0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 1 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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
0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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: 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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) 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 1 × 2 = 0 + 0.239 191 465 468 837 731 611 454 239 191 465 468 837 731 611 478 2;
  • 2) 0.239 191 465 468 837 731 611 454 239 191 465 468 837 731 611 478 2 × 2 = 0 + 0.478 382 930 937 675 463 222 908 478 382 930 937 675 463 222 956 4;
  • 3) 0.478 382 930 937 675 463 222 908 478 382 930 937 675 463 222 956 4 × 2 = 0 + 0.956 765 861 875 350 926 445 816 956 765 861 875 350 926 445 912 8;
  • 4) 0.956 765 861 875 350 926 445 816 956 765 861 875 350 926 445 912 8 × 2 = 1 + 0.913 531 723 750 701 852 891 633 913 531 723 750 701 852 891 825 6;
  • 5) 0.913 531 723 750 701 852 891 633 913 531 723 750 701 852 891 825 6 × 2 = 1 + 0.827 063 447 501 403 705 783 267 827 063 447 501 403 705 783 651 2;
  • 6) 0.827 063 447 501 403 705 783 267 827 063 447 501 403 705 783 651 2 × 2 = 1 + 0.654 126 895 002 807 411 566 535 654 126 895 002 807 411 567 302 4;
  • 7) 0.654 126 895 002 807 411 566 535 654 126 895 002 807 411 567 302 4 × 2 = 1 + 0.308 253 790 005 614 823 133 071 308 253 790 005 614 823 134 604 8;
  • 8) 0.308 253 790 005 614 823 133 071 308 253 790 005 614 823 134 604 8 × 2 = 0 + 0.616 507 580 011 229 646 266 142 616 507 580 011 229 646 269 209 6;
  • 9) 0.616 507 580 011 229 646 266 142 616 507 580 011 229 646 269 209 6 × 2 = 1 + 0.233 015 160 022 459 292 532 285 233 015 160 022 459 292 538 419 2;
  • 10) 0.233 015 160 022 459 292 532 285 233 015 160 022 459 292 538 419 2 × 2 = 0 + 0.466 030 320 044 918 585 064 570 466 030 320 044 918 585 076 838 4;
  • 11) 0.466 030 320 044 918 585 064 570 466 030 320 044 918 585 076 838 4 × 2 = 0 + 0.932 060 640 089 837 170 129 140 932 060 640 089 837 170 153 676 8;
  • 12) 0.932 060 640 089 837 170 129 140 932 060 640 089 837 170 153 676 8 × 2 = 1 + 0.864 121 280 179 674 340 258 281 864 121 280 179 674 340 307 353 6;
  • 13) 0.864 121 280 179 674 340 258 281 864 121 280 179 674 340 307 353 6 × 2 = 1 + 0.728 242 560 359 348 680 516 563 728 242 560 359 348 680 614 707 2;
  • 14) 0.728 242 560 359 348 680 516 563 728 242 560 359 348 680 614 707 2 × 2 = 1 + 0.456 485 120 718 697 361 033 127 456 485 120 718 697 361 229 414 4;
  • 15) 0.456 485 120 718 697 361 033 127 456 485 120 718 697 361 229 414 4 × 2 = 0 + 0.912 970 241 437 394 722 066 254 912 970 241 437 394 722 458 828 8;
  • 16) 0.912 970 241 437 394 722 066 254 912 970 241 437 394 722 458 828 8 × 2 = 1 + 0.825 940 482 874 789 444 132 509 825 940 482 874 789 444 917 657 6;
  • 17) 0.825 940 482 874 789 444 132 509 825 940 482 874 789 444 917 657 6 × 2 = 1 + 0.651 880 965 749 578 888 265 019 651 880 965 749 578 889 835 315 2;
  • 18) 0.651 880 965 749 578 888 265 019 651 880 965 749 578 889 835 315 2 × 2 = 1 + 0.303 761 931 499 157 776 530 039 303 761 931 499 157 779 670 630 4;
  • 19) 0.303 761 931 499 157 776 530 039 303 761 931 499 157 779 670 630 4 × 2 = 0 + 0.607 523 862 998 315 553 060 078 607 523 862 998 315 559 341 260 8;
  • 20) 0.607 523 862 998 315 553 060 078 607 523 862 998 315 559 341 260 8 × 2 = 1 + 0.215 047 725 996 631 106 120 157 215 047 725 996 631 118 682 521 6;
  • 21) 0.215 047 725 996 631 106 120 157 215 047 725 996 631 118 682 521 6 × 2 = 0 + 0.430 095 451 993 262 212 240 314 430 095 451 993 262 237 365 043 2;
  • 22) 0.430 095 451 993 262 212 240 314 430 095 451 993 262 237 365 043 2 × 2 = 0 + 0.860 190 903 986 524 424 480 628 860 190 903 986 524 474 730 086 4;
  • 23) 0.860 190 903 986 524 424 480 628 860 190 903 986 524 474 730 086 4 × 2 = 1 + 0.720 381 807 973 048 848 961 257 720 381 807 973 048 949 460 172 8;
  • 24) 0.720 381 807 973 048 848 961 257 720 381 807 973 048 949 460 172 8 × 2 = 1 + 0.440 763 615 946 097 697 922 515 440 763 615 946 097 898 920 345 6;
  • 25) 0.440 763 615 946 097 697 922 515 440 763 615 946 097 898 920 345 6 × 2 = 0 + 0.881 527 231 892 195 395 845 030 881 527 231 892 195 797 840 691 2;
  • 26) 0.881 527 231 892 195 395 845 030 881 527 231 892 195 797 840 691 2 × 2 = 1 + 0.763 054 463 784 390 791 690 061 763 054 463 784 391 595 681 382 4;
  • 27) 0.763 054 463 784 390 791 690 061 763 054 463 784 391 595 681 382 4 × 2 = 1 + 0.526 108 927 568 781 583 380 123 526 108 927 568 783 191 362 764 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:


0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 1(10) =

0.0001 1110 1001 1101 1101 0011 011(2)

5. Positive number before normalization:

0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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:


0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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 0.119 595 732 734 418 865 805 727 119 595 732 734 418 865 805 739 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