940 730 572 8 × 2 = 0 + 0.000 000 000 019 605 881 461 145 6;
  • 43) 0.000 000 000 019 605 881 461 145 6 × 2 = 0 + 0.000 000 000 039 211 762 922 291 2;
  • 44) 0.000 000 000 039 211 762 922 291 2 × 2 = 0 + 0.000 000 000 078 423 525 844 582 4;
  • 45) 0.000 000 000 078 423 525 844 582 4 × 2 = 0 + 0.000 000 000 156 847 051 689 164 8;
  • 46) 0.000 000 000 156 847 051 689 164 8 × 2 = 0 + 0.000 000 000 313 694 103 378 329 6;
  • 47) 0.000 000 000 313 694 103 378 329 6 × 2 = 0 + 0.000 000 000 627 388 206 756 659 2;
  • 48) 0.000 000 000 627 388 206 756 659 2 × 2 = 0 + 0.000 000 001 254 776 413 513 318 4;
  • 49) 0.000 000 001 254 776 413 513 318 4 × 2 = 0 + 0.000 000 002 509 552 827 026 636 8;
  • 50) 0.000 000 002 509 552 827 026 636 8 × 2 = 0 + 0.000 000 005 019 105 654 053 273 6;
  • 51) 0.000 000 005 019 105 654 053 273 6 × 2 = 0 + 0.000 000 010 038 211 308 106 547 2;
  • 52) 0.000 000 010 038 211 308 106 547 2 × 2 = 0 + 0.000 000 020 076 422 616 213 094 4;
  • 53) 0.000 000 020 076 422 616 213 094 4 × 2 = 0 + 0.000 000 040 152 845 232 426 188 8;
  • 54) 0.000 000 040 152 845 232 426 188 8 × 2 = 0 + 0.000 000 080 305 690 464 852 377 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:


0.000 000 000 742 147 676 646 713 9(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 713 9(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:


0.000 000 000 742 147 676 646 713 9(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 -0.000 000 000 742 147 676 646 713 9 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