32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 1.463 422 085 898 363 477 004 651 08 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 1.463 422 085 898 363 477 004 651 08(10) converted and written in 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: 1.
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;
  • 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.


1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.463 422 085 898 363 477 004 651 08.

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) 0.463 422 085 898 363 477 004 651 08 × 2 = 0 + 0.926 844 171 796 726 954 009 302 16;
  • 2) 0.926 844 171 796 726 954 009 302 16 × 2 = 1 + 0.853 688 343 593 453 908 018 604 32;
  • 3) 0.853 688 343 593 453 908 018 604 32 × 2 = 1 + 0.707 376 687 186 907 816 037 208 64;
  • 4) 0.707 376 687 186 907 816 037 208 64 × 2 = 1 + 0.414 753 374 373 815 632 074 417 28;
  • 5) 0.414 753 374 373 815 632 074 417 28 × 2 = 0 + 0.829 506 748 747 631 264 148 834 56;
  • 6) 0.829 506 748 747 631 264 148 834 56 × 2 = 1 + 0.659 013 497 495 262 528 297 669 12;
  • 7) 0.659 013 497 495 262 528 297 669 12 × 2 = 1 + 0.318 026 994 990 525 056 595 338 24;
  • 8) 0.318 026 994 990 525 056 595 338 24 × 2 = 0 + 0.636 053 989 981 050 113 190 676 48;
  • 9) 0.636 053 989 981 050 113 190 676 48 × 2 = 1 + 0.272 107 979 962 100 226 381 352 96;
  • 10) 0.272 107 979 962 100 226 381 352 96 × 2 = 0 + 0.544 215 959 924 200 452 762 705 92;
  • 11) 0.544 215 959 924 200 452 762 705 92 × 2 = 1 + 0.088 431 919 848 400 905 525 411 84;
  • 12) 0.088 431 919 848 400 905 525 411 84 × 2 = 0 + 0.176 863 839 696 801 811 050 823 68;
  • 13) 0.176 863 839 696 801 811 050 823 68 × 2 = 0 + 0.353 727 679 393 603 622 101 647 36;
  • 14) 0.353 727 679 393 603 622 101 647 36 × 2 = 0 + 0.707 455 358 787 207 244 203 294 72;
  • 15) 0.707 455 358 787 207 244 203 294 72 × 2 = 1 + 0.414 910 717 574 414 488 406 589 44;
  • 16) 0.414 910 717 574 414 488 406 589 44 × 2 = 0 + 0.829 821 435 148 828 976 813 178 88;
  • 17) 0.829 821 435 148 828 976 813 178 88 × 2 = 1 + 0.659 642 870 297 657 953 626 357 76;
  • 18) 0.659 642 870 297 657 953 626 357 76 × 2 = 1 + 0.319 285 740 595 315 907 252 715 52;
  • 19) 0.319 285 740 595 315 907 252 715 52 × 2 = 0 + 0.638 571 481 190 631 814 505 431 04;
  • 20) 0.638 571 481 190 631 814 505 431 04 × 2 = 1 + 0.277 142 962 381 263 629 010 862 08;
  • 21) 0.277 142 962 381 263 629 010 862 08 × 2 = 0 + 0.554 285 924 762 527 258 021 724 16;
  • 22) 0.554 285 924 762 527 258 021 724 16 × 2 = 1 + 0.108 571 849 525 054 516 043 448 32;
  • 23) 0.108 571 849 525 054 516 043 448 32 × 2 = 0 + 0.217 143 699 050 109 032 086 896 64;
  • 24) 0.217 143 699 050 109 032 086 896 64 × 2 = 0 + 0.434 287 398 100 218 064 173 793 28;

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...)


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:


0.463 422 085 898 363 477 004 651 08(10) =

0.0111 0110 1010 0010 1101 0100(2)


5. Positive number before normalization:

1.463 422 085 898 363 477 004 651 08(10) =

1.0111 0110 1010 0010 1101 0100(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.463 422 085 898 363 477 004 651 08(10) =

1.0111 0110 1010 0010 1101 0100(2) =

1.0111 0110 1010 0010 1101 0100(2) × 20


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): 0

Mantissa (not normalized):
1.0111 0110 1010 0010 1101 0100


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(8-1) - 1 =

(0 + 127)(10) =

127(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;
  • 127 ÷ 2 = 63 + 1;
  • 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) =

127(10) =

0111 1111(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. 011 1011 0101 0001 0110 1010 0 =

011 1011 0101 0001 0110 1010


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 1111

Mantissa (23 bits) =
011 1011 0101 0001 0110 1010


The base ten decimal number 1.463 422 085 898 363 477 004 651 08 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 0111 1111 - 011 1011 0101 0001 0110 1010

More operations with decimal numbers converted to 32 bit single precision IEEE 754 binary floating point representation:

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» Number 1.463 422 085 898 363 477 004 651 09 converted from decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point representation = ?

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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