-0.000 000 000 742 147 676 377 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 377(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
-0.000 000 000 742 147 676 377(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:

|-0.000 000 000 742 147 676 377| = 0.000 000 000 742 147 676 377


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: 0.000 000 000 742 147 676 377.

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) 0.000 000 000 742 147 676 377 × 2 = 0 + 0.000 000 001 484 295 352 754;
  • 2) 0.000 000 001 484 295 352 754 × 2 = 0 + 0.000 000 002 968 590 705 508;
  • 3) 0.000 000 002 968 590 705 508 × 2 = 0 + 0.000 000 005 937 181 411 016;
  • 4) 0.000 000 005 937 181 411 016 × 2 = 0 + 0.000 000 011 874 362 822 032;
  • 5) 0.000 000 011 874 362 822 032 × 2 = 0 + 0.000 000 023 748 725 644 064;
  • 6) 0.000 000 023 748 725 644 064 × 2 = 0 + 0.000 000 047 497 451 288 128;
  • 7) 0.000 000 047 497 451 288 128 × 2 = 0 + 0.000 000 094 994 902 576 256;
  • 8) 0.000 000 094 994 902 576 256 × 2 = 0 + 0.000 000 189 989 805 152 512;
  • 9) 0.000 000 189 989 805 152 512 × 2 = 0 + 0.000 000 379 979 610 305 024;
  • 10) 0.000 000 379 979 610 305 024 × 2 = 0 + 0.000 000 759 959 220 610 048;
  • 11) 0.000 000 759 959 220 610 048 × 2 = 0 + 0.000 001 519 918 441 220 096;
  • 12) 0.000 001 519 918 441 220 096 × 2 = 0 + 0.000 003 039 836 882 440 192;
  • 13) 0.000 003 039 836 882 440 192 × 2 = 0 + 0.000 006 079 673 764 880 384;
  • 14) 0.000 006 079 673 764 880 384 × 2 = 0 + 0.000 012 159 347 529 760 768;
  • 15) 0.000 012 159 347 529 760 768 × 2 = 0 + 0.000 024 318 695 059 521 536;
  • 16) 0.000 024 318 695 059 521 536 × 2 = 0 + 0.000 048 637 390 119 043 072;
  • 17) 0.000 048 637 390 119 043 072 × 2 = 0 + 0.000 097 274 780 238 086 144;
  • 18) 0.000 097 274 780 238 086 144 × 2 = 0 + 0.000 194 549 560 476 172 288;
  • 19) 0.000 194 549 560 476 172 288 × 2 = 0 + 0.000 389 099 120 952 344 576;
  • 20) 0.000 389 099 120 952 344 576 × 2 = 0 + 0.000 778 198 241 904 689 152;
  • 21) 0.000 778 198 241 904 689 152 × 2 = 0 + 0.001 556 396 483 809 378 304;
  • 22) 0.001 556 396 483 809 378 304 × 2 = 0 + 0.003 112 792 967 618 756 608;
  • 23) 0.003 112 792 967 618 756 608 × 2 = 0 + 0.006 225 585 935 237 513 216;
  • 24) 0.006 225 585 935 237 513 216 × 2 = 0 + 0.012 451 171 870 475 026 432;
  • 25) 0.012 451 171 870 475 026 432 × 2 = 0 + 0.024 902 343 740 950 052 864;
  • 26) 0.024 902 343 740 950 052 864 × 2 = 0 + 0.049 804 687 481 900 105 728;
  • 27) 0.049 804 687 481 900 105 728 × 2 = 0 + 0.099 609 374 963 800 211 456;
  • 28) 0.099 609 374 963 800 211 456 × 2 = 0 + 0.199 218 749 927 600 422 912;
  • 29) 0.199 218 749 927 600 422 912 × 2 = 0 + 0.398 437 499 855 200 845 824;
  • 30) 0.398 437 499 855 200 845 824 × 2 = 0 + 0.796 874 999 710 401 691 648;
  • 31) 0.796 874 999 710 401 691 648 × 2 = 1 + 0.593 749 999 420 803 383 296;
  • 32) 0.593 749 999 420 803 383 296 × 2 = 1 + 0.187 499 998 841 606 766 592;
  • 33) 0.187 499 998 841 606 766 592 × 2 = 0 + 0.374 999 997 683 213 533 184;
  • 34) 0.374 999 997 683 213 533 184 × 2 = 0 + 0.749 999 995 366 427 066 368;
  • 35) 0.749 999 995 366 427 066 368 × 2 = 1 + 0.499 999 990 732 854 132 736;
  • 36) 0.499 999 990 732 854 132 736 × 2 = 0 + 0.999 999 981 465 708 265 472;
  • 37) 0.999 999 981 465 708 265 472 × 2 = 1 + 0.999 999 962 931 416 530 944;
  • 38) 0.999 999 962 931 416 530 944 × 2 = 1 + 0.999 999 925 862 833 061 888;
  • 39) 0.999 999 925 862 833 061 888 × 2 = 1 + 0.999 999 851 725 666 123 776;
  • 40) 0.999 999 851 725 666 123 776 × 2 = 1 + 0.999 999 703 451 332 247 552;
  • 41) 0.999 999 703 451 332 247 552 × 2 = 1 + 0.999 999 406 902 664 495 104;
  • 42) 0.999 999 406 902 664 495 104 × 2 = 1 + 0.999 998 813 805 328 990 208;
  • 43) 0.999 998 813 805 328 990 208 × 2 = 1 + 0.999 997 627 610 657 980 416;
  • 44) 0.999 997 627 610 657 980 416 × 2 = 1 + 0.999 995 255 221 315 960 832;
  • 45) 0.999 995 255 221 315 960 832 × 2 = 1 + 0.999 990 510 442 631 921 664;
  • 46) 0.999 990 510 442 631 921 664 × 2 = 1 + 0.999 981 020 885 263 843 328;
  • 47) 0.999 981 020 885 263 843 328 × 2 = 1 + 0.999 962 041 770 527 686 656;
  • 48) 0.999 962 041 770 527 686 656 × 2 = 1 + 0.999 924 083 541 055 373 312;
  • 49) 0.999 924 083 541 055 373 312 × 2 = 1 + 0.999 848 167 082 110 746 624;
  • 50) 0.999 848 167 082 110 746 624 × 2 = 1 + 0.999 696 334 164 221 493 248;
  • 51) 0.999 696 334 164 221 493 248 × 2 = 1 + 0.999 392 668 328 442 986 496;
  • 52) 0.999 392 668 328 442 986 496 × 2 = 1 + 0.998 785 336 656 885 972 992;
  • 53) 0.998 785 336 656 885 972 992 × 2 = 1 + 0.997 570 673 313 771 945 984;
  • 54) 0.997 570 673 313 771 945 984 × 2 = 1 + 0.995 141 346 627 543 891 968;

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:


0.000 000 000 742 147 676 377(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 377(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:


0.000 000 000 742 147 676 377(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 -0.000 000 000 742 147 676 377 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