-0.000 000 000 742 147 676 646 708 76 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 708 76(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
-0.000 000 000 742 147 676 646 708 76(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:

|-0.000 000 000 742 147 676 646 708 76| = 0.000 000 000 742 147 676 646 708 76


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: 0.000 000 000 742 147 676 646 708 76.

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) 0.000 000 000 742 147 676 646 708 76 × 2 = 0 + 0.000 000 001 484 295 353 293 417 52;
  • 2) 0.000 000 001 484 295 353 293 417 52 × 2 = 0 + 0.000 000 002 968 590 706 586 835 04;
  • 3) 0.000 000 002 968 590 706 586 835 04 × 2 = 0 + 0.000 000 005 937 181 413 173 670 08;
  • 4) 0.000 000 005 937 181 413 173 670 08 × 2 = 0 + 0.000 000 011 874 362 826 347 340 16;
  • 5) 0.000 000 011 874 362 826 347 340 16 × 2 = 0 + 0.000 000 023 748 725 652 694 680 32;
  • 6) 0.000 000 023 748 725 652 694 680 32 × 2 = 0 + 0.000 000 047 497 451 305 389 360 64;
  • 7) 0.000 000 047 497 451 305 389 360 64 × 2 = 0 + 0.000 000 094 994 902 610 778 721 28;
  • 8) 0.000 000 094 994 902 610 778 721 28 × 2 = 0 + 0.000 000 189 989 805 221 557 442 56;
  • 9) 0.000 000 189 989 805 221 557 442 56 × 2 = 0 + 0.000 000 379 979 610 443 114 885 12;
  • 10) 0.000 000 379 979 610 443 114 885 12 × 2 = 0 + 0.000 000 759 959 220 886 229 770 24;
  • 11) 0.000 000 759 959 220 886 229 770 24 × 2 = 0 + 0.000 001 519 918 441 772 459 540 48;
  • 12) 0.000 001 519 918 441 772 459 540 48 × 2 = 0 + 0.000 003 039 836 883 544 919 080 96;
  • 13) 0.000 003 039 836 883 544 919 080 96 × 2 = 0 + 0.000 006 079 673 767 089 838 161 92;
  • 14) 0.000 006 079 673 767 089 838 161 92 × 2 = 0 + 0.000 012 159 347 534 179 676 323 84;
  • 15) 0.000 012 159 347 534 179 676 323 84 × 2 = 0 + 0.000 024 318 695 068 359 352 647 68;
  • 16) 0.000 024 318 695 068 359 352 647 68 × 2 = 0 + 0.000 048 637 390 136 718 705 295 36;
  • 17) 0.000 048 637 390 136 718 705 295 36 × 2 = 0 + 0.000 097 274 780 273 437 410 590 72;
  • 18) 0.000 097 274 780 273 437 410 590 72 × 2 = 0 + 0.000 194 549 560 546 874 821 181 44;
  • 19) 0.000 194 549 560 546 874 821 181 44 × 2 = 0 + 0.000 389 099 121 093 749 642 362 88;
  • 20) 0.000 389 099 121 093 749 642 362 88 × 2 = 0 + 0.000 778 198 242 187 499 284 725 76;
  • 21) 0.000 778 198 242 187 499 284 725 76 × 2 = 0 + 0.001 556 396 484 374 998 569 451 52;
  • 22) 0.001 556 396 484 374 998 569 451 52 × 2 = 0 + 0.003 112 792 968 749 997 138 903 04;
  • 23) 0.003 112 792 968 749 997 138 903 04 × 2 = 0 + 0.006 225 585 937 499 994 277 806 08;
  • 24) 0.006 225 585 937 499 994 277 806 08 × 2 = 0 + 0.012 451 171 874 999 988 555 612 16;
  • 25) 0.012 451 171 874 999 988 555 612 16 × 2 = 0 + 0.024 902 343 749 999 977 111 224 32;
  • 26) 0.024 902 343 749 999 977 111 224 32 × 2 = 0 + 0.049 804 687 499 999 954 222 448 64;
  • 27) 0.049 804 687 499 999 954 222 448 64 × 2 = 0 + 0.099 609 374 999 999 908 444 897 28;
  • 28) 0.099 609 374 999 999 908 444 897 28 × 2 = 0 + 0.199 218 749 999 999 816 889 794 56;
  • 29) 0.199 218 749 999 999 816 889 794 56 × 2 = 0 + 0.398 437 499 999 999 633 779 589 12;
  • 30) 0.398 437 499 999 999 633 779 589 12 × 2 = 0 + 0.796 874 999 999 999 267 559 178 24;
  • 31) 0.796 874 999 999 999 267 559 178 24 × 2 = 1 + 0.593 749 999 999 998 535 118 356 48;
  • 32) 0.593 749 999 999 998 535 118 356 48 × 2 = 1 + 0.187 499 999 999 997 070 236 712 96;
  • 33) 0.187 499 999 999 997 070 236 712 96 × 2 = 0 + 0.374 999 999 999 994 140 473 425 92;
  • 34) 0.374 999 999 999 994 140 473 425 92 × 2 = 0 + 0.749 999 999 999 988 280 946 851 84;
  • 35) 0.749 999 999 999 988 280 946 851 84 × 2 = 1 + 0.499 999 999 999 976 561 893 703 68;
  • 36) 0.499 999 999 999 976 561 893 703 68 × 2 = 0 + 0.999 999 999 999 953 123 787 407 36;
  • 37) 0.999 999 999 999 953 123 787 407 36 × 2 = 1 + 0.999 999 999 999 906 247 574 814 72;
  • 38) 0.999 999 999 999 906 247 574 814 72 × 2 = 1 + 0.999 999 999 999 812 495 149 629 44;
  • 39) 0.999 999 999 999 812 495 149 629 44 × 2 = 1 + 0.999 999 999 999 624 990 299 258 88;
  • 40) 0.999 999 999 999 624 990 299 258 88 × 2 = 1 + 0.999 999 999 999 249 980 598 517 76;
  • 41) 0.999 999 999 999 249 980 598 517 76 × 2 = 1 + 0.999 999 999 998 499 961 197 035 52;
  • 42) 0.999 999 999 998 499 961 197 035 52 × 2 = 1 + 0.999 999 999 996 999 922 394 071 04;
  • 43) 0.999 999 999 996 999 922 394 071 04 × 2 = 1 + 0.999 999 999 993 999 844 788 142 08;
  • 44) 0.999 999 999 993 999 844 788 142 08 × 2 = 1 + 0.999 999 999 987 999 689 576 284 16;
  • 45) 0.999 999 999 987 999 689 576 284 16 × 2 = 1 + 0.999 999 999 975 999 379 152 568 32;
  • 46) 0.999 999 999 975 999 379 152 568 32 × 2 = 1 + 0.999 999 999 951 998 758 305 136 64;
  • 47) 0.999 999 999 951 998 758 305 136 64 × 2 = 1 + 0.999 999 999 903 997 516 610 273 28;
  • 48) 0.999 999 999 903 997 516 610 273 28 × 2 = 1 + 0.999 999 999 807 995 033 220 546 56;
  • 49) 0.999 999 999 807 995 033 220 546 56 × 2 = 1 + 0.999 999 999 615 990 066 441 093 12;
  • 50) 0.999 999 999 615 990 066 441 093 12 × 2 = 1 + 0.999 999 999 231 980 132 882 186 24;
  • 51) 0.999 999 999 231 980 132 882 186 24 × 2 = 1 + 0.999 999 998 463 960 265 764 372 48;
  • 52) 0.999 999 998 463 960 265 764 372 48 × 2 = 1 + 0.999 999 996 927 920 531 528 744 96;
  • 53) 0.999 999 996 927 920 531 528 744 96 × 2 = 1 + 0.999 999 993 855 841 063 057 489 92;
  • 54) 0.999 999 993 855 841 063 057 489 92 × 2 = 1 + 0.999 999 987 711 682 126 114 979 84;

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:


0.000 000 000 742 147 676 646 708 76(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 708 76(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:


0.000 000 000 742 147 676 646 708 76(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 -0.000 000 000 742 147 676 646 708 76 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