0.866 025 403 784 438 646 763 723 170 752 936 171 9 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.866 025 403 784 438 646 763 723 170 752 936 171 9(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
0.866 025 403 784 438 646 763 723 170 752 936 171 9(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: 0.866 025 403 784 438 646 763 723 170 752 936 171 9.

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) 0.866 025 403 784 438 646 763 723 170 752 936 171 9 × 2 = 1 + 0.732 050 807 568 877 293 527 446 341 505 872 343 8;
  • 2) 0.732 050 807 568 877 293 527 446 341 505 872 343 8 × 2 = 1 + 0.464 101 615 137 754 587 054 892 683 011 744 687 6;
  • 3) 0.464 101 615 137 754 587 054 892 683 011 744 687 6 × 2 = 0 + 0.928 203 230 275 509 174 109 785 366 023 489 375 2;
  • 4) 0.928 203 230 275 509 174 109 785 366 023 489 375 2 × 2 = 1 + 0.856 406 460 551 018 348 219 570 732 046 978 750 4;
  • 5) 0.856 406 460 551 018 348 219 570 732 046 978 750 4 × 2 = 1 + 0.712 812 921 102 036 696 439 141 464 093 957 500 8;
  • 6) 0.712 812 921 102 036 696 439 141 464 093 957 500 8 × 2 = 1 + 0.425 625 842 204 073 392 878 282 928 187 915 001 6;
  • 7) 0.425 625 842 204 073 392 878 282 928 187 915 001 6 × 2 = 0 + 0.851 251 684 408 146 785 756 565 856 375 830 003 2;
  • 8) 0.851 251 684 408 146 785 756 565 856 375 830 003 2 × 2 = 1 + 0.702 503 368 816 293 571 513 131 712 751 660 006 4;
  • 9) 0.702 503 368 816 293 571 513 131 712 751 660 006 4 × 2 = 1 + 0.405 006 737 632 587 143 026 263 425 503 320 012 8;
  • 10) 0.405 006 737 632 587 143 026 263 425 503 320 012 8 × 2 = 0 + 0.810 013 475 265 174 286 052 526 851 006 640 025 6;
  • 11) 0.810 013 475 265 174 286 052 526 851 006 640 025 6 × 2 = 1 + 0.620 026 950 530 348 572 105 053 702 013 280 051 2;
  • 12) 0.620 026 950 530 348 572 105 053 702 013 280 051 2 × 2 = 1 + 0.240 053 901 060 697 144 210 107 404 026 560 102 4;
  • 13) 0.240 053 901 060 697 144 210 107 404 026 560 102 4 × 2 = 0 + 0.480 107 802 121 394 288 420 214 808 053 120 204 8;
  • 14) 0.480 107 802 121 394 288 420 214 808 053 120 204 8 × 2 = 0 + 0.960 215 604 242 788 576 840 429 616 106 240 409 6;
  • 15) 0.960 215 604 242 788 576 840 429 616 106 240 409 6 × 2 = 1 + 0.920 431 208 485 577 153 680 859 232 212 480 819 2;
  • 16) 0.920 431 208 485 577 153 680 859 232 212 480 819 2 × 2 = 1 + 0.840 862 416 971 154 307 361 718 464 424 961 638 4;
  • 17) 0.840 862 416 971 154 307 361 718 464 424 961 638 4 × 2 = 1 + 0.681 724 833 942 308 614 723 436 928 849 923 276 8;
  • 18) 0.681 724 833 942 308 614 723 436 928 849 923 276 8 × 2 = 1 + 0.363 449 667 884 617 229 446 873 857 699 846 553 6;
  • 19) 0.363 449 667 884 617 229 446 873 857 699 846 553 6 × 2 = 0 + 0.726 899 335 769 234 458 893 747 715 399 693 107 2;
  • 20) 0.726 899 335 769 234 458 893 747 715 399 693 107 2 × 2 = 1 + 0.453 798 671 538 468 917 787 495 430 799 386 214 4;
  • 21) 0.453 798 671 538 468 917 787 495 430 799 386 214 4 × 2 = 0 + 0.907 597 343 076 937 835 574 990 861 598 772 428 8;
  • 22) 0.907 597 343 076 937 835 574 990 861 598 772 428 8 × 2 = 1 + 0.815 194 686 153 875 671 149 981 723 197 544 857 6;
  • 23) 0.815 194 686 153 875 671 149 981 723 197 544 857 6 × 2 = 1 + 0.630 389 372 307 751 342 299 963 446 395 089 715 2;
  • 24) 0.630 389 372 307 751 342 299 963 446 395 089 715 2 × 2 = 1 + 0.260 778 744 615 502 684 599 926 892 790 179 430 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).


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:


0.866 025 403 784 438 646 763 723 170 752 936 171 9(10) =

0.1101 1101 1011 0011 1101 0111(2)

5. Positive number before normalization:

0.866 025 403 784 438 646 763 723 170 752 936 171 9(10) =

0.1101 1101 1011 0011 1101 0111(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.866 025 403 784 438 646 763 723 170 752 936 171 9(10) =

0.1101 1101 1011 0011 1101 0111(2) =

0.1101 1101 1011 0011 1101 0111(2) × 20 =

1.1011 1011 0110 0111 1010 111(2) × 2-1


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): -1

Mantissa (not normalized):
1.1011 1011 0110 0111 1010 111


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-1 + 127)(10) =

126(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;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

126(10) =

0111 1110(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. 101 1101 1011 0011 1101 0111 =

101 1101 1011 0011 1101 0111


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 1110

Mantissa (23 bits) =
101 1101 1011 0011 1101 0111


Decimal number 0.866 025 403 784 438 646 763 723 170 752 936 171 9 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0111 1110 - 101 1101 1011 0011 1101 0111

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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