0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1(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
0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1(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: 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1.

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) 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1 × 2 = 0 + 0.239 191 465 468 837 731 611 454 239 191 465 468 837 731 611 506 2;
  • 2) 0.239 191 465 468 837 731 611 454 239 191 465 468 837 731 611 506 2 × 2 = 0 + 0.478 382 930 937 675 463 222 908 478 382 930 937 675 463 223 012 4;
  • 3) 0.478 382 930 937 675 463 222 908 478 382 930 937 675 463 223 012 4 × 2 = 0 + 0.956 765 861 875 350 926 445 816 956 765 861 875 350 926 446 024 8;
  • 4) 0.956 765 861 875 350 926 445 816 956 765 861 875 350 926 446 024 8 × 2 = 1 + 0.913 531 723 750 701 852 891 633 913 531 723 750 701 852 892 049 6;
  • 5) 0.913 531 723 750 701 852 891 633 913 531 723 750 701 852 892 049 6 × 2 = 1 + 0.827 063 447 501 403 705 783 267 827 063 447 501 403 705 784 099 2;
  • 6) 0.827 063 447 501 403 705 783 267 827 063 447 501 403 705 784 099 2 × 2 = 1 + 0.654 126 895 002 807 411 566 535 654 126 895 002 807 411 568 198 4;
  • 7) 0.654 126 895 002 807 411 566 535 654 126 895 002 807 411 568 198 4 × 2 = 1 + 0.308 253 790 005 614 823 133 071 308 253 790 005 614 823 136 396 8;
  • 8) 0.308 253 790 005 614 823 133 071 308 253 790 005 614 823 136 396 8 × 2 = 0 + 0.616 507 580 011 229 646 266 142 616 507 580 011 229 646 272 793 6;
  • 9) 0.616 507 580 011 229 646 266 142 616 507 580 011 229 646 272 793 6 × 2 = 1 + 0.233 015 160 022 459 292 532 285 233 015 160 022 459 292 545 587 2;
  • 10) 0.233 015 160 022 459 292 532 285 233 015 160 022 459 292 545 587 2 × 2 = 0 + 0.466 030 320 044 918 585 064 570 466 030 320 044 918 585 091 174 4;
  • 11) 0.466 030 320 044 918 585 064 570 466 030 320 044 918 585 091 174 4 × 2 = 0 + 0.932 060 640 089 837 170 129 140 932 060 640 089 837 170 182 348 8;
  • 12) 0.932 060 640 089 837 170 129 140 932 060 640 089 837 170 182 348 8 × 2 = 1 + 0.864 121 280 179 674 340 258 281 864 121 280 179 674 340 364 697 6;
  • 13) 0.864 121 280 179 674 340 258 281 864 121 280 179 674 340 364 697 6 × 2 = 1 + 0.728 242 560 359 348 680 516 563 728 242 560 359 348 680 729 395 2;
  • 14) 0.728 242 560 359 348 680 516 563 728 242 560 359 348 680 729 395 2 × 2 = 1 + 0.456 485 120 718 697 361 033 127 456 485 120 718 697 361 458 790 4;
  • 15) 0.456 485 120 718 697 361 033 127 456 485 120 718 697 361 458 790 4 × 2 = 0 + 0.912 970 241 437 394 722 066 254 912 970 241 437 394 722 917 580 8;
  • 16) 0.912 970 241 437 394 722 066 254 912 970 241 437 394 722 917 580 8 × 2 = 1 + 0.825 940 482 874 789 444 132 509 825 940 482 874 789 445 835 161 6;
  • 17) 0.825 940 482 874 789 444 132 509 825 940 482 874 789 445 835 161 6 × 2 = 1 + 0.651 880 965 749 578 888 265 019 651 880 965 749 578 891 670 323 2;
  • 18) 0.651 880 965 749 578 888 265 019 651 880 965 749 578 891 670 323 2 × 2 = 1 + 0.303 761 931 499 157 776 530 039 303 761 931 499 157 783 340 646 4;
  • 19) 0.303 761 931 499 157 776 530 039 303 761 931 499 157 783 340 646 4 × 2 = 0 + 0.607 523 862 998 315 553 060 078 607 523 862 998 315 566 681 292 8;
  • 20) 0.607 523 862 998 315 553 060 078 607 523 862 998 315 566 681 292 8 × 2 = 1 + 0.215 047 725 996 631 106 120 157 215 047 725 996 631 133 362 585 6;
  • 21) 0.215 047 725 996 631 106 120 157 215 047 725 996 631 133 362 585 6 × 2 = 0 + 0.430 095 451 993 262 212 240 314 430 095 451 993 262 266 725 171 2;
  • 22) 0.430 095 451 993 262 212 240 314 430 095 451 993 262 266 725 171 2 × 2 = 0 + 0.860 190 903 986 524 424 480 628 860 190 903 986 524 533 450 342 4;
  • 23) 0.860 190 903 986 524 424 480 628 860 190 903 986 524 533 450 342 4 × 2 = 1 + 0.720 381 807 973 048 848 961 257 720 381 807 973 049 066 900 684 8;
  • 24) 0.720 381 807 973 048 848 961 257 720 381 807 973 049 066 900 684 8 × 2 = 1 + 0.440 763 615 946 097 697 922 515 440 763 615 946 098 133 801 369 6;
  • 25) 0.440 763 615 946 097 697 922 515 440 763 615 946 098 133 801 369 6 × 2 = 0 + 0.881 527 231 892 195 395 845 030 881 527 231 892 196 267 602 739 2;
  • 26) 0.881 527 231 892 195 395 845 030 881 527 231 892 196 267 602 739 2 × 2 = 1 + 0.763 054 463 784 390 791 690 061 763 054 463 784 392 535 205 478 4;
  • 27) 0.763 054 463 784 390 791 690 061 763 054 463 784 392 535 205 478 4 × 2 = 1 + 0.526 108 927 568 781 583 380 123 526 108 927 568 785 070 410 956 8;

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:


0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1(10) =

0.0001 1110 1001 1101 1101 0011 011(2)

5. Positive number before normalization:

0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1(10) =

0.0001 1110 1001 1101 1101 0011 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the right, so that only one non zero digit remains to the left of it:


0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1(10) =

0.0001 1110 1001 1101 1101 0011 011(2) =

0.0001 1110 1001 1101 1101 0011 011(2) × 20 =

1.1110 1001 1101 1101 0011 011(2) × 2-4


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): -4

Mantissa (not normalized):
1.1110 1001 1101 1101 0011 011


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-4 + 2(8-1) - 1 =

(-4 + 127)(10) =

123(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;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

123(10) =

0111 1011(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. 111 0100 1110 1110 1001 1011 =

111 0100 1110 1110 1001 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 1011

Mantissa (23 bits) =
111 0100 1110 1110 1001 1011


Decimal number 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 753 1 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0111 1011 - 111 0100 1110 1110 1001 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