-0.000 000 000 742 147 676 647 1 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 647 1(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
-0.000 000 000 742 147 676 647 1(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:

|-0.000 000 000 742 147 676 647 1| = 0.000 000 000 742 147 676 647 1


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: 0.000 000 000 742 147 676 647 1.

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) 0.000 000 000 742 147 676 647 1 × 2 = 0 + 0.000 000 001 484 295 353 294 2;
  • 2) 0.000 000 001 484 295 353 294 2 × 2 = 0 + 0.000 000 002 968 590 706 588 4;
  • 3) 0.000 000 002 968 590 706 588 4 × 2 = 0 + 0.000 000 005 937 181 413 176 8;
  • 4) 0.000 000 005 937 181 413 176 8 × 2 = 0 + 0.000 000 011 874 362 826 353 6;
  • 5) 0.000 000 011 874 362 826 353 6 × 2 = 0 + 0.000 000 023 748 725 652 707 2;
  • 6) 0.000 000 023 748 725 652 707 2 × 2 = 0 + 0.000 000 047 497 451 305 414 4;
  • 7) 0.000 000 047 497 451 305 414 4 × 2 = 0 + 0.000 000 094 994 902 610 828 8;
  • 8) 0.000 000 094 994 902 610 828 8 × 2 = 0 + 0.000 000 189 989 805 221 657 6;
  • 9) 0.000 000 189 989 805 221 657 6 × 2 = 0 + 0.000 000 379 979 610 443 315 2;
  • 10) 0.000 000 379 979 610 443 315 2 × 2 = 0 + 0.000 000 759 959 220 886 630 4;
  • 11) 0.000 000 759 959 220 886 630 4 × 2 = 0 + 0.000 001 519 918 441 773 260 8;
  • 12) 0.000 001 519 918 441 773 260 8 × 2 = 0 + 0.000 003 039 836 883 546 521 6;
  • 13) 0.000 003 039 836 883 546 521 6 × 2 = 0 + 0.000 006 079 673 767 093 043 2;
  • 14) 0.000 006 079 673 767 093 043 2 × 2 = 0 + 0.000 012 159 347 534 186 086 4;
  • 15) 0.000 012 159 347 534 186 086 4 × 2 = 0 + 0.000 024 318 695 068 372 172 8;
  • 16) 0.000 024 318 695 068 372 172 8 × 2 = 0 + 0.000 048 637 390 136 744 345 6;
  • 17) 0.000 048 637 390 136 744 345 6 × 2 = 0 + 0.000 097 274 780 273 488 691 2;
  • 18) 0.000 097 274 780 273 488 691 2 × 2 = 0 + 0.000 194 549 560 546 977 382 4;
  • 19) 0.000 194 549 560 546 977 382 4 × 2 = 0 + 0.000 389 099 121 093 954 764 8;
  • 20) 0.000 389 099 121 093 954 764 8 × 2 = 0 + 0.000 778 198 242 187 909 529 6;
  • 21) 0.000 778 198 242 187 909 529 6 × 2 = 0 + 0.001 556 396 484 375 819 059 2;
  • 22) 0.001 556 396 484 375 819 059 2 × 2 = 0 + 0.003 112 792 968 751 638 118 4;
  • 23) 0.003 112 792 968 751 638 118 4 × 2 = 0 + 0.006 225 585 937 503 276 236 8;
  • 24) 0.006 225 585 937 503 276 236 8 × 2 = 0 + 0.012 451 171 875 006 552 473 6;
  • 25) 0.012 451 171 875 006 552 473 6 × 2 = 0 + 0.024 902 343 750 013 104 947 2;
  • 26) 0.024 902 343 750 013 104 947 2 × 2 = 0 + 0.049 804 687 500 026 209 894 4;
  • 27) 0.049 804 687 500 026 209 894 4 × 2 = 0 + 0.099 609 375 000 052 419 788 8;
  • 28) 0.099 609 375 000 052 419 788 8 × 2 = 0 + 0.199 218 750 000 104 839 577 6;
  • 29) 0.199 218 750 000 104 839 577 6 × 2 = 0 + 0.398 437 500 000 209 679 155 2;
  • 30) 0.398 437 500 000 209 679 155 2 × 2 = 0 + 0.796 875 000 000 419 358 310 4;
  • 31) 0.796 875 000 000 419 358 310 4 × 2 = 1 + 0.593 750 000 000 838 716 620 8;
  • 32) 0.593 750 000 000 838 716 620 8 × 2 = 1 + 0.187 500 000 001 677 433 241 6;
  • 33) 0.187 500 000 001 677 433 241 6 × 2 = 0 + 0.375 000 000 003 354 866 483 2;
  • 34) 0.375 000 000 003 354 866 483 2 × 2 = 0 + 0.750 000 000 006 709 732 966 4;
  • 35) 0.750 000 000 006 709 732 966 4 × 2 = 1 + 0.500 000 000 013 419 465 932 8;
  • 36) 0.500 000 000 013 419 465 932 8 × 2 = 1 + 0.000 000 000 026 838 931 865 6;
  • 37) 0.000 000 000 026 838 931 865 6 × 2 = 0 + 0.000 000 000 053 677 863 731 2;
  • 38) 0.000 000 000 053 677 863 731 2 × 2 = 0 + 0.000 000 000 107 355 727 462 4;
  • 39) 0.000 000 000 107 355 727 462 4 × 2 = 0 + 0.000 000 000 214 711 454 924 8;
  • 40) 0.000 000 000 214 711 454 924 8 × 2 = 0 + 0.000 000 000 429 422 909 849 6;
  • 41) 0.000 000 000 429 422 909 849 6 × 2 = 0 + 0.000 000 000 858 845 819 699 2;
  • 42) 0.000 000 000 858 845 819 699 2 × 2 = 0 + 0.000 000 001 717 691 639 398 4;
  • 43) 0.000 000 001 717 691 639 398 4 × 2 = 0 + 0.000 000 003 435 383 278 796 8;
  • 44) 0.000 000 003 435 383 278 796 8 × 2 = 0 + 0.000 000 006 870 766 557 593 6;
  • 45) 0.000 000 006 870 766 557 593 6 × 2 = 0 + 0.000 000 013 741 533 115 187 2;
  • 46) 0.000 000 013 741 533 115 187 2 × 2 = 0 + 0.000 000 027 483 066 230 374 4;
  • 47) 0.000 000 027 483 066 230 374 4 × 2 = 0 + 0.000 000 054 966 132 460 748 8;
  • 48) 0.000 000 054 966 132 460 748 8 × 2 = 0 + 0.000 000 109 932 264 921 497 6;
  • 49) 0.000 000 109 932 264 921 497 6 × 2 = 0 + 0.000 000 219 864 529 842 995 2;
  • 50) 0.000 000 219 864 529 842 995 2 × 2 = 0 + 0.000 000 439 729 059 685 990 4;
  • 51) 0.000 000 439 729 059 685 990 4 × 2 = 0 + 0.000 000 879 458 119 371 980 8;
  • 52) 0.000 000 879 458 119 371 980 8 × 2 = 0 + 0.000 001 758 916 238 743 961 6;
  • 53) 0.000 001 758 916 238 743 961 6 × 2 = 0 + 0.000 003 517 832 477 487 923 2;
  • 54) 0.000 003 517 832 477 487 923 2 × 2 = 0 + 0.000 007 035 664 954 975 846 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:


0.000 000 000 742 147 676 647 1(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 647 1(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:


0.000 000 000 742 147 676 647 1(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 -0.000 000 000 742 147 676 647 1 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