-0.000 000 000 742 147 676 646 509 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 509(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
-0.000 000 000 742 147 676 646 509(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:

|-0.000 000 000 742 147 676 646 509| = 0.000 000 000 742 147 676 646 509


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: 0.000 000 000 742 147 676 646 509.

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) 0.000 000 000 742 147 676 646 509 × 2 = 0 + 0.000 000 001 484 295 353 293 018;
  • 2) 0.000 000 001 484 295 353 293 018 × 2 = 0 + 0.000 000 002 968 590 706 586 036;
  • 3) 0.000 000 002 968 590 706 586 036 × 2 = 0 + 0.000 000 005 937 181 413 172 072;
  • 4) 0.000 000 005 937 181 413 172 072 × 2 = 0 + 0.000 000 011 874 362 826 344 144;
  • 5) 0.000 000 011 874 362 826 344 144 × 2 = 0 + 0.000 000 023 748 725 652 688 288;
  • 6) 0.000 000 023 748 725 652 688 288 × 2 = 0 + 0.000 000 047 497 451 305 376 576;
  • 7) 0.000 000 047 497 451 305 376 576 × 2 = 0 + 0.000 000 094 994 902 610 753 152;
  • 8) 0.000 000 094 994 902 610 753 152 × 2 = 0 + 0.000 000 189 989 805 221 506 304;
  • 9) 0.000 000 189 989 805 221 506 304 × 2 = 0 + 0.000 000 379 979 610 443 012 608;
  • 10) 0.000 000 379 979 610 443 012 608 × 2 = 0 + 0.000 000 759 959 220 886 025 216;
  • 11) 0.000 000 759 959 220 886 025 216 × 2 = 0 + 0.000 001 519 918 441 772 050 432;
  • 12) 0.000 001 519 918 441 772 050 432 × 2 = 0 + 0.000 003 039 836 883 544 100 864;
  • 13) 0.000 003 039 836 883 544 100 864 × 2 = 0 + 0.000 006 079 673 767 088 201 728;
  • 14) 0.000 006 079 673 767 088 201 728 × 2 = 0 + 0.000 012 159 347 534 176 403 456;
  • 15) 0.000 012 159 347 534 176 403 456 × 2 = 0 + 0.000 024 318 695 068 352 806 912;
  • 16) 0.000 024 318 695 068 352 806 912 × 2 = 0 + 0.000 048 637 390 136 705 613 824;
  • 17) 0.000 048 637 390 136 705 613 824 × 2 = 0 + 0.000 097 274 780 273 411 227 648;
  • 18) 0.000 097 274 780 273 411 227 648 × 2 = 0 + 0.000 194 549 560 546 822 455 296;
  • 19) 0.000 194 549 560 546 822 455 296 × 2 = 0 + 0.000 389 099 121 093 644 910 592;
  • 20) 0.000 389 099 121 093 644 910 592 × 2 = 0 + 0.000 778 198 242 187 289 821 184;
  • 21) 0.000 778 198 242 187 289 821 184 × 2 = 0 + 0.001 556 396 484 374 579 642 368;
  • 22) 0.001 556 396 484 374 579 642 368 × 2 = 0 + 0.003 112 792 968 749 159 284 736;
  • 23) 0.003 112 792 968 749 159 284 736 × 2 = 0 + 0.006 225 585 937 498 318 569 472;
  • 24) 0.006 225 585 937 498 318 569 472 × 2 = 0 + 0.012 451 171 874 996 637 138 944;
  • 25) 0.012 451 171 874 996 637 138 944 × 2 = 0 + 0.024 902 343 749 993 274 277 888;
  • 26) 0.024 902 343 749 993 274 277 888 × 2 = 0 + 0.049 804 687 499 986 548 555 776;
  • 27) 0.049 804 687 499 986 548 555 776 × 2 = 0 + 0.099 609 374 999 973 097 111 552;
  • 28) 0.099 609 374 999 973 097 111 552 × 2 = 0 + 0.199 218 749 999 946 194 223 104;
  • 29) 0.199 218 749 999 946 194 223 104 × 2 = 0 + 0.398 437 499 999 892 388 446 208;
  • 30) 0.398 437 499 999 892 388 446 208 × 2 = 0 + 0.796 874 999 999 784 776 892 416;
  • 31) 0.796 874 999 999 784 776 892 416 × 2 = 1 + 0.593 749 999 999 569 553 784 832;
  • 32) 0.593 749 999 999 569 553 784 832 × 2 = 1 + 0.187 499 999 999 139 107 569 664;
  • 33) 0.187 499 999 999 139 107 569 664 × 2 = 0 + 0.374 999 999 998 278 215 139 328;
  • 34) 0.374 999 999 998 278 215 139 328 × 2 = 0 + 0.749 999 999 996 556 430 278 656;
  • 35) 0.749 999 999 996 556 430 278 656 × 2 = 1 + 0.499 999 999 993 112 860 557 312;
  • 36) 0.499 999 999 993 112 860 557 312 × 2 = 0 + 0.999 999 999 986 225 721 114 624;
  • 37) 0.999 999 999 986 225 721 114 624 × 2 = 1 + 0.999 999 999 972 451 442 229 248;
  • 38) 0.999 999 999 972 451 442 229 248 × 2 = 1 + 0.999 999 999 944 902 884 458 496;
  • 39) 0.999 999 999 944 902 884 458 496 × 2 = 1 + 0.999 999 999 889 805 768 916 992;
  • 40) 0.999 999 999 889 805 768 916 992 × 2 = 1 + 0.999 999 999 779 611 537 833 984;
  • 41) 0.999 999 999 779 611 537 833 984 × 2 = 1 + 0.999 999 999 559 223 075 667 968;
  • 42) 0.999 999 999 559 223 075 667 968 × 2 = 1 + 0.999 999 999 118 446 151 335 936;
  • 43) 0.999 999 999 118 446 151 335 936 × 2 = 1 + 0.999 999 998 236 892 302 671 872;
  • 44) 0.999 999 998 236 892 302 671 872 × 2 = 1 + 0.999 999 996 473 784 605 343 744;
  • 45) 0.999 999 996 473 784 605 343 744 × 2 = 1 + 0.999 999 992 947 569 210 687 488;
  • 46) 0.999 999 992 947 569 210 687 488 × 2 = 1 + 0.999 999 985 895 138 421 374 976;
  • 47) 0.999 999 985 895 138 421 374 976 × 2 = 1 + 0.999 999 971 790 276 842 749 952;
  • 48) 0.999 999 971 790 276 842 749 952 × 2 = 1 + 0.999 999 943 580 553 685 499 904;
  • 49) 0.999 999 943 580 553 685 499 904 × 2 = 1 + 0.999 999 887 161 107 370 999 808;
  • 50) 0.999 999 887 161 107 370 999 808 × 2 = 1 + 0.999 999 774 322 214 741 999 616;
  • 51) 0.999 999 774 322 214 741 999 616 × 2 = 1 + 0.999 999 548 644 429 483 999 232;
  • 52) 0.999 999 548 644 429 483 999 232 × 2 = 1 + 0.999 999 097 288 858 967 998 464;
  • 53) 0.999 999 097 288 858 967 998 464 × 2 = 1 + 0.999 998 194 577 717 935 996 928;
  • 54) 0.999 998 194 577 717 935 996 928 × 2 = 1 + 0.999 996 389 155 435 871 993 856;

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:


0.000 000 000 742 147 676 646 509(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 646 509(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(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:


0.000 000 000 742 147 676 646 509(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) × 20 =

1.1001 0111 1111 1111 1111 111(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 0111 1111 1111 1111 111


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 1011 1111 1111 1111 1111 =

100 1011 1111 1111 1111 1111


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 1011 1111 1111 1111 1111


Decimal number -0.000 000 000 742 147 676 646 509 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1011 1111 1111 1111 1111

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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