-0.000 000 000 742 147 676 646 719 6 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 719 6(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
-0.000 000 000 742 147 676 646 719 6(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:

|-0.000 000 000 742 147 676 646 719 6| = 0.000 000 000 742 147 676 646 719 6


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: 0.000 000 000 742 147 676 646 719 6.

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) 0.000 000 000 742 147 676 646 719 6 × 2 = 0 + 0.000 000 001 484 295 353 293 439 2;
  • 2) 0.000 000 001 484 295 353 293 439 2 × 2 = 0 + 0.000 000 002 968 590 706 586 878 4;
  • 3) 0.000 000 002 968 590 706 586 878 4 × 2 = 0 + 0.000 000 005 937 181 413 173 756 8;
  • 4) 0.000 000 005 937 181 413 173 756 8 × 2 = 0 + 0.000 000 011 874 362 826 347 513 6;
  • 5) 0.000 000 011 874 362 826 347 513 6 × 2 = 0 + 0.000 000 023 748 725 652 695 027 2;
  • 6) 0.000 000 023 748 725 652 695 027 2 × 2 = 0 + 0.000 000 047 497 451 305 390 054 4;
  • 7) 0.000 000 047 497 451 305 390 054 4 × 2 = 0 + 0.000 000 094 994 902 610 780 108 8;
  • 8) 0.000 000 094 994 902 610 780 108 8 × 2 = 0 + 0.000 000 189 989 805 221 560 217 6;
  • 9) 0.000 000 189 989 805 221 560 217 6 × 2 = 0 + 0.000 000 379 979 610 443 120 435 2;
  • 10) 0.000 000 379 979 610 443 120 435 2 × 2 = 0 + 0.000 000 759 959 220 886 240 870 4;
  • 11) 0.000 000 759 959 220 886 240 870 4 × 2 = 0 + 0.000 001 519 918 441 772 481 740 8;
  • 12) 0.000 001 519 918 441 772 481 740 8 × 2 = 0 + 0.000 003 039 836 883 544 963 481 6;
  • 13) 0.000 003 039 836 883 544 963 481 6 × 2 = 0 + 0.000 006 079 673 767 089 926 963 2;
  • 14) 0.000 006 079 673 767 089 926 963 2 × 2 = 0 + 0.000 012 159 347 534 179 853 926 4;
  • 15) 0.000 012 159 347 534 179 853 926 4 × 2 = 0 + 0.000 024 318 695 068 359 707 852 8;
  • 16) 0.000 024 318 695 068 359 707 852 8 × 2 = 0 + 0.000 048 637 390 136 719 415 705 6;
  • 17) 0.000 048 637 390 136 719 415 705 6 × 2 = 0 + 0.000 097 274 780 273 438 831 411 2;
  • 18) 0.000 097 274 780 273 438 831 411 2 × 2 = 0 + 0.000 194 549 560 546 877 662 822 4;
  • 19) 0.000 194 549 560 546 877 662 822 4 × 2 = 0 + 0.000 389 099 121 093 755 325 644 8;
  • 20) 0.000 389 099 121 093 755 325 644 8 × 2 = 0 + 0.000 778 198 242 187 510 651 289 6;
  • 21) 0.000 778 198 242 187 510 651 289 6 × 2 = 0 + 0.001 556 396 484 375 021 302 579 2;
  • 22) 0.001 556 396 484 375 021 302 579 2 × 2 = 0 + 0.003 112 792 968 750 042 605 158 4;
  • 23) 0.003 112 792 968 750 042 605 158 4 × 2 = 0 + 0.006 225 585 937 500 085 210 316 8;
  • 24) 0.006 225 585 937 500 085 210 316 8 × 2 = 0 + 0.012 451 171 875 000 170 420 633 6;
  • 25) 0.012 451 171 875 000 170 420 633 6 × 2 = 0 + 0.024 902 343 750 000 340 841 267 2;
  • 26) 0.024 902 343 750 000 340 841 267 2 × 2 = 0 + 0.049 804 687 500 000 681 682 534 4;
  • 27) 0.049 804 687 500 000 681 682 534 4 × 2 = 0 + 0.099 609 375 000 001 363 365 068 8;
  • 28) 0.099 609 375 000 001 363 365 068 8 × 2 = 0 + 0.199 218 750 000 002 726 730 137 6;
  • 29) 0.199 218 750 000 002 726 730 137 6 × 2 = 0 + 0.398 437 500 000 005 453 460 275 2;
  • 30) 0.398 437 500 000 005 453 460 275 2 × 2 = 0 + 0.796 875 000 000 010 906 920 550 4;
  • 31) 0.796 875 000 000 010 906 920 550 4 × 2 = 1 + 0.593 750 000 000 021 813 841 100 8;
  • 32) 0.593 750 000 000 021 813 841 100 8 × 2 = 1 + 0.187 500 000 000 043 627 682 201 6;
  • 33) 0.187 500 000 000 043 627 682 201 6 × 2 = 0 + 0.375 000 000 000 087 255 364 403 2;
  • 34) 0.375 000 000 000 087 255 364 403 2 × 2 = 0 + 0.750 000 000 000 174 510 728 806 4;
  • 35) 0.750 000 000 000 174 510 728 806 4 × 2 = 1 + 0.500 000 000 000 349 021 457 612 8;
  • 36) 0.500 000 000 000 349 021 457 612 8 × 2 = 1 + 0.000 000 000 000 698 042 915 225 6;
  • 37) 0.000 000 000 000 698 042 915 225 6 × 2 = 0 + 0.000 000 000 001 396 085 830 451 2;
  • 38) 0.000 000 000 001 396 085 830 451 2 × 2 = 0 + 0.000 000 000 002 792 171 660 902 4;
  • 39) 0.000 000 000 002 792 171 660 902 4 × 2 = 0 + 0.000 000 000 005 584 343 321 804 8;
  • 40) 0.000 000 000 005 584 343 321 804 8 × 2 = 0 + 0.000 000 000 011 168 686 643 609 6;
  • 41) 0.000 000 000 011 168 686 643 609 6 × 2 = 0 + 0.000 000 000 022 337 373 287 219 2;
  • 42) 0.000 000 000 022 337 373 287 219 2 × 2 = 0 + 0.000 000 000 044 674 746 574 438 4;
  • 43) 0.000 000 000 044 674 746 574 438 4 × 2 = 0 + 0.000 000 000 089 349 493 148 876 8;
  • 44) 0.000 000 000 089 349 493 148 876 8 × 2 = 0 + 0.000 000 000 178 698 986 297 753 6;
  • 45) 0.000 000 000 178 698 986 297 753 6 × 2 = 0 + 0.000 000 000 357 397 972 595 507 2;
  • 46) 0.000 000 000 357 397 972 595 507 2 × 2 = 0 + 0.000 000 000 714 795 945 191 014 4;
  • 47) 0.000 000 000 714 795 945 191 014 4 × 2 = 0 + 0.000 000 001 429 591 890 382 028 8;
  • 48) 0.000 000 001 429 591 890 382 028 8 × 2 = 0 + 0.000 000 002 859 183 780 764 057 6;
  • 49) 0.000 000 002 859 183 780 764 057 6 × 2 = 0 + 0.000 000 005 718 367 561 528 115 2;
  • 50) 0.000 000 005 718 367 561 528 115 2 × 2 = 0 + 0.000 000 011 436 735 123 056 230 4;
  • 51) 0.000 000 011 436 735 123 056 230 4 × 2 = 0 + 0.000 000 022 873 470 246 112 460 8;
  • 52) 0.000 000 022 873 470 246 112 460 8 × 2 = 0 + 0.000 000 045 746 940 492 224 921 6;
  • 53) 0.000 000 045 746 940 492 224 921 6 × 2 = 0 + 0.000 000 091 493 880 984 449 843 2;
  • 54) 0.000 000 091 493 880 984 449 843 2 × 2 = 0 + 0.000 000 182 987 761 968 899 686 4;

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:


0.000 000 000 742 147 676 646 719 6(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 719 6(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:


0.000 000 000 742 147 676 646 719 6(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 -0.000 000 000 742 147 676 646 719 6 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