-0.000 000 077 194 272 307 679 057 121 276 855 454 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 077 194 272 307 679 057 121 276 855 454(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
-0.000 000 077 194 272 307 679 057 121 276 855 454(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:

|-0.000 000 077 194 272 307 679 057 121 276 855 454| = 0.000 000 077 194 272 307 679 057 121 276 855 454


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: 0.000 000 077 194 272 307 679 057 121 276 855 454.

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) 0.000 000 077 194 272 307 679 057 121 276 855 454 × 2 = 0 + 0.000 000 154 388 544 615 358 114 242 553 710 908;
  • 2) 0.000 000 154 388 544 615 358 114 242 553 710 908 × 2 = 0 + 0.000 000 308 777 089 230 716 228 485 107 421 816;
  • 3) 0.000 000 308 777 089 230 716 228 485 107 421 816 × 2 = 0 + 0.000 000 617 554 178 461 432 456 970 214 843 632;
  • 4) 0.000 000 617 554 178 461 432 456 970 214 843 632 × 2 = 0 + 0.000 001 235 108 356 922 864 913 940 429 687 264;
  • 5) 0.000 001 235 108 356 922 864 913 940 429 687 264 × 2 = 0 + 0.000 002 470 216 713 845 729 827 880 859 374 528;
  • 6) 0.000 002 470 216 713 845 729 827 880 859 374 528 × 2 = 0 + 0.000 004 940 433 427 691 459 655 761 718 749 056;
  • 7) 0.000 004 940 433 427 691 459 655 761 718 749 056 × 2 = 0 + 0.000 009 880 866 855 382 919 311 523 437 498 112;
  • 8) 0.000 009 880 866 855 382 919 311 523 437 498 112 × 2 = 0 + 0.000 019 761 733 710 765 838 623 046 874 996 224;
  • 9) 0.000 019 761 733 710 765 838 623 046 874 996 224 × 2 = 0 + 0.000 039 523 467 421 531 677 246 093 749 992 448;
  • 10) 0.000 039 523 467 421 531 677 246 093 749 992 448 × 2 = 0 + 0.000 079 046 934 843 063 354 492 187 499 984 896;
  • 11) 0.000 079 046 934 843 063 354 492 187 499 984 896 × 2 = 0 + 0.000 158 093 869 686 126 708 984 374 999 969 792;
  • 12) 0.000 158 093 869 686 126 708 984 374 999 969 792 × 2 = 0 + 0.000 316 187 739 372 253 417 968 749 999 939 584;
  • 13) 0.000 316 187 739 372 253 417 968 749 999 939 584 × 2 = 0 + 0.000 632 375 478 744 506 835 937 499 999 879 168;
  • 14) 0.000 632 375 478 744 506 835 937 499 999 879 168 × 2 = 0 + 0.001 264 750 957 489 013 671 874 999 999 758 336;
  • 15) 0.001 264 750 957 489 013 671 874 999 999 758 336 × 2 = 0 + 0.002 529 501 914 978 027 343 749 999 999 516 672;
  • 16) 0.002 529 501 914 978 027 343 749 999 999 516 672 × 2 = 0 + 0.005 059 003 829 956 054 687 499 999 999 033 344;
  • 17) 0.005 059 003 829 956 054 687 499 999 999 033 344 × 2 = 0 + 0.010 118 007 659 912 109 374 999 999 998 066 688;
  • 18) 0.010 118 007 659 912 109 374 999 999 998 066 688 × 2 = 0 + 0.020 236 015 319 824 218 749 999 999 996 133 376;
  • 19) 0.020 236 015 319 824 218 749 999 999 996 133 376 × 2 = 0 + 0.040 472 030 639 648 437 499 999 999 992 266 752;
  • 20) 0.040 472 030 639 648 437 499 999 999 992 266 752 × 2 = 0 + 0.080 944 061 279 296 874 999 999 999 984 533 504;
  • 21) 0.080 944 061 279 296 874 999 999 999 984 533 504 × 2 = 0 + 0.161 888 122 558 593 749 999 999 999 969 067 008;
  • 22) 0.161 888 122 558 593 749 999 999 999 969 067 008 × 2 = 0 + 0.323 776 245 117 187 499 999 999 999 938 134 016;
  • 23) 0.323 776 245 117 187 499 999 999 999 938 134 016 × 2 = 0 + 0.647 552 490 234 374 999 999 999 999 876 268 032;
  • 24) 0.647 552 490 234 374 999 999 999 999 876 268 032 × 2 = 1 + 0.295 104 980 468 749 999 999 999 999 752 536 064;
  • 25) 0.295 104 980 468 749 999 999 999 999 752 536 064 × 2 = 0 + 0.590 209 960 937 499 999 999 999 999 505 072 128;
  • 26) 0.590 209 960 937 499 999 999 999 999 505 072 128 × 2 = 1 + 0.180 419 921 874 999 999 999 999 999 010 144 256;
  • 27) 0.180 419 921 874 999 999 999 999 999 010 144 256 × 2 = 0 + 0.360 839 843 749 999 999 999 999 998 020 288 512;
  • 28) 0.360 839 843 749 999 999 999 999 998 020 288 512 × 2 = 0 + 0.721 679 687 499 999 999 999 999 996 040 577 024;
  • 29) 0.721 679 687 499 999 999 999 999 996 040 577 024 × 2 = 1 + 0.443 359 374 999 999 999 999 999 992 081 154 048;
  • 30) 0.443 359 374 999 999 999 999 999 992 081 154 048 × 2 = 0 + 0.886 718 749 999 999 999 999 999 984 162 308 096;
  • 31) 0.886 718 749 999 999 999 999 999 984 162 308 096 × 2 = 1 + 0.773 437 499 999 999 999 999 999 968 324 616 192;
  • 32) 0.773 437 499 999 999 999 999 999 968 324 616 192 × 2 = 1 + 0.546 874 999 999 999 999 999 999 936 649 232 384;
  • 33) 0.546 874 999 999 999 999 999 999 936 649 232 384 × 2 = 1 + 0.093 749 999 999 999 999 999 999 873 298 464 768;
  • 34) 0.093 749 999 999 999 999 999 999 873 298 464 768 × 2 = 0 + 0.187 499 999 999 999 999 999 999 746 596 929 536;
  • 35) 0.187 499 999 999 999 999 999 999 746 596 929 536 × 2 = 0 + 0.374 999 999 999 999 999 999 999 493 193 859 072;
  • 36) 0.374 999 999 999 999 999 999 999 493 193 859 072 × 2 = 0 + 0.749 999 999 999 999 999 999 998 986 387 718 144;
  • 37) 0.749 999 999 999 999 999 999 998 986 387 718 144 × 2 = 1 + 0.499 999 999 999 999 999 999 997 972 775 436 288;
  • 38) 0.499 999 999 999 999 999 999 997 972 775 436 288 × 2 = 0 + 0.999 999 999 999 999 999 999 995 945 550 872 576;
  • 39) 0.999 999 999 999 999 999 999 995 945 550 872 576 × 2 = 1 + 0.999 999 999 999 999 999 999 991 891 101 745 152;
  • 40) 0.999 999 999 999 999 999 999 991 891 101 745 152 × 2 = 1 + 0.999 999 999 999 999 999 999 983 782 203 490 304;
  • 41) 0.999 999 999 999 999 999 999 983 782 203 490 304 × 2 = 1 + 0.999 999 999 999 999 999 999 967 564 406 980 608;
  • 42) 0.999 999 999 999 999 999 999 967 564 406 980 608 × 2 = 1 + 0.999 999 999 999 999 999 999 935 128 813 961 216;
  • 43) 0.999 999 999 999 999 999 999 935 128 813 961 216 × 2 = 1 + 0.999 999 999 999 999 999 999 870 257 627 922 432;
  • 44) 0.999 999 999 999 999 999 999 870 257 627 922 432 × 2 = 1 + 0.999 999 999 999 999 999 999 740 515 255 844 864;
  • 45) 0.999 999 999 999 999 999 999 740 515 255 844 864 × 2 = 1 + 0.999 999 999 999 999 999 999 481 030 511 689 728;
  • 46) 0.999 999 999 999 999 999 999 481 030 511 689 728 × 2 = 1 + 0.999 999 999 999 999 999 998 962 061 023 379 456;
  • 47) 0.999 999 999 999 999 999 998 962 061 023 379 456 × 2 = 1 + 0.999 999 999 999 999 999 997 924 122 046 758 912;

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:


0.000 000 077 194 272 307 679 057 121 276 855 454(10) =

0.0000 0000 0000 0000 0000 0001 0100 1011 1000 1011 1111 111(2)

6. Positive number before normalization:

0.000 000 077 194 272 307 679 057 121 276 855 454(10) =

0.0000 0000 0000 0000 0000 0001 0100 1011 1000 1011 1111 111(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 24 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 077 194 272 307 679 057 121 276 855 454(10) =

0.0000 0000 0000 0000 0000 0001 0100 1011 1000 1011 1111 111(2) =

0.0000 0000 0000 0000 0000 0001 0100 1011 1000 1011 1111 111(2) × 20 =

1.0100 1011 1000 1011 1111 111(2) × 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 representation:

Sign 1 (a negative number)

Exponent (unadjusted): -24

Mantissa (not normalized):
1.0100 1011 1000 1011 1111 111


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-24 + 2(8-1) - 1 =

(-24 + 127)(10) =

103(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;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 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) =

103(10) =

0110 0111(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. 010 0101 1100 0101 1111 1111 =

010 0101 1100 0101 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 0111

Mantissa (23 bits) =
010 0101 1100 0101 1111 1111


Decimal number -0.000 000 077 194 272 307 679 057 121 276 855 454 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0111 - 010 0101 1100 0101 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