-0.000 000 000 742 147 676 646 707 87 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 707 87(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
-0.000 000 000 742 147 676 646 707 87(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:

|-0.000 000 000 742 147 676 646 707 87| = 0.000 000 000 742 147 676 646 707 87


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: 0.000 000 000 742 147 676 646 707 87.

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) 0.000 000 000 742 147 676 646 707 87 × 2 = 0 + 0.000 000 001 484 295 353 293 415 74;
  • 2) 0.000 000 001 484 295 353 293 415 74 × 2 = 0 + 0.000 000 002 968 590 706 586 831 48;
  • 3) 0.000 000 002 968 590 706 586 831 48 × 2 = 0 + 0.000 000 005 937 181 413 173 662 96;
  • 4) 0.000 000 005 937 181 413 173 662 96 × 2 = 0 + 0.000 000 011 874 362 826 347 325 92;
  • 5) 0.000 000 011 874 362 826 347 325 92 × 2 = 0 + 0.000 000 023 748 725 652 694 651 84;
  • 6) 0.000 000 023 748 725 652 694 651 84 × 2 = 0 + 0.000 000 047 497 451 305 389 303 68;
  • 7) 0.000 000 047 497 451 305 389 303 68 × 2 = 0 + 0.000 000 094 994 902 610 778 607 36;
  • 8) 0.000 000 094 994 902 610 778 607 36 × 2 = 0 + 0.000 000 189 989 805 221 557 214 72;
  • 9) 0.000 000 189 989 805 221 557 214 72 × 2 = 0 + 0.000 000 379 979 610 443 114 429 44;
  • 10) 0.000 000 379 979 610 443 114 429 44 × 2 = 0 + 0.000 000 759 959 220 886 228 858 88;
  • 11) 0.000 000 759 959 220 886 228 858 88 × 2 = 0 + 0.000 001 519 918 441 772 457 717 76;
  • 12) 0.000 001 519 918 441 772 457 717 76 × 2 = 0 + 0.000 003 039 836 883 544 915 435 52;
  • 13) 0.000 003 039 836 883 544 915 435 52 × 2 = 0 + 0.000 006 079 673 767 089 830 871 04;
  • 14) 0.000 006 079 673 767 089 830 871 04 × 2 = 0 + 0.000 012 159 347 534 179 661 742 08;
  • 15) 0.000 012 159 347 534 179 661 742 08 × 2 = 0 + 0.000 024 318 695 068 359 323 484 16;
  • 16) 0.000 024 318 695 068 359 323 484 16 × 2 = 0 + 0.000 048 637 390 136 718 646 968 32;
  • 17) 0.000 048 637 390 136 718 646 968 32 × 2 = 0 + 0.000 097 274 780 273 437 293 936 64;
  • 18) 0.000 097 274 780 273 437 293 936 64 × 2 = 0 + 0.000 194 549 560 546 874 587 873 28;
  • 19) 0.000 194 549 560 546 874 587 873 28 × 2 = 0 + 0.000 389 099 121 093 749 175 746 56;
  • 20) 0.000 389 099 121 093 749 175 746 56 × 2 = 0 + 0.000 778 198 242 187 498 351 493 12;
  • 21) 0.000 778 198 242 187 498 351 493 12 × 2 = 0 + 0.001 556 396 484 374 996 702 986 24;
  • 22) 0.001 556 396 484 374 996 702 986 24 × 2 = 0 + 0.003 112 792 968 749 993 405 972 48;
  • 23) 0.003 112 792 968 749 993 405 972 48 × 2 = 0 + 0.006 225 585 937 499 986 811 944 96;
  • 24) 0.006 225 585 937 499 986 811 944 96 × 2 = 0 + 0.012 451 171 874 999 973 623 889 92;
  • 25) 0.012 451 171 874 999 973 623 889 92 × 2 = 0 + 0.024 902 343 749 999 947 247 779 84;
  • 26) 0.024 902 343 749 999 947 247 779 84 × 2 = 0 + 0.049 804 687 499 999 894 495 559 68;
  • 27) 0.049 804 687 499 999 894 495 559 68 × 2 = 0 + 0.099 609 374 999 999 788 991 119 36;
  • 28) 0.099 609 374 999 999 788 991 119 36 × 2 = 0 + 0.199 218 749 999 999 577 982 238 72;
  • 29) 0.199 218 749 999 999 577 982 238 72 × 2 = 0 + 0.398 437 499 999 999 155 964 477 44;
  • 30) 0.398 437 499 999 999 155 964 477 44 × 2 = 0 + 0.796 874 999 999 998 311 928 954 88;
  • 31) 0.796 874 999 999 998 311 928 954 88 × 2 = 1 + 0.593 749 999 999 996 623 857 909 76;
  • 32) 0.593 749 999 999 996 623 857 909 76 × 2 = 1 + 0.187 499 999 999 993 247 715 819 52;
  • 33) 0.187 499 999 999 993 247 715 819 52 × 2 = 0 + 0.374 999 999 999 986 495 431 639 04;
  • 34) 0.374 999 999 999 986 495 431 639 04 × 2 = 0 + 0.749 999 999 999 972 990 863 278 08;
  • 35) 0.749 999 999 999 972 990 863 278 08 × 2 = 1 + 0.499 999 999 999 945 981 726 556 16;
  • 36) 0.499 999 999 999 945 981 726 556 16 × 2 = 0 + 0.999 999 999 999 891 963 453 112 32;
  • 37) 0.999 999 999 999 891 963 453 112 32 × 2 = 1 + 0.999 999 999 999 783 926 906 224 64;
  • 38) 0.999 999 999 999 783 926 906 224 64 × 2 = 1 + 0.999 999 999 999 567 853 812 449 28;
  • 39) 0.999 999 999 999 567 853 812 449 28 × 2 = 1 + 0.999 999 999 999 135 707 624 898 56;
  • 40) 0.999 999 999 999 135 707 624 898 56 × 2 = 1 + 0.999 999 999 998 271 415 249 797 12;
  • 41) 0.999 999 999 998 271 415 249 797 12 × 2 = 1 + 0.999 999 999 996 542 830 499 594 24;
  • 42) 0.999 999 999 996 542 830 499 594 24 × 2 = 1 + 0.999 999 999 993 085 660 999 188 48;
  • 43) 0.999 999 999 993 085 660 999 188 48 × 2 = 1 + 0.999 999 999 986 171 321 998 376 96;
  • 44) 0.999 999 999 986 171 321 998 376 96 × 2 = 1 + 0.999 999 999 972 342 643 996 753 92;
  • 45) 0.999 999 999 972 342 643 996 753 92 × 2 = 1 + 0.999 999 999 944 685 287 993 507 84;
  • 46) 0.999 999 999 944 685 287 993 507 84 × 2 = 1 + 0.999 999 999 889 370 575 987 015 68;
  • 47) 0.999 999 999 889 370 575 987 015 68 × 2 = 1 + 0.999 999 999 778 741 151 974 031 36;
  • 48) 0.999 999 999 778 741 151 974 031 36 × 2 = 1 + 0.999 999 999 557 482 303 948 062 72;
  • 49) 0.999 999 999 557 482 303 948 062 72 × 2 = 1 + 0.999 999 999 114 964 607 896 125 44;
  • 50) 0.999 999 999 114 964 607 896 125 44 × 2 = 1 + 0.999 999 998 229 929 215 792 250 88;
  • 51) 0.999 999 998 229 929 215 792 250 88 × 2 = 1 + 0.999 999 996 459 858 431 584 501 76;
  • 52) 0.999 999 996 459 858 431 584 501 76 × 2 = 1 + 0.999 999 992 919 716 863 169 003 52;
  • 53) 0.999 999 992 919 716 863 169 003 52 × 2 = 1 + 0.999 999 985 839 433 726 338 007 04;
  • 54) 0.999 999 985 839 433 726 338 007 04 × 2 = 1 + 0.999 999 971 678 867 452 676 014 08;

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:


0.000 000 000 742 147 676 646 707 87(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 707 87(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:


0.000 000 000 742 147 676 646 707 87(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 -0.000 000 000 742 147 676 646 707 87 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