Why 0.1 + 0.2
is not equal to 0.3
Converting decimal to binary
Converting fraction
base-q expansion form
0.625 = x_1\cdot 2^{-1}+x_2\cdot 2^{-2}+x_3\cdot 2^{-3}
0.625 = x_1\cdot \frac{1}{2}+x_2\cdot \frac{1}{2^2}+x_3\cdot \frac{1}{2^3}
Factor out 1/2
0.625 = \frac{1}{2}(x_1+\frac{1}{2}x_2+\frac{1}{2^2}x_3)
What is ?
x_1
Single out first digit
If are at most equal to 1, and the right side is greater than 1, then must be equal to 1
x_2\cdot\frac{1}{2^2}+x_3\cdot\frac{1}{2^3}
1.25 = x_1+x_2\cdot 2^{-1}+x_3\cdot 2^{-2}
x_1
Factor out
0.25 = \frac{1}{2}\cdot(x_2+\frac{1}{2}\cdot x_3)
What is ?
x_2
Repeat until no digits with left
0.5 = x_2+\frac{1}{2}\cdot x_3
1 = x_3
0.5 = x_3\cdot\frac{1}{2}
\frac{1}{2}
\frac{1}{2}
So the algorithm is:
continue multiplying by 2 until the remainder is zero
0.625_{10} = 0.101_2
0.375_{10} = ?_2
Converting integer
base-q expansion form
12 = x_3\cdot 2^3+x_2\cdot 2^2+x_1\cdot 2^1+x_0\cdot 2^0
12 = x_3\cdot 2^3+x_2\cdot 2^2+x_1\cdot 2^1+x_0
Factor out 2
12 = 2\cdot(x_3\cdot 2^2+x_2\cdot 2^1+x_1\cdot 2^0) +x_0
What is ?
x_0
Single out last digits
If and are both even numbers, than must be
x_0
2y + x_0
x_0
0
Factor out 2
6 = 2\cdot(x_2\cdot 2^1+x_2\cdot 2^0) + x_1
What is ?
x_1
Repeat until no digits with 2 left
Continue...
So the algorithm is:
continue factoring out 2 until the remainder is zero
12_{10}=1100_2
Converting binary to decimal
Converting fraction
base-q expansion form
0.625 = x_1\cdot \frac{1}{2}+x_2\cdot \frac{1}{2^2}+x_3\cdot \frac{1}{2^3}
0.625_{10} = 0.101_2
So, the algorithm is...
start from 0, add the current result to the current digit and divide by 2
0.625_{10} = 0.101_2
0.1011_2=?_{10}
Converting integer
base-q expansion form
12 = x_3\cdot 2^3+x_2\cdot 2^2+x_1\cdot 2^1+x_0\cdot 2^0
So, the algorithm is...
start from 0, multiply the current result by 2 and add the current digit
12_{10} = 1100_2
Converting
0.1 and 0.2
to binary
Converting 0.1
0.1\cdot2=0.2\enspace\enspace\enspace0.0...
0.2\cdot2=0.4\enspace\enspace\enspace0.00...
0.4\cdot2=0.8\enspace\enspace\enspace0.000...
0.8\cdot2=1.6\enspace\enspace\enspace0.0001...
0.6\cdot2=1.2\enspace\enspace\enspace0.00011...
0.2\cdot2=0.4\enspace\enspace\enspace0.000110...
Converting 0.2
0.2\cdot2=0.4\enspace\enspace\enspace0.00...
0.4\cdot2=0.8\enspace\enspace\enspace0.00...
0.8\cdot2=1.6\enspace\enspace\enspace0.001...
0.6\cdot2=1.2\enspace\enspace\enspace0.0011...
0.2\cdot2=0.4\enspace\enspace\enspace0.00110...
Representinga number in a
scientific notation
Why scientific notation?
Scientific notation is a way to work easily with
very large or small numbers.
General form
2 = 2\cdot10^0
Normalized form
1100.101
1100.101 = 1.100101\cdot2^3
?
Representing 0.1 and 0.2 using normalized form
Rounding binary numbers
Possible rouding options
Round 0.42385 to 2 places. What numbers can it be round to?
0.42 and 0.43
Round 0.110110 to 2 places. What numbers can it be round to?
- for a lower number we simply discard the remaining places
- for a larger number we add 1 to the number in the last place
0.11 and 1.00 (0.11+0.01)
How we define that
Defining shortest distance
x_1 < x_2
x_2 < x_1
Is or ?
But!
Subtraction is cumbersome and
won't work for infinite fractions
x_1 = |0.11-0.110110|=0.000110
x_2 = |1.00-0.110110|=0.001010
So, and we should round to
x_1 < x_2
0.11
Comparing with the middle
So, and we should round to
0.11
0.110110 > 0.111
0.110110 < 0.111
Is or ?
0.110110
0.111000
0.110110 < 0.111
larger
General rule for rounding
- If X is 0,
Round 0.UVXYZ... to 2 places, the middle is 0.UV1, so
- If X is 1 and and any remaining digit is 1,
round down
- If X is 1 and all remaining digits are 0,
apply tie breaker rule (round to even)
0.UV0YZ...
0.UV1 (middle)
0.UV100...
0.UV1 (middle)
round up
0.UV101...
0.UV1 (middle)
Rounding 0.1 and 0.2
Rounding 0.1 to 52 bits
Finding middle
>
Rounding up
Comparing
Rounding 0.2 to 52 bits
Finding middle
>
Rounding up
Comparing
Representing negative numbers with offset binary (Excess-K, biased representation)
Offset-K
How many numbers can represent with 4 bits?
2^4 = 16
[0;15]
For positive numbers the range is
But if we include negative numbers, it can be
[-8;7]
[0000;1111]
[0000;1111]
K = 2^{n-1}
where n, is the number of bits
[-7;8]
[0000;1111]
Offset defines the range
K = 2^{n-1}-1
(floating point standard)
8 and 7 is an offset, referred to as `K` (excess-K, excess-7)
(common usage)
Calculating number
How to store the number 3 in 4 bits?
1. Calculating offset (K)
K = 2^{4-1}-1 = 7
If 0000 is -7, then what number should we add to get 3?
-7+10=3\rightarrow10=3+7
representation = number to store + K
3+K=3+7=10_{10}=1010_2
number to store = representation - K
1010_2=10_{10};\enspace\enspace10-K=10-7=3
First bit defines the sign
-8_{10} = 0000_2
8_{10} = 1000_2
-8_{10} + 8_{10} = 0000_{2}+1000_{2} = 0_{10} = 1000_{2}
0 - negative number
1 - positive number
Floating point according to the IEEE754
Format
| Name | Total bits | Exponent | Significand |
|---|---|---|---|
| Single precision | 32 | 8 | 23 |
| Double precision | 64 | 11 | 52 |
0 - positive value
1- negative value
Sign
Exponent
offset binary (biased representation)
Converting 0.1
Calculating exponent
to EEE-754 double precision
K = 2^{11-1}-1 = 1023
-4+1023=1019_{10}=01111111011_2
Converting 0.2
Calculating exponent
to EEE-754 double precision
K = 2^{11-1}-1 = 1023
-3+1023=1020_{10}=01111111100_2
Validating conversion
function to64bitFloat(number) {
var f = new Float64Array(1);
f[0] = number;
var view = new Uint8Array(f.buffer);
var i, result = "";
for (i = view.length - 1; i >= 0; i--) {
var bits = view[i].toString(2);
if (bits.length < 8) {
bits = new Array(8 - bits.length).fill('0').join("") + bits;
}
result += bits;
}
return result;
}to64bitFloat(0.1);
// 0 01111111011 1001100110011001100110011001100110011001100110011010
to64bitFloat(0.2);
// 0 01111111100 1001100110011001100110011001100110011001100110011010Calculating 0.1 + 0.2
Adjusting the exponent
1.100110011001100110011001100110011001100110011001101\cdot2^{-3} =
11.00110011001100110011001100110011001100110011001101\cdot2^{-4}
Adding numbers
0.1100110011001100110011001100110011001100110011001101
1.1001100110011001100110011001100110011001100110011010
+
10.0110011001100110011001100110011001100110011001100111
Normalizing
Rounded 0.1
Rounded 0.2
1.100110011001100110011001100110011001100110011001101\cdot2^{-4}
10.0110011001100110011001100110011001100110011001100111\cdot2^{-3}=
Rounding
Finding middle
=
Round to even (round up here)
Comparing
1.00110011...001100110011
0.00000000...000000000001
+
1.00110011...001100110100
Converting to decimal
1.00110011001100110011001100110011001100110011001101 \cdot 2^{-2}=
0.0100110011001100110011001100110011001100110011001101
Big.DP = 52;
var numbers = "0100110011001100110011001100110011001100110011001101".split("").reverse();
var sum = Big(0);
numbers.forEach(function (number) {
sum = sum.add(number).div(2);
console.log(sum.toString());
});Output
0.5
0.25
0.625
0.8125
0.40625
0.203125
0.6015625
0.80078125
0.400390625
0.2001953125
0.60009765625
0.800048828125
0.4000244140625
0.20001220703125
0.600006103515625
0.8000030517578125
0.40000152587890625
0.200000762939453125
0.6000003814697265625
0.80000019073486328125
0.400000095367431640625
0.2000000476837158203125
0.60000002384185791015625
0.800000011920928955078125
0.4000000059604644775390625
0.20000000298023223876953125
0.600000001490116119384765625
0.8000000007450580596923828125
0.40000000037252902984619140625
0.200000000186264514923095703125
0.6000000000931322574615478515625
0.80000000004656612873077392578125
0.400000000023283064365386962890625
0.2000000000116415321826934814453125
0.60000000000582076609134674072265625
0.800000000002910383045673370361328125
0.4000000000014551915228366851806640625
0.20000000000072759576141834259033203125
0.600000000000363797880709171295166015625
0.8000000000001818989403545856475830078125
0.40000000000009094947017729282379150390625
0.200000000000045474735088646411895751953125
0.6000000000000227373675443232059478759765625
0.80000000000001136868377216160297393798828125
0.400000000000005684341886080801486968994140625
0.2000000000000028421709430404007434844970703125
0.60000000000000142108547152020037174224853515625
0.800000000000000710542735760100185871124267578125
0.4000000000000003552713678800500929355621337890625
0.20000000000000017763568394002504646778106689453125
0.600000000000000088817841970012523233890533447265625
0.30000000000000004440892098500626161694526672363281250.3000000000000000444089209850062616169452667236328125
Result
Calculating
Resources
http://www.exploringbinary.com/binary-converter/
http://blog.chewxy.com/2014/02/24/what-every-javascript-developer-should-know-about-floating-point-numbers/
https://en.wikipedia.org/wiki/Offset_binary
Why 0.1+0.2 =
By maximk
