12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821(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
12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821(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: 12.
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;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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.

12(10) =

1100(2)


3. Convert to binary (base 2) the fractional part: 0.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821.

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) 0.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821 × 2 = 1 + 0.315 979 375 555 555 550 746 750 675 075 075 075 075 071 475 642;
  • 2) 0.315 979 375 555 555 550 746 750 675 075 075 075 075 071 475 642 × 2 = 0 + 0.631 958 751 111 111 101 493 501 350 150 150 150 150 142 951 284;
  • 3) 0.631 958 751 111 111 101 493 501 350 150 150 150 150 142 951 284 × 2 = 1 + 0.263 917 502 222 222 202 987 002 700 300 300 300 300 285 902 568;
  • 4) 0.263 917 502 222 222 202 987 002 700 300 300 300 300 285 902 568 × 2 = 0 + 0.527 835 004 444 444 405 974 005 400 600 600 600 600 571 805 136;
  • 5) 0.527 835 004 444 444 405 974 005 400 600 600 600 600 571 805 136 × 2 = 1 + 0.055 670 008 888 888 811 948 010 801 201 201 201 201 143 610 272;
  • 6) 0.055 670 008 888 888 811 948 010 801 201 201 201 201 143 610 272 × 2 = 0 + 0.111 340 017 777 777 623 896 021 602 402 402 402 402 287 220 544;
  • 7) 0.111 340 017 777 777 623 896 021 602 402 402 402 402 287 220 544 × 2 = 0 + 0.222 680 035 555 555 247 792 043 204 804 804 804 804 574 441 088;
  • 8) 0.222 680 035 555 555 247 792 043 204 804 804 804 804 574 441 088 × 2 = 0 + 0.445 360 071 111 110 495 584 086 409 609 609 609 609 148 882 176;
  • 9) 0.445 360 071 111 110 495 584 086 409 609 609 609 609 148 882 176 × 2 = 0 + 0.890 720 142 222 220 991 168 172 819 219 219 219 218 297 764 352;
  • 10) 0.890 720 142 222 220 991 168 172 819 219 219 219 218 297 764 352 × 2 = 1 + 0.781 440 284 444 441 982 336 345 638 438 438 438 436 595 528 704;
  • 11) 0.781 440 284 444 441 982 336 345 638 438 438 438 436 595 528 704 × 2 = 1 + 0.562 880 568 888 883 964 672 691 276 876 876 876 873 191 057 408;
  • 12) 0.562 880 568 888 883 964 672 691 276 876 876 876 873 191 057 408 × 2 = 1 + 0.125 761 137 777 767 929 345 382 553 753 753 753 746 382 114 816;
  • 13) 0.125 761 137 777 767 929 345 382 553 753 753 753 746 382 114 816 × 2 = 0 + 0.251 522 275 555 535 858 690 765 107 507 507 507 492 764 229 632;
  • 14) 0.251 522 275 555 535 858 690 765 107 507 507 507 492 764 229 632 × 2 = 0 + 0.503 044 551 111 071 717 381 530 215 015 015 014 985 528 459 264;
  • 15) 0.503 044 551 111 071 717 381 530 215 015 015 014 985 528 459 264 × 2 = 1 + 0.006 089 102 222 143 434 763 060 430 030 030 029 971 056 918 528;
  • 16) 0.006 089 102 222 143 434 763 060 430 030 030 029 971 056 918 528 × 2 = 0 + 0.012 178 204 444 286 869 526 120 860 060 060 059 942 113 837 056;
  • 17) 0.012 178 204 444 286 869 526 120 860 060 060 059 942 113 837 056 × 2 = 0 + 0.024 356 408 888 573 739 052 241 720 120 120 119 884 227 674 112;
  • 18) 0.024 356 408 888 573 739 052 241 720 120 120 119 884 227 674 112 × 2 = 0 + 0.048 712 817 777 147 478 104 483 440 240 240 239 768 455 348 224;
  • 19) 0.048 712 817 777 147 478 104 483 440 240 240 239 768 455 348 224 × 2 = 0 + 0.097 425 635 554 294 956 208 966 880 480 480 479 536 910 696 448;
  • 20) 0.097 425 635 554 294 956 208 966 880 480 480 479 536 910 696 448 × 2 = 0 + 0.194 851 271 108 589 912 417 933 760 960 960 959 073 821 392 896;
  • 21) 0.194 851 271 108 589 912 417 933 760 960 960 959 073 821 392 896 × 2 = 0 + 0.389 702 542 217 179 824 835 867 521 921 921 918 147 642 785 792;
  • 22) 0.389 702 542 217 179 824 835 867 521 921 921 918 147 642 785 792 × 2 = 0 + 0.779 405 084 434 359 649 671 735 043 843 843 836 295 285 571 584;
  • 23) 0.779 405 084 434 359 649 671 735 043 843 843 836 295 285 571 584 × 2 = 1 + 0.558 810 168 868 719 299 343 470 087 687 687 672 590 571 143 168;
  • 24) 0.558 810 168 868 719 299 343 470 087 687 687 672 590 571 143 168 × 2 = 1 + 0.117 620 337 737 438 598 686 940 175 375 375 345 181 142 286 336;

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:


0.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821(10) =

0.1010 1000 0111 0010 0000 0011(2)

5. Positive number before normalization:

12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821(10) =

1100.1010 1000 0111 0010 0000 0011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 3 positions to the left, so that only one non zero digit remains to the left of it:


12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821(10) =

1100.1010 1000 0111 0010 0000 0011(2) =

1100.1010 1000 0111 0010 0000 0011(2) × 20 =

1.1001 0101 0000 1110 0100 0000 011(2) × 23


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): 3

Mantissa (not normalized):
1.1001 0101 0000 1110 0100 0000 011


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

3 + 2(8-1) - 1 =

(3 + 127)(10) =

130(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;
  • 130 ÷ 2 = 65 + 0;
  • 65 ÷ 2 = 32 + 1;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

130(10) =

1000 0010(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 100 1010 1000 0111 0010 0000 0011 =

100 1010 1000 0111 0010 0000


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) =
1000 0010

Mantissa (23 bits) =
100 1010 1000 0111 0010 0000


Decimal number 12.657 989 687 777 777 775 373 375 337 537 537 537 537 535 737 821 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 1000 0010 - 100 1010 1000 0111 0010 0000

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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