#### Have you ever discovered a scenario wherever you probably did some arithmetic computation on decimal numbers in a computer and it returned an unexpected weird result?

**That one question you’d have asked is why 0.1 + 0.2 does not equal to 0.3?**

**Examples:**

- 0.1 + 0.2 = 0.3 but computer says 0.30000000000000004
- 6 * 0.1 = 0.6 but computer says 0.6000000000000001
- 0.11 + 0.12 = 0.23 but again computer says 0.22999999999999998
- 0.1 + 0.7 = 0.7999999999999999
- 0.3+0.6 = 0.8999999999999999 and so on… There are several other such cases.

**Note:** It is also a interview questions that an interviewer asks “output of 0.1 + 0.2 == 0.3”? true or false? Simply to see how well you know arithmetic computations on decimal numbers by computer.

**In straightforward sentences, the explanation behind this is:**

- The computer uses base-2 floating-point number whereas Human(We) use base-10 floating-point numbers.
- In binary, decimal numbers cannot represent all decimals precisely and have infinite representations.
- So computers can not accurately represent numbers like 0.1, 0.2 or 0.3 at all as it uses binary floating-point format.

**Now let’s understand the elaborated reason behind this:**

- In base 10 system (Used by human/us), fractions can be expressed precisely if it uses a prime factor of the base (10).
- 2 and 5 are the prime factors of 10.
- 1/2, 1/4, 1/5 (0.2), 1/8, and 1/10 (0.1) can be expressed precisely as a result of denominators use prime factors of 10.
- Whereas, 1/3, 1/6, and 1/7 are repeating decimals as a result of denominators use a prime factor of 3 or 7.

- In base 2 (binary) system on the other hand (Used by computer), fractions can be expressed precisely if it uses a prime factor of the base (2).
- 2 is the only prime factor of 2.
- So 1/2, 1/4, 1/8 can all be expressed precisely because the denominators use prime factors of 2.
- Whereas 1/5 (0.2) or 1/10 (0.1) are repeating decimals.

- So we end up with leftovers for these repeating decimals and that carries over when we convert the computer’s base 2 number into a human-readable base 10 number.

**Let’s take 0.1 as an example to understand binary and decimal representations:**

- 0.1 is one-tenth(1/10) and to get binary representation (bicimal) of 0.1, we need to use the binary long division that is to divide binary 1 by binary 1010 (1/1010) like below:

- You can see above that the division process would never end and repeats forever with the digits in the quotient because 100 reappears as the dividend.
- so the binary representation of our decimal number 0.1 will be as below:

- Now above result can be slightly greater or Less than 0.1 which depends upon how many bits of precision are used.
- In half-precision which uses 11 significant bits, the floating-point approximation of 0.1 could be less than 0.1

1234so 0.1 rounds to 0.0001100110011 in binary and 0.0999755859375 in decimal.That's why the result becomes slightly less than 0.10.0999755859375 < 0.1 - In double-precision floating-point which uses 53 bits, the floating-point approximation of 0.1 could be greater than 0.1

12345so 0.1 rounds to 0.0001100110011001100110011001100110011001100110011001101 in binaryand 0.1000000000000000055511151231257827021181583404541015625 in decimal.That's why the result becomes slightly greater than 0.10.1000000000000000055511151231257827021181583404541015625 > 0.1

- In half-precision which uses 11 significant bits, the floating-point approximation of 0.1 could be less than 0.1

**Due to all this, a decimal point may not give exact floating-point result as**

- In pure math, every decimal has an equivalent bicimal (decimal converted to binary)
- In floating-point math, this is can be simply not true and that we do not get precise value for every decimal number.

**Note:**In most calculators, we see 0.1 + 0.2 = 0.3 despite the fact that that’s also a computing device. The explanation is that it uses additional guard digits to get around this downside by rounding off the final few bits.

**References:**

Below are some useful links you can refer to get even additional dipper into the floating-point arithmetics in computer.

- https://0.30000000000000004.com
- https://stackoverflow.com/a/28679423/1651334
- https://www.exploringbinary.com/why-0-point-1-does-not-exist-in-floating-point
- https://www.quora.com/Why-is-0-1+0-2-not-equal-to-0-3-in-most-programming-languages
- https://www.exploringbinary.com/floating-point-questions-are-endless-on-stackoverflow-com
- https://stackoverflow.com/questions/588004/is-floating-point-math-broken/27030789#27030789
- https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

I hope this can clear a number of your queries on why the computer does not return an expected output when we do the mathematical operation with decimal numbers.

Also, I believe you would be in a better position now to reply back if someone asks you why 0.1 + 0.2 does not equal to 0.3?

Happy Learning!!!

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