-0.000 000 000 742 147 676 646 16 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 16(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 16(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 16| = 0.000 000 000 742 147 676 646 16


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 16.

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 16 × 2 = 0 + 0.000 000 001 484 295 353 292 32;
  • 2) 0.000 000 001 484 295 353 292 32 × 2 = 0 + 0.000 000 002 968 590 706 584 64;
  • 3) 0.000 000 002 968 590 706 584 64 × 2 = 0 + 0.000 000 005 937 181 413 169 28;
  • 4) 0.000 000 005 937 181 413 169 28 × 2 = 0 + 0.000 000 011 874 362 826 338 56;
  • 5) 0.000 000 011 874 362 826 338 56 × 2 = 0 + 0.000 000 023 748 725 652 677 12;
  • 6) 0.000 000 023 748 725 652 677 12 × 2 = 0 + 0.000 000 047 497 451 305 354 24;
  • 7) 0.000 000 047 497 451 305 354 24 × 2 = 0 + 0.000 000 094 994 902 610 708 48;
  • 8) 0.000 000 094 994 902 610 708 48 × 2 = 0 + 0.000 000 189 989 805 221 416 96;
  • 9) 0.000 000 189 989 805 221 416 96 × 2 = 0 + 0.000 000 379 979 610 442 833 92;
  • 10) 0.000 000 379 979 610 442 833 92 × 2 = 0 + 0.000 000 759 959 220 885 667 84;
  • 11) 0.000 000 759 959 220 885 667 84 × 2 = 0 + 0.000 001 519 918 441 771 335 68;
  • 12) 0.000 001 519 918 441 771 335 68 × 2 = 0 + 0.000 003 039 836 883 542 671 36;
  • 13) 0.000 003 039 836 883 542 671 36 × 2 = 0 + 0.000 006 079 673 767 085 342 72;
  • 14) 0.000 006 079 673 767 085 342 72 × 2 = 0 + 0.000 012 159 347 534 170 685 44;
  • 15) 0.000 012 159 347 534 170 685 44 × 2 = 0 + 0.000 024 318 695 068 341 370 88;
  • 16) 0.000 024 318 695 068 341 370 88 × 2 = 0 + 0.000 048 637 390 136 682 741 76;
  • 17) 0.000 048 637 390 136 682 741 76 × 2 = 0 + 0.000 097 274 780 273 365 483 52;
  • 18) 0.000 097 274 780 273 365 483 52 × 2 = 0 + 0.000 194 549 560 546 730 967 04;
  • 19) 0.000 194 549 560 546 730 967 04 × 2 = 0 + 0.000 389 099 121 093 461 934 08;
  • 20) 0.000 389 099 121 093 461 934 08 × 2 = 0 + 0.000 778 198 242 186 923 868 16;
  • 21) 0.000 778 198 242 186 923 868 16 × 2 = 0 + 0.001 556 396 484 373 847 736 32;
  • 22) 0.001 556 396 484 373 847 736 32 × 2 = 0 + 0.003 112 792 968 747 695 472 64;
  • 23) 0.003 112 792 968 747 695 472 64 × 2 = 0 + 0.006 225 585 937 495 390 945 28;
  • 24) 0.006 225 585 937 495 390 945 28 × 2 = 0 + 0.012 451 171 874 990 781 890 56;
  • 25) 0.012 451 171 874 990 781 890 56 × 2 = 0 + 0.024 902 343 749 981 563 781 12;
  • 26) 0.024 902 343 749 981 563 781 12 × 2 = 0 + 0.049 804 687 499 963 127 562 24;
  • 27) 0.049 804 687 499 963 127 562 24 × 2 = 0 + 0.099 609 374 999 926 255 124 48;
  • 28) 0.099 609 374 999 926 255 124 48 × 2 = 0 + 0.199 218 749 999 852 510 248 96;
  • 29) 0.199 218 749 999 852 510 248 96 × 2 = 0 + 0.398 437 499 999 705 020 497 92;
  • 30) 0.398 437 499 999 705 020 497 92 × 2 = 0 + 0.796 874 999 999 410 040 995 84;
  • 31) 0.796 874 999 999 410 040 995 84 × 2 = 1 + 0.593 749 999 998 820 081 991 68;
  • 32) 0.593 749 999 998 820 081 991 68 × 2 = 1 + 0.187 499 999 997 640 163 983 36;
  • 33) 0.187 499 999 997 640 163 983 36 × 2 = 0 + 0.374 999 999 995 280 327 966 72;
  • 34) 0.374 999 999 995 280 327 966 72 × 2 = 0 + 0.749 999 999 990 560 655 933 44;
  • 35) 0.749 999 999 990 560 655 933 44 × 2 = 1 + 0.499 999 999 981 121 311 866 88;
  • 36) 0.499 999 999 981 121 311 866 88 × 2 = 0 + 0.999 999 999 962 242 623 733 76;
  • 37) 0.999 999 999 962 242 623 733 76 × 2 = 1 + 0.999 999 999 924 485 247 467 52;
  • 38) 0.999 999 999 924 485 247 467 52 × 2 = 1 + 0.999 999 999 848 970 494 935 04;
  • 39) 0.999 999 999 848 970 494 935 04 × 2 = 1 + 0.999 999 999 697 940 989 870 08;
  • 40) 0.999 999 999 697 940 989 870 08 × 2 = 1 + 0.999 999 999 395 881 979 740 16;
  • 41) 0.999 999 999 395 881 979 740 16 × 2 = 1 + 0.999 999 998 791 763 959 480 32;
  • 42) 0.999 999 998 791 763 959 480 32 × 2 = 1 + 0.999 999 997 583 527 918 960 64;
  • 43) 0.999 999 997 583 527 918 960 64 × 2 = 1 + 0.999 999 995 167 055 837 921 28;
  • 44) 0.999 999 995 167 055 837 921 28 × 2 = 1 + 0.999 999 990 334 111 675 842 56;
  • 45) 0.999 999 990 334 111 675 842 56 × 2 = 1 + 0.999 999 980 668 223 351 685 12;
  • 46) 0.999 999 980 668 223 351 685 12 × 2 = 1 + 0.999 999 961 336 446 703 370 24;
  • 47) 0.999 999 961 336 446 703 370 24 × 2 = 1 + 0.999 999 922 672 893 406 740 48;
  • 48) 0.999 999 922 672 893 406 740 48 × 2 = 1 + 0.999 999 845 345 786 813 480 96;
  • 49) 0.999 999 845 345 786 813 480 96 × 2 = 1 + 0.999 999 690 691 573 626 961 92;
  • 50) 0.999 999 690 691 573 626 961 92 × 2 = 1 + 0.999 999 381 383 147 253 923 84;
  • 51) 0.999 999 381 383 147 253 923 84 × 2 = 1 + 0.999 998 762 766 294 507 847 68;
  • 52) 0.999 998 762 766 294 507 847 68 × 2 = 1 + 0.999 997 525 532 589 015 695 36;
  • 53) 0.999 997 525 532 589 015 695 36 × 2 = 1 + 0.999 995 051 065 178 031 390 72;
  • 54) 0.999 995 051 065 178 031 390 72 × 2 = 1 + 0.999 990 102 130 356 062 781 44;

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 16(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 16(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 16(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 16 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