32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 3.141 592 653 589 793 238 462 643 383 27 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 3.141 592 653 589 793 238 462 643 383 27(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: 3.
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;
  • 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.


3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.141 592 653 589 793 238 462 643 383 27.

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) 0.141 592 653 589 793 238 462 643 383 27 × 2 = 0 + 0.283 185 307 179 586 476 925 286 766 54;
  • 2) 0.283 185 307 179 586 476 925 286 766 54 × 2 = 0 + 0.566 370 614 359 172 953 850 573 533 08;
  • 3) 0.566 370 614 359 172 953 850 573 533 08 × 2 = 1 + 0.132 741 228 718 345 907 701 147 066 16;
  • 4) 0.132 741 228 718 345 907 701 147 066 16 × 2 = 0 + 0.265 482 457 436 691 815 402 294 132 32;
  • 5) 0.265 482 457 436 691 815 402 294 132 32 × 2 = 0 + 0.530 964 914 873 383 630 804 588 264 64;
  • 6) 0.530 964 914 873 383 630 804 588 264 64 × 2 = 1 + 0.061 929 829 746 767 261 609 176 529 28;
  • 7) 0.061 929 829 746 767 261 609 176 529 28 × 2 = 0 + 0.123 859 659 493 534 523 218 353 058 56;
  • 8) 0.123 859 659 493 534 523 218 353 058 56 × 2 = 0 + 0.247 719 318 987 069 046 436 706 117 12;
  • 9) 0.247 719 318 987 069 046 436 706 117 12 × 2 = 0 + 0.495 438 637 974 138 092 873 412 234 24;
  • 10) 0.495 438 637 974 138 092 873 412 234 24 × 2 = 0 + 0.990 877 275 948 276 185 746 824 468 48;
  • 11) 0.990 877 275 948 276 185 746 824 468 48 × 2 = 1 + 0.981 754 551 896 552 371 493 648 936 96;
  • 12) 0.981 754 551 896 552 371 493 648 936 96 × 2 = 1 + 0.963 509 103 793 104 742 987 297 873 92;
  • 13) 0.963 509 103 793 104 742 987 297 873 92 × 2 = 1 + 0.927 018 207 586 209 485 974 595 747 84;
  • 14) 0.927 018 207 586 209 485 974 595 747 84 × 2 = 1 + 0.854 036 415 172 418 971 949 191 495 68;
  • 15) 0.854 036 415 172 418 971 949 191 495 68 × 2 = 1 + 0.708 072 830 344 837 943 898 382 991 36;
  • 16) 0.708 072 830 344 837 943 898 382 991 36 × 2 = 1 + 0.416 145 660 689 675 887 796 765 982 72;
  • 17) 0.416 145 660 689 675 887 796 765 982 72 × 2 = 0 + 0.832 291 321 379 351 775 593 531 965 44;
  • 18) 0.832 291 321 379 351 775 593 531 965 44 × 2 = 1 + 0.664 582 642 758 703 551 187 063 930 88;
  • 19) 0.664 582 642 758 703 551 187 063 930 88 × 2 = 1 + 0.329 165 285 517 407 102 374 127 861 76;
  • 20) 0.329 165 285 517 407 102 374 127 861 76 × 2 = 0 + 0.658 330 571 034 814 204 748 255 723 52;
  • 21) 0.658 330 571 034 814 204 748 255 723 52 × 2 = 1 + 0.316 661 142 069 628 409 496 511 447 04;
  • 22) 0.316 661 142 069 628 409 496 511 447 04 × 2 = 0 + 0.633 322 284 139 256 818 993 022 894 08;
  • 23) 0.633 322 284 139 256 818 993 022 894 08 × 2 = 1 + 0.266 644 568 278 513 637 986 045 788 16;
  • 24) 0.266 644 568 278 513 637 986 045 788 16 × 2 = 0 + 0.533 289 136 557 027 275 972 091 576 32;

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:


0.141 592 653 589 793 238 462 643 383 27(10) =

0.0010 0100 0011 1111 0110 1010(2)


5. Positive number before normalization:

3.141 592 653 589 793 238 462 643 383 27(10) =

11.0010 0100 0011 1111 0110 1010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.141 592 653 589 793 238 462 643 383 27(10) =

11.0010 0100 0011 1111 0110 1010(2) =

11.0010 0100 0011 1111 0110 1010(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0(2) × 21


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): 1

Mantissa (not normalized):
1.1001 0010 0001 1111 1011 0101 0


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(8-1) - 1 =

(1 + 127)(10) =

128(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;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 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) =

128(10) =

1000 0000(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 1001 0000 1111 1101 1010 10 =

100 1001 0000 1111 1101 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) =
1000 0000

Mantissa (23 bits) =
100 1001 0000 1111 1101 1010


The base ten decimal number 3.141 592 653 589 793 238 462 643 383 27 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1000 0000 - 100 1001 0000 1111 1101 1010

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