-0.000 000 000 742 147 676 617 6 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 617 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
-0.000 000 000 742 147 676 617 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:

|-0.000 000 000 742 147 676 617 6| = 0.000 000 000 742 147 676 617 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: 0.000 000 000 742 147 676 617 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) 0.000 000 000 742 147 676 617 6 × 2 = 0 + 0.000 000 001 484 295 353 235 2;
  • 2) 0.000 000 001 484 295 353 235 2 × 2 = 0 + 0.000 000 002 968 590 706 470 4;
  • 3) 0.000 000 002 968 590 706 470 4 × 2 = 0 + 0.000 000 005 937 181 412 940 8;
  • 4) 0.000 000 005 937 181 412 940 8 × 2 = 0 + 0.000 000 011 874 362 825 881 6;
  • 5) 0.000 000 011 874 362 825 881 6 × 2 = 0 + 0.000 000 023 748 725 651 763 2;
  • 6) 0.000 000 023 748 725 651 763 2 × 2 = 0 + 0.000 000 047 497 451 303 526 4;
  • 7) 0.000 000 047 497 451 303 526 4 × 2 = 0 + 0.000 000 094 994 902 607 052 8;
  • 8) 0.000 000 094 994 902 607 052 8 × 2 = 0 + 0.000 000 189 989 805 214 105 6;
  • 9) 0.000 000 189 989 805 214 105 6 × 2 = 0 + 0.000 000 379 979 610 428 211 2;
  • 10) 0.000 000 379 979 610 428 211 2 × 2 = 0 + 0.000 000 759 959 220 856 422 4;
  • 11) 0.000 000 759 959 220 856 422 4 × 2 = 0 + 0.000 001 519 918 441 712 844 8;
  • 12) 0.000 001 519 918 441 712 844 8 × 2 = 0 + 0.000 003 039 836 883 425 689 6;
  • 13) 0.000 003 039 836 883 425 689 6 × 2 = 0 + 0.000 006 079 673 766 851 379 2;
  • 14) 0.000 006 079 673 766 851 379 2 × 2 = 0 + 0.000 012 159 347 533 702 758 4;
  • 15) 0.000 012 159 347 533 702 758 4 × 2 = 0 + 0.000 024 318 695 067 405 516 8;
  • 16) 0.000 024 318 695 067 405 516 8 × 2 = 0 + 0.000 048 637 390 134 811 033 6;
  • 17) 0.000 048 637 390 134 811 033 6 × 2 = 0 + 0.000 097 274 780 269 622 067 2;
  • 18) 0.000 097 274 780 269 622 067 2 × 2 = 0 + 0.000 194 549 560 539 244 134 4;
  • 19) 0.000 194 549 560 539 244 134 4 × 2 = 0 + 0.000 389 099 121 078 488 268 8;
  • 20) 0.000 389 099 121 078 488 268 8 × 2 = 0 + 0.000 778 198 242 156 976 537 6;
  • 21) 0.000 778 198 242 156 976 537 6 × 2 = 0 + 0.001 556 396 484 313 953 075 2;
  • 22) 0.001 556 396 484 313 953 075 2 × 2 = 0 + 0.003 112 792 968 627 906 150 4;
  • 23) 0.003 112 792 968 627 906 150 4 × 2 = 0 + 0.006 225 585 937 255 812 300 8;
  • 24) 0.006 225 585 937 255 812 300 8 × 2 = 0 + 0.012 451 171 874 511 624 601 6;
  • 25) 0.012 451 171 874 511 624 601 6 × 2 = 0 + 0.024 902 343 749 023 249 203 2;
  • 26) 0.024 902 343 749 023 249 203 2 × 2 = 0 + 0.049 804 687 498 046 498 406 4;
  • 27) 0.049 804 687 498 046 498 406 4 × 2 = 0 + 0.099 609 374 996 092 996 812 8;
  • 28) 0.099 609 374 996 092 996 812 8 × 2 = 0 + 0.199 218 749 992 185 993 625 6;
  • 29) 0.199 218 749 992 185 993 625 6 × 2 = 0 + 0.398 437 499 984 371 987 251 2;
  • 30) 0.398 437 499 984 371 987 251 2 × 2 = 0 + 0.796 874 999 968 743 974 502 4;
  • 31) 0.796 874 999 968 743 974 502 4 × 2 = 1 + 0.593 749 999 937 487 949 004 8;
  • 32) 0.593 749 999 937 487 949 004 8 × 2 = 1 + 0.187 499 999 874 975 898 009 6;
  • 33) 0.187 499 999 874 975 898 009 6 × 2 = 0 + 0.374 999 999 749 951 796 019 2;
  • 34) 0.374 999 999 749 951 796 019 2 × 2 = 0 + 0.749 999 999 499 903 592 038 4;
  • 35) 0.749 999 999 499 903 592 038 4 × 2 = 1 + 0.499 999 998 999 807 184 076 8;
  • 36) 0.499 999 998 999 807 184 076 8 × 2 = 0 + 0.999 999 997 999 614 368 153 6;
  • 37) 0.999 999 997 999 614 368 153 6 × 2 = 1 + 0.999 999 995 999 228 736 307 2;
  • 38) 0.999 999 995 999 228 736 307 2 × 2 = 1 + 0.999 999 991 998 457 472 614 4;
  • 39) 0.999 999 991 998 457 472 614 4 × 2 = 1 + 0.999 999 983 996 914 945 228 8;
  • 40) 0.999 999 983 996 914 945 228 8 × 2 = 1 + 0.999 999 967 993 829 890 457 6;
  • 41) 0.999 999 967 993 829 890 457 6 × 2 = 1 + 0.999 999 935 987 659 780 915 2;
  • 42) 0.999 999 935 987 659 780 915 2 × 2 = 1 + 0.999 999 871 975 319 561 830 4;
  • 43) 0.999 999 871 975 319 561 830 4 × 2 = 1 + 0.999 999 743 950 639 123 660 8;
  • 44) 0.999 999 743 950 639 123 660 8 × 2 = 1 + 0.999 999 487 901 278 247 321 6;
  • 45) 0.999 999 487 901 278 247 321 6 × 2 = 1 + 0.999 998 975 802 556 494 643 2;
  • 46) 0.999 998 975 802 556 494 643 2 × 2 = 1 + 0.999 997 951 605 112 989 286 4;
  • 47) 0.999 997 951 605 112 989 286 4 × 2 = 1 + 0.999 995 903 210 225 978 572 8;
  • 48) 0.999 995 903 210 225 978 572 8 × 2 = 1 + 0.999 991 806 420 451 957 145 6;
  • 49) 0.999 991 806 420 451 957 145 6 × 2 = 1 + 0.999 983 612 840 903 914 291 2;
  • 50) 0.999 983 612 840 903 914 291 2 × 2 = 1 + 0.999 967 225 681 807 828 582 4;
  • 51) 0.999 967 225 681 807 828 582 4 × 2 = 1 + 0.999 934 451 363 615 657 164 8;
  • 52) 0.999 934 451 363 615 657 164 8 × 2 = 1 + 0.999 868 902 727 231 314 329 6;
  • 53) 0.999 868 902 727 231 314 329 6 × 2 = 1 + 0.999 737 805 454 462 628 659 2;
  • 54) 0.999 737 805 454 462 628 659 2 × 2 = 1 + 0.999 475 610 908 925 257 318 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:


0.000 000 000 742 147 676 617 6(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 617 6(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:


0.000 000 000 742 147 676 617 6(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 -0.000 000 000 742 147 676 617 6 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