0.002 394 522 356 732 212 773 426 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.002 394 522 356 732 212 773 426(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
0.002 394 522 356 732 212 773 426(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. 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;

2. 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)


3. Convert to binary (base 2) the fractional part: 0.002 394 522 356 732 212 773 426.

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) 0.002 394 522 356 732 212 773 426 × 2 = 0 + 0.004 789 044 713 464 425 546 852;
  • 2) 0.004 789 044 713 464 425 546 852 × 2 = 0 + 0.009 578 089 426 928 851 093 704;
  • 3) 0.009 578 089 426 928 851 093 704 × 2 = 0 + 0.019 156 178 853 857 702 187 408;
  • 4) 0.019 156 178 853 857 702 187 408 × 2 = 0 + 0.038 312 357 707 715 404 374 816;
  • 5) 0.038 312 357 707 715 404 374 816 × 2 = 0 + 0.076 624 715 415 430 808 749 632;
  • 6) 0.076 624 715 415 430 808 749 632 × 2 = 0 + 0.153 249 430 830 861 617 499 264;
  • 7) 0.153 249 430 830 861 617 499 264 × 2 = 0 + 0.306 498 861 661 723 234 998 528;
  • 8) 0.306 498 861 661 723 234 998 528 × 2 = 0 + 0.612 997 723 323 446 469 997 056;
  • 9) 0.612 997 723 323 446 469 997 056 × 2 = 1 + 0.225 995 446 646 892 939 994 112;
  • 10) 0.225 995 446 646 892 939 994 112 × 2 = 0 + 0.451 990 893 293 785 879 988 224;
  • 11) 0.451 990 893 293 785 879 988 224 × 2 = 0 + 0.903 981 786 587 571 759 976 448;
  • 12) 0.903 981 786 587 571 759 976 448 × 2 = 1 + 0.807 963 573 175 143 519 952 896;
  • 13) 0.807 963 573 175 143 519 952 896 × 2 = 1 + 0.615 927 146 350 287 039 905 792;
  • 14) 0.615 927 146 350 287 039 905 792 × 2 = 1 + 0.231 854 292 700 574 079 811 584;
  • 15) 0.231 854 292 700 574 079 811 584 × 2 = 0 + 0.463 708 585 401 148 159 623 168;
  • 16) 0.463 708 585 401 148 159 623 168 × 2 = 0 + 0.927 417 170 802 296 319 246 336;
  • 17) 0.927 417 170 802 296 319 246 336 × 2 = 1 + 0.854 834 341 604 592 638 492 672;
  • 18) 0.854 834 341 604 592 638 492 672 × 2 = 1 + 0.709 668 683 209 185 276 985 344;
  • 19) 0.709 668 683 209 185 276 985 344 × 2 = 1 + 0.419 337 366 418 370 553 970 688;
  • 20) 0.419 337 366 418 370 553 970 688 × 2 = 0 + 0.838 674 732 836 741 107 941 376;
  • 21) 0.838 674 732 836 741 107 941 376 × 2 = 1 + 0.677 349 465 673 482 215 882 752;
  • 22) 0.677 349 465 673 482 215 882 752 × 2 = 1 + 0.354 698 931 346 964 431 765 504;
  • 23) 0.354 698 931 346 964 431 765 504 × 2 = 0 + 0.709 397 862 693 928 863 531 008;
  • 24) 0.709 397 862 693 928 863 531 008 × 2 = 1 + 0.418 795 725 387 857 727 062 016;
  • 25) 0.418 795 725 387 857 727 062 016 × 2 = 0 + 0.837 591 450 775 715 454 124 032;
  • 26) 0.837 591 450 775 715 454 124 032 × 2 = 1 + 0.675 182 901 551 430 908 248 064;
  • 27) 0.675 182 901 551 430 908 248 064 × 2 = 1 + 0.350 365 803 102 861 816 496 128;
  • 28) 0.350 365 803 102 861 816 496 128 × 2 = 0 + 0.700 731 606 205 723 632 992 256;
  • 29) 0.700 731 606 205 723 632 992 256 × 2 = 1 + 0.401 463 212 411 447 265 984 512;
  • 30) 0.401 463 212 411 447 265 984 512 × 2 = 0 + 0.802 926 424 822 894 531 969 024;
  • 31) 0.802 926 424 822 894 531 969 024 × 2 = 1 + 0.605 852 849 645 789 063 938 048;
  • 32) 0.605 852 849 645 789 063 938 048 × 2 = 1 + 0.211 705 699 291 578 127 876 096;

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).


4. 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:


0.002 394 522 356 732 212 773 426(10) =

0.0000 0000 1001 1100 1110 1101 0110 1011(2)

5. Positive number before normalization:

0.002 394 522 356 732 212 773 426(10) =

0.0000 0000 1001 1100 1110 1101 0110 1011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 9 positions to the right, so that only one non zero digit remains to the left of it:


0.002 394 522 356 732 212 773 426(10) =

0.0000 0000 1001 1100 1110 1101 0110 1011(2) =

0.0000 0000 1001 1100 1110 1101 0110 1011(2) × 20 =

1.0011 1001 1101 1010 1101 011(2) × 2-9


7. 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 0 (a positive number)

Exponent (unadjusted): -9

Mantissa (not normalized):
1.0011 1001 1101 1010 1101 011


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-9 + 2(8-1) - 1 =

(-9 + 127)(10) =

118(10)


9. 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;
  • 118 ÷ 2 = 59 + 0;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

118(10) =

0111 0110(2)


11. 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. 001 1100 1110 1101 0110 1011 =

001 1100 1110 1101 0110 1011


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (8 bits) =
0111 0110

Mantissa (23 bits) =
001 1100 1110 1101 0110 1011


Decimal number 0.002 394 522 356 732 212 773 426 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0111 0110 - 001 1100 1110 1101 0110 1011

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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