-0.000 000 000 742 147 676 649 53 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 649 53(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
-0.000 000 000 742 147 676 649 53(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. Start with the positive version of the number:

|-0.000 000 000 742 147 676 649 53| = 0.000 000 000 742 147 676 649 53


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 742 147 676 649 53.

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) 0.000 000 000 742 147 676 649 53 × 2 = 0 + 0.000 000 001 484 295 353 299 06;
  • 2) 0.000 000 001 484 295 353 299 06 × 2 = 0 + 0.000 000 002 968 590 706 598 12;
  • 3) 0.000 000 002 968 590 706 598 12 × 2 = 0 + 0.000 000 005 937 181 413 196 24;
  • 4) 0.000 000 005 937 181 413 196 24 × 2 = 0 + 0.000 000 011 874 362 826 392 48;
  • 5) 0.000 000 011 874 362 826 392 48 × 2 = 0 + 0.000 000 023 748 725 652 784 96;
  • 6) 0.000 000 023 748 725 652 784 96 × 2 = 0 + 0.000 000 047 497 451 305 569 92;
  • 7) 0.000 000 047 497 451 305 569 92 × 2 = 0 + 0.000 000 094 994 902 611 139 84;
  • 8) 0.000 000 094 994 902 611 139 84 × 2 = 0 + 0.000 000 189 989 805 222 279 68;
  • 9) 0.000 000 189 989 805 222 279 68 × 2 = 0 + 0.000 000 379 979 610 444 559 36;
  • 10) 0.000 000 379 979 610 444 559 36 × 2 = 0 + 0.000 000 759 959 220 889 118 72;
  • 11) 0.000 000 759 959 220 889 118 72 × 2 = 0 + 0.000 001 519 918 441 778 237 44;
  • 12) 0.000 001 519 918 441 778 237 44 × 2 = 0 + 0.000 003 039 836 883 556 474 88;
  • 13) 0.000 003 039 836 883 556 474 88 × 2 = 0 + 0.000 006 079 673 767 112 949 76;
  • 14) 0.000 006 079 673 767 112 949 76 × 2 = 0 + 0.000 012 159 347 534 225 899 52;
  • 15) 0.000 012 159 347 534 225 899 52 × 2 = 0 + 0.000 024 318 695 068 451 799 04;
  • 16) 0.000 024 318 695 068 451 799 04 × 2 = 0 + 0.000 048 637 390 136 903 598 08;
  • 17) 0.000 048 637 390 136 903 598 08 × 2 = 0 + 0.000 097 274 780 273 807 196 16;
  • 18) 0.000 097 274 780 273 807 196 16 × 2 = 0 + 0.000 194 549 560 547 614 392 32;
  • 19) 0.000 194 549 560 547 614 392 32 × 2 = 0 + 0.000 389 099 121 095 228 784 64;
  • 20) 0.000 389 099 121 095 228 784 64 × 2 = 0 + 0.000 778 198 242 190 457 569 28;
  • 21) 0.000 778 198 242 190 457 569 28 × 2 = 0 + 0.001 556 396 484 380 915 138 56;
  • 22) 0.001 556 396 484 380 915 138 56 × 2 = 0 + 0.003 112 792 968 761 830 277 12;
  • 23) 0.003 112 792 968 761 830 277 12 × 2 = 0 + 0.006 225 585 937 523 660 554 24;
  • 24) 0.006 225 585 937 523 660 554 24 × 2 = 0 + 0.012 451 171 875 047 321 108 48;
  • 25) 0.012 451 171 875 047 321 108 48 × 2 = 0 + 0.024 902 343 750 094 642 216 96;
  • 26) 0.024 902 343 750 094 642 216 96 × 2 = 0 + 0.049 804 687 500 189 284 433 92;
  • 27) 0.049 804 687 500 189 284 433 92 × 2 = 0 + 0.099 609 375 000 378 568 867 84;
  • 28) 0.099 609 375 000 378 568 867 84 × 2 = 0 + 0.199 218 750 000 757 137 735 68;
  • 29) 0.199 218 750 000 757 137 735 68 × 2 = 0 + 0.398 437 500 001 514 275 471 36;
  • 30) 0.398 437 500 001 514 275 471 36 × 2 = 0 + 0.796 875 000 003 028 550 942 72;
  • 31) 0.796 875 000 003 028 550 942 72 × 2 = 1 + 0.593 750 000 006 057 101 885 44;
  • 32) 0.593 750 000 006 057 101 885 44 × 2 = 1 + 0.187 500 000 012 114 203 770 88;
  • 33) 0.187 500 000 012 114 203 770 88 × 2 = 0 + 0.375 000 000 024 228 407 541 76;
  • 34) 0.375 000 000 024 228 407 541 76 × 2 = 0 + 0.750 000 000 048 456 815 083 52;
  • 35) 0.750 000 000 048 456 815 083 52 × 2 = 1 + 0.500 000 000 096 913 630 167 04;
  • 36) 0.500 000 000 096 913 630 167 04 × 2 = 1 + 0.000 000 000 193 827 260 334 08;
  • 37) 0.000 000 000 193 827 260 334 08 × 2 = 0 + 0.000 000 000 387 654 520 668 16;
  • 38) 0.000 000 000 387 654 520 668 16 × 2 = 0 + 0.000 000 000 775 309 041 336 32;
  • 39) 0.000 000 000 775 309 041 336 32 × 2 = 0 + 0.000 000 001 550 618 082 672 64;
  • 40) 0.000 000 001 550 618 082 672 64 × 2 = 0 + 0.000 000 003 101 236 165 345 28;
  • 41) 0.000 000 003 101 236 165 345 28 × 2 = 0 + 0.000 000 006 202 472 330 690 56;
  • 42) 0.000 000 006 202 472 330 690 56 × 2 = 0 + 0.000 000 012 404 944 661 381 12;
  • 43) 0.000 000 012 404 944 661 381 12 × 2 = 0 + 0.000 000 024 809 889 322 762 24;
  • 44) 0.000 000 024 809 889 322 762 24 × 2 = 0 + 0.000 000 049 619 778 645 524 48;
  • 45) 0.000 000 049 619 778 645 524 48 × 2 = 0 + 0.000 000 099 239 557 291 048 96;
  • 46) 0.000 000 099 239 557 291 048 96 × 2 = 0 + 0.000 000 198 479 114 582 097 92;
  • 47) 0.000 000 198 479 114 582 097 92 × 2 = 0 + 0.000 000 396 958 229 164 195 84;
  • 48) 0.000 000 396 958 229 164 195 84 × 2 = 0 + 0.000 000 793 916 458 328 391 68;
  • 49) 0.000 000 793 916 458 328 391 68 × 2 = 0 + 0.000 001 587 832 916 656 783 36;
  • 50) 0.000 001 587 832 916 656 783 36 × 2 = 0 + 0.000 003 175 665 833 313 566 72;
  • 51) 0.000 003 175 665 833 313 566 72 × 2 = 0 + 0.000 006 351 331 666 627 133 44;
  • 52) 0.000 006 351 331 666 627 133 44 × 2 = 0 + 0.000 012 702 663 333 254 266 88;
  • 53) 0.000 012 702 663 333 254 266 88 × 2 = 0 + 0.000 025 405 326 666 508 533 76;
  • 54) 0.000 025 405 326 666 508 533 76 × 2 = 0 + 0.000 050 810 653 333 017 067 52;

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).


5. 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:


0.000 000 000 742 147 676 649 53(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 649 53(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 742 147 676 649 53(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(2) × 20 =

1.1001 1000 0000 0000 0000 000(2) × 2-31


8. 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 1 (a negative number)

Exponent (unadjusted): -31

Mantissa (not normalized):
1.1001 1000 0000 0000 0000 000


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(8-1) - 1 =

(-31 + 127)(10) =

96(10)


10. 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;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

96(10) =

0110 0000(2)


12. 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. 100 1100 0000 0000 0000 0000 =

100 1100 0000 0000 0000 0000


13. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (8 bits) =
0110 0000

Mantissa (23 bits) =
100 1100 0000 0000 0000 0000


Decimal number -0.000 000 000 742 147 676 649 53 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1100 0000 0000 0000 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