-0.000 000 000 742 147 676 49 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 49(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 49(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 49| = 0.000 000 000 742 147 676 49


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 49.

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 49 × 2 = 0 + 0.000 000 001 484 295 352 98;
  • 2) 0.000 000 001 484 295 352 98 × 2 = 0 + 0.000 000 002 968 590 705 96;
  • 3) 0.000 000 002 968 590 705 96 × 2 = 0 + 0.000 000 005 937 181 411 92;
  • 4) 0.000 000 005 937 181 411 92 × 2 = 0 + 0.000 000 011 874 362 823 84;
  • 5) 0.000 000 011 874 362 823 84 × 2 = 0 + 0.000 000 023 748 725 647 68;
  • 6) 0.000 000 023 748 725 647 68 × 2 = 0 + 0.000 000 047 497 451 295 36;
  • 7) 0.000 000 047 497 451 295 36 × 2 = 0 + 0.000 000 094 994 902 590 72;
  • 8) 0.000 000 094 994 902 590 72 × 2 = 0 + 0.000 000 189 989 805 181 44;
  • 9) 0.000 000 189 989 805 181 44 × 2 = 0 + 0.000 000 379 979 610 362 88;
  • 10) 0.000 000 379 979 610 362 88 × 2 = 0 + 0.000 000 759 959 220 725 76;
  • 11) 0.000 000 759 959 220 725 76 × 2 = 0 + 0.000 001 519 918 441 451 52;
  • 12) 0.000 001 519 918 441 451 52 × 2 = 0 + 0.000 003 039 836 882 903 04;
  • 13) 0.000 003 039 836 882 903 04 × 2 = 0 + 0.000 006 079 673 765 806 08;
  • 14) 0.000 006 079 673 765 806 08 × 2 = 0 + 0.000 012 159 347 531 612 16;
  • 15) 0.000 012 159 347 531 612 16 × 2 = 0 + 0.000 024 318 695 063 224 32;
  • 16) 0.000 024 318 695 063 224 32 × 2 = 0 + 0.000 048 637 390 126 448 64;
  • 17) 0.000 048 637 390 126 448 64 × 2 = 0 + 0.000 097 274 780 252 897 28;
  • 18) 0.000 097 274 780 252 897 28 × 2 = 0 + 0.000 194 549 560 505 794 56;
  • 19) 0.000 194 549 560 505 794 56 × 2 = 0 + 0.000 389 099 121 011 589 12;
  • 20) 0.000 389 099 121 011 589 12 × 2 = 0 + 0.000 778 198 242 023 178 24;
  • 21) 0.000 778 198 242 023 178 24 × 2 = 0 + 0.001 556 396 484 046 356 48;
  • 22) 0.001 556 396 484 046 356 48 × 2 = 0 + 0.003 112 792 968 092 712 96;
  • 23) 0.003 112 792 968 092 712 96 × 2 = 0 + 0.006 225 585 936 185 425 92;
  • 24) 0.006 225 585 936 185 425 92 × 2 = 0 + 0.012 451 171 872 370 851 84;
  • 25) 0.012 451 171 872 370 851 84 × 2 = 0 + 0.024 902 343 744 741 703 68;
  • 26) 0.024 902 343 744 741 703 68 × 2 = 0 + 0.049 804 687 489 483 407 36;
  • 27) 0.049 804 687 489 483 407 36 × 2 = 0 + 0.099 609 374 978 966 814 72;
  • 28) 0.099 609 374 978 966 814 72 × 2 = 0 + 0.199 218 749 957 933 629 44;
  • 29) 0.199 218 749 957 933 629 44 × 2 = 0 + 0.398 437 499 915 867 258 88;
  • 30) 0.398 437 499 915 867 258 88 × 2 = 0 + 0.796 874 999 831 734 517 76;
  • 31) 0.796 874 999 831 734 517 76 × 2 = 1 + 0.593 749 999 663 469 035 52;
  • 32) 0.593 749 999 663 469 035 52 × 2 = 1 + 0.187 499 999 326 938 071 04;
  • 33) 0.187 499 999 326 938 071 04 × 2 = 0 + 0.374 999 998 653 876 142 08;
  • 34) 0.374 999 998 653 876 142 08 × 2 = 0 + 0.749 999 997 307 752 284 16;
  • 35) 0.749 999 997 307 752 284 16 × 2 = 1 + 0.499 999 994 615 504 568 32;
  • 36) 0.499 999 994 615 504 568 32 × 2 = 0 + 0.999 999 989 231 009 136 64;
  • 37) 0.999 999 989 231 009 136 64 × 2 = 1 + 0.999 999 978 462 018 273 28;
  • 38) 0.999 999 978 462 018 273 28 × 2 = 1 + 0.999 999 956 924 036 546 56;
  • 39) 0.999 999 956 924 036 546 56 × 2 = 1 + 0.999 999 913 848 073 093 12;
  • 40) 0.999 999 913 848 073 093 12 × 2 = 1 + 0.999 999 827 696 146 186 24;
  • 41) 0.999 999 827 696 146 186 24 × 2 = 1 + 0.999 999 655 392 292 372 48;
  • 42) 0.999 999 655 392 292 372 48 × 2 = 1 + 0.999 999 310 784 584 744 96;
  • 43) 0.999 999 310 784 584 744 96 × 2 = 1 + 0.999 998 621 569 169 489 92;
  • 44) 0.999 998 621 569 169 489 92 × 2 = 1 + 0.999 997 243 138 338 979 84;
  • 45) 0.999 997 243 138 338 979 84 × 2 = 1 + 0.999 994 486 276 677 959 68;
  • 46) 0.999 994 486 276 677 959 68 × 2 = 1 + 0.999 988 972 553 355 919 36;
  • 47) 0.999 988 972 553 355 919 36 × 2 = 1 + 0.999 977 945 106 711 838 72;
  • 48) 0.999 977 945 106 711 838 72 × 2 = 1 + 0.999 955 890 213 423 677 44;
  • 49) 0.999 955 890 213 423 677 44 × 2 = 1 + 0.999 911 780 426 847 354 88;
  • 50) 0.999 911 780 426 847 354 88 × 2 = 1 + 0.999 823 560 853 694 709 76;
  • 51) 0.999 823 560 853 694 709 76 × 2 = 1 + 0.999 647 121 707 389 419 52;
  • 52) 0.999 647 121 707 389 419 52 × 2 = 1 + 0.999 294 243 414 778 839 04;
  • 53) 0.999 294 243 414 778 839 04 × 2 = 1 + 0.998 588 486 829 557 678 08;
  • 54) 0.998 588 486 829 557 678 08 × 2 = 1 + 0.997 176 973 659 115 356 16;

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 49(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 49(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 49(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 49 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