Convert the Number -187.512 205 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number. Detailed Explanations
Number -187.512 205(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)
The first steps we'll go through to make the conversion:
Convert to binary (to base 2) the integer part of the number.
Convert to binary the fractional part of the number.
1. Start with the positive version of the number:
|-187.512 205| = 187.512 205
2. First, convert to binary (in base 2) the integer part: 187.
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;
- 187 ÷ 2 = 93 + 1;
- 93 ÷ 2 = 46 + 1;
- 46 ÷ 2 = 23 + 0;
- 23 ÷ 2 = 11 + 1;
- 11 ÷ 2 = 5 + 1;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 1 ÷ 2 = 0 + 1;
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.
187(10) =
1011 1011(2)
4. Convert to binary (base 2) the fractional part: 0.512 205.
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) 0.512 205 × 2 = 1 + 0.024 41;
- 2) 0.024 41 × 2 = 0 + 0.048 82;
- 3) 0.048 82 × 2 = 0 + 0.097 64;
- 4) 0.097 64 × 2 = 0 + 0.195 28;
- 5) 0.195 28 × 2 = 0 + 0.390 56;
- 6) 0.390 56 × 2 = 0 + 0.781 12;
- 7) 0.781 12 × 2 = 1 + 0.562 24;
- 8) 0.562 24 × 2 = 1 + 0.124 48;
- 9) 0.124 48 × 2 = 0 + 0.248 96;
- 10) 0.248 96 × 2 = 0 + 0.497 92;
- 11) 0.497 92 × 2 = 0 + 0.995 84;
- 12) 0.995 84 × 2 = 1 + 0.991 68;
- 13) 0.991 68 × 2 = 1 + 0.983 36;
- 14) 0.983 36 × 2 = 1 + 0.966 72;
- 15) 0.966 72 × 2 = 1 + 0.933 44;
- 16) 0.933 44 × 2 = 1 + 0.866 88;
- 17) 0.866 88 × 2 = 1 + 0.733 76;
- 18) 0.733 76 × 2 = 1 + 0.467 52;
- 19) 0.467 52 × 2 = 0 + 0.935 04;
- 20) 0.935 04 × 2 = 1 + 0.870 08;
- 21) 0.870 08 × 2 = 1 + 0.740 16;
- 22) 0.740 16 × 2 = 1 + 0.480 32;
- 23) 0.480 32 × 2 = 0 + 0.960 64;
- 24) 0.960 64 × 2 = 1 + 0.921 28;
- 25) 0.921 28 × 2 = 1 + 0.842 56;
- 26) 0.842 56 × 2 = 1 + 0.685 12;
- 27) 0.685 12 × 2 = 1 + 0.370 24;
- 28) 0.370 24 × 2 = 0 + 0.740 48;
- 29) 0.740 48 × 2 = 1 + 0.480 96;
- 30) 0.480 96 × 2 = 0 + 0.961 92;
- 31) 0.961 92 × 2 = 1 + 0.923 84;
- 32) 0.923 84 × 2 = 1 + 0.847 68;
- 33) 0.847 68 × 2 = 1 + 0.695 36;
- 34) 0.695 36 × 2 = 1 + 0.390 72;
- 35) 0.390 72 × 2 = 0 + 0.781 44;
- 36) 0.781 44 × 2 = 1 + 0.562 88;
- 37) 0.562 88 × 2 = 1 + 0.125 76;
- 38) 0.125 76 × 2 = 0 + 0.251 52;
- 39) 0.251 52 × 2 = 0 + 0.503 04;
- 40) 0.503 04 × 2 = 1 + 0.006 08;
- 41) 0.006 08 × 2 = 0 + 0.012 16;
- 42) 0.012 16 × 2 = 0 + 0.024 32;
- 43) 0.024 32 × 2 = 0 + 0.048 64;
- 44) 0.048 64 × 2 = 0 + 0.097 28;
- 45) 0.097 28 × 2 = 0 + 0.194 56;
- 46) 0.194 56 × 2 = 0 + 0.389 12;
- 47) 0.389 12 × 2 = 0 + 0.778 24;
- 48) 0.778 24 × 2 = 1 + 0.556 48;
- 49) 0.556 48 × 2 = 1 + 0.112 96;
- 50) 0.112 96 × 2 = 0 + 0.225 92;
- 51) 0.225 92 × 2 = 0 + 0.451 84;
- 52) 0.451 84 × 2 = 0 + 0.903 68;
- 53) 0.903 68 × 2 = 1 + 0.807 36;
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...)
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:
0.512 205(10) =
0.1000 0011 0001 1111 1101 1101 1110 1011 1101 1001 0000 0001 1000 1(2)
6. Positive number before normalization:
187.512 205(10) =
1011 1011.1000 0011 0001 1111 1101 1101 1110 1011 1101 1001 0000 0001 1000 1(2)
The last steps we'll go through to make the conversion:
Normalize the binary representation of the number.
Adjust the exponent.
Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.
Normalize the mantissa.
7. 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:
187.512 205(10) =
1011 1011.1000 0011 0001 1111 1101 1101 1110 1011 1101 1001 0000 0001 1000 1(2) =
1011 1011.1000 0011 0001 1111 1101 1101 1110 1011 1101 1001 0000 0001 1000 1(2) × 20 =
1.0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000 0011 0001(2) × 27
8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:
Sign 1 (a negative number)
Exponent (unadjusted): 7
Mantissa (not normalized):
1.0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000 0011 0001
9. Adjust the exponent.
Use the 11 bit excess/bias notation:
Exponent (adjusted) =
Exponent (unadjusted) + 2(11-1) - 1 =
7 + 2(11-1) - 1 =
(7 + 1 023)(10) =
1 030(10)
10. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.
Use the same technique of repeatedly dividing by 2:
- division = quotient + remainder;
- 1 030 ÷ 2 = 515 + 0;
- 515 ÷ 2 = 257 + 1;
- 257 ÷ 2 = 128 + 1;
- 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;
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) =
1030(10) =
100 0000 0110(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 52 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. 0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000 0011 0001 =
0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000
13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:
Sign (1 bit) =
1 (a negative number)
Exponent (11 bits) =
100 0000 0110
Mantissa (52 bits) =
0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000
The base ten decimal number -187.512 205 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 100 0000 0110 - 0111 0111 0000 0110 0011 1111 1011 1011 1101 0111 1011 0010 0000
Sign (1 bit):
Exponent (11 bits):
1
620
610
600
590
580
570
560
551
541
530
52
Mantissa (52 bits):
0
511
501
491
480
471
461
451
440
430
420
410
400
391
381
370
360
350
341
331
321
311
301
291
281
270
261
251
241
230
221
211
201
191
180
171
160
151
141
131
121
110
101
91
80
70
61
50
40
30
20
10
0
Convert to 64 bit double precision IEEE 754 binary floating point representation standard
A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)
The latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation
How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard
Follow the steps below to convert a base 10 decimal number to 64 bit double 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 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 from the previous 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 multiplying operations, starting from the top of the list constructed 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, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
- 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 - 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' 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 -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:
- 1. Start with the positive version of the number:
|-31.640 215| = 31.640 215
- 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
- division = quotient + remainder;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 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:
31(10) = 1 1111(2)
- 4. Then, convert the fractional part, 0.640 215. 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) 0.640 215 × 2 = 1 + 0.280 43;
- 2) 0.280 43 × 2 = 0 + 0.560 86;
- 3) 0.560 86 × 2 = 1 + 0.121 72;
- 4) 0.121 72 × 2 = 0 + 0.243 44;
- 5) 0.243 44 × 2 = 0 + 0.486 88;
- 6) 0.486 88 × 2 = 0 + 0.973 76;
- 7) 0.973 76 × 2 = 1 + 0.947 52;
- 8) 0.947 52 × 2 = 1 + 0.895 04;
- 9) 0.895 04 × 2 = 1 + 0.790 08;
- 10) 0.790 08 × 2 = 1 + 0.580 16;
- 11) 0.580 16 × 2 = 1 + 0.160 32;
- 12) 0.160 32 × 2 = 0 + 0.320 64;
- 13) 0.320 64 × 2 = 0 + 0.641 28;
- 14) 0.641 28 × 2 = 1 + 0.282 56;
- 15) 0.282 56 × 2 = 0 + 0.565 12;
- 16) 0.565 12 × 2 = 1 + 0.130 24;
- 17) 0.130 24 × 2 = 0 + 0.260 48;
- 18) 0.260 48 × 2 = 0 + 0.520 96;
- 19) 0.520 96 × 2 = 1 + 0.041 92;
- 20) 0.041 92 × 2 = 0 + 0.083 84;
- 21) 0.083 84 × 2 = 0 + 0.167 68;
- 22) 0.167 68 × 2 = 0 + 0.335 36;
- 23) 0.335 36 × 2 = 0 + 0.670 72;
- 24) 0.670 72 × 2 = 1 + 0.341 44;
- 25) 0.341 44 × 2 = 0 + 0.682 88;
- 26) 0.682 88 × 2 = 1 + 0.365 76;
- 27) 0.365 76 × 2 = 0 + 0.731 52;
- 28) 0.731 52 × 2 = 1 + 0.463 04;
- 29) 0.463 04 × 2 = 0 + 0.926 08;
- 30) 0.926 08 × 2 = 1 + 0.852 16;
- 31) 0.852 16 × 2 = 1 + 0.704 32;
- 32) 0.704 32 × 2 = 1 + 0.408 64;
- 33) 0.408 64 × 2 = 0 + 0.817 28;
- 34) 0.817 28 × 2 = 1 + 0.634 56;
- 35) 0.634 56 × 2 = 1 + 0.269 12;
- 36) 0.269 12 × 2 = 0 + 0.538 24;
- 37) 0.538 24 × 2 = 1 + 0.076 48;
- 38) 0.076 48 × 2 = 0 + 0.152 96;
- 39) 0.152 96 × 2 = 0 + 0.305 92;
- 40) 0.305 92 × 2 = 0 + 0.611 84;
- 41) 0.611 84 × 2 = 1 + 0.223 68;
- 42) 0.223 68 × 2 = 0 + 0.447 36;
- 43) 0.447 36 × 2 = 0 + 0.894 72;
- 44) 0.894 72 × 2 = 1 + 0.789 44;
- 45) 0.789 44 × 2 = 1 + 0.578 88;
- 46) 0.578 88 × 2 = 1 + 0.157 76;
- 47) 0.157 76 × 2 = 0 + 0.315 52;
- 48) 0.315 52 × 2 = 0 + 0.631 04;
- 49) 0.631 04 × 2 = 1 + 0.262 08;
- 50) 0.262 08 × 2 = 0 + 0.524 16;
- 51) 0.524 16 × 2 = 1 + 0.048 32;
- 52) 0.048 32 × 2 = 0 + 0.096 64;
- 53) 0.096 64 × 2 = 0 + 0.193 28;
- We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) 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:
0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)
- 6. Summarizing - the positive number before normalization:
31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)
- 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:
31.640 215(10) =
1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24
- 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:
Sign: 1 (a negative number)
Exponent (unadjusted): 4
Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0
- 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:
Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
100 0000 0011(2)
- 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):
Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0
Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100
- Conclusion:
Sign (1 bit) = 1 (a negative number)
Exponent (8 bits) = 100 0000 0011
Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100
Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100
Available Base Conversions Between Decimal and Binary Systems
Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):
1. Integer -> Binary
2. Decimal -> Binary
3. Binary -> Integer
4. Binary -> Decimal