165.395 862 948 928 595 415 054 587 647 318 840 021 1 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 165.395 862 948 928 595 415 054 587 647 318 840 021 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
165.395 862 948 928 595 415 054 587 647 318 840 021 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: 165.
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;
  • 165 ÷ 2 = 82 + 1;
  • 82 ÷ 2 = 41 + 0;
  • 41 ÷ 2 = 20 + 1;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

165(10) =

1010 0101(2)


3. Convert to binary (base 2) the fractional part: 0.395 862 948 928 595 415 054 587 647 318 840 021 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) 0.395 862 948 928 595 415 054 587 647 318 840 021 1 × 2 = 0 + 0.791 725 897 857 190 830 109 175 294 637 680 042 2;
  • 2) 0.791 725 897 857 190 830 109 175 294 637 680 042 2 × 2 = 1 + 0.583 451 795 714 381 660 218 350 589 275 360 084 4;
  • 3) 0.583 451 795 714 381 660 218 350 589 275 360 084 4 × 2 = 1 + 0.166 903 591 428 763 320 436 701 178 550 720 168 8;
  • 4) 0.166 903 591 428 763 320 436 701 178 550 720 168 8 × 2 = 0 + 0.333 807 182 857 526 640 873 402 357 101 440 337 6;
  • 5) 0.333 807 182 857 526 640 873 402 357 101 440 337 6 × 2 = 0 + 0.667 614 365 715 053 281 746 804 714 202 880 675 2;
  • 6) 0.667 614 365 715 053 281 746 804 714 202 880 675 2 × 2 = 1 + 0.335 228 731 430 106 563 493 609 428 405 761 350 4;
  • 7) 0.335 228 731 430 106 563 493 609 428 405 761 350 4 × 2 = 0 + 0.670 457 462 860 213 126 987 218 856 811 522 700 8;
  • 8) 0.670 457 462 860 213 126 987 218 856 811 522 700 8 × 2 = 1 + 0.340 914 925 720 426 253 974 437 713 623 045 401 6;
  • 9) 0.340 914 925 720 426 253 974 437 713 623 045 401 6 × 2 = 0 + 0.681 829 851 440 852 507 948 875 427 246 090 803 2;
  • 10) 0.681 829 851 440 852 507 948 875 427 246 090 803 2 × 2 = 1 + 0.363 659 702 881 705 015 897 750 854 492 181 606 4;
  • 11) 0.363 659 702 881 705 015 897 750 854 492 181 606 4 × 2 = 0 + 0.727 319 405 763 410 031 795 501 708 984 363 212 8;
  • 12) 0.727 319 405 763 410 031 795 501 708 984 363 212 8 × 2 = 1 + 0.454 638 811 526 820 063 591 003 417 968 726 425 6;
  • 13) 0.454 638 811 526 820 063 591 003 417 968 726 425 6 × 2 = 0 + 0.909 277 623 053 640 127 182 006 835 937 452 851 2;
  • 14) 0.909 277 623 053 640 127 182 006 835 937 452 851 2 × 2 = 1 + 0.818 555 246 107 280 254 364 013 671 874 905 702 4;
  • 15) 0.818 555 246 107 280 254 364 013 671 874 905 702 4 × 2 = 1 + 0.637 110 492 214 560 508 728 027 343 749 811 404 8;
  • 16) 0.637 110 492 214 560 508 728 027 343 749 811 404 8 × 2 = 1 + 0.274 220 984 429 121 017 456 054 687 499 622 809 6;
  • 17) 0.274 220 984 429 121 017 456 054 687 499 622 809 6 × 2 = 0 + 0.548 441 968 858 242 034 912 109 374 999 245 619 2;
  • 18) 0.548 441 968 858 242 034 912 109 374 999 245 619 2 × 2 = 1 + 0.096 883 937 716 484 069 824 218 749 998 491 238 4;
  • 19) 0.096 883 937 716 484 069 824 218 749 998 491 238 4 × 2 = 0 + 0.193 767 875 432 968 139 648 437 499 996 982 476 8;
  • 20) 0.193 767 875 432 968 139 648 437 499 996 982 476 8 × 2 = 0 + 0.387 535 750 865 936 279 296 874 999 993 964 953 6;
  • 21) 0.387 535 750 865 936 279 296 874 999 993 964 953 6 × 2 = 0 + 0.775 071 501 731 872 558 593 749 999 987 929 907 2;
  • 22) 0.775 071 501 731 872 558 593 749 999 987 929 907 2 × 2 = 1 + 0.550 143 003 463 745 117 187 499 999 975 859 814 4;
  • 23) 0.550 143 003 463 745 117 187 499 999 975 859 814 4 × 2 = 1 + 0.100 286 006 927 490 234 374 999 999 951 719 628 8;
  • 24) 0.100 286 006 927 490 234 374 999 999 951 719 628 8 × 2 = 0 + 0.200 572 013 854 980 468 749 999 999 903 439 257 6;

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:


0.395 862 948 928 595 415 054 587 647 318 840 021 1(10) =

0.0110 0101 0101 0111 0100 0110(2)

5. Positive number before normalization:

165.395 862 948 928 595 415 054 587 647 318 840 021 1(10) =

1010 0101.0110 0101 0101 0111 0100 0110(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the left, so that only one non zero digit remains to the left of it:


165.395 862 948 928 595 415 054 587 647 318 840 021 1(10) =

1010 0101.0110 0101 0101 0111 0100 0110(2) =

1010 0101.0110 0101 0101 0111 0100 0110(2) × 20 =

1.0100 1010 1100 1010 1010 1110 1000 110(2) × 27


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): 7

Mantissa (not normalized):
1.0100 1010 1100 1010 1010 1110 1000 110


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

7 + 2(8-1) - 1 =

(7 + 127)(10) =

134(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;
  • 134 ÷ 2 = 67 + 0;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 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) =

134(10) =

1000 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, 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. 010 0101 0110 0101 0101 0111 0100 0110 =

010 0101 0110 0101 0101 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) =
1000 0110

Mantissa (23 bits) =
010 0101 0110 0101 0101 0111


Decimal number 165.395 862 948 928 595 415 054 587 647 318 840 021 1 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 1000 0110 - 010 0101 0110 0101 0101 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