How can a computer deal with real numbers

Computers are an exceptionally powerful tool for various computations, but they don't excel at storing decimal numbers. However, people have managed to overcome these issues: not storing the number in a decimal format, which is limited to very few decimal places, but as an integer instead, while keeping track of the number's precision.

Still, how can a computer simplify computations just like humans do? Take a look at this basic example

$$\sqrt{3} \times (\frac{4}{\sqrt{3}} - \sqrt{3}) = \sqrt{3} \times \frac{4}{\sqrt{3}} - \sqrt{3} \times \sqrt{3} = 4 - 3 = 1$$

That's how a human would solve it. Meanwhile, a computer would have a fun time calculating the square root of 3, diving 4 by it, subtracting the square root of 3 from the result and multiplying everything again by the square root of 3.

It would surely defeat a human in terms of speed, but it would lack in terms of accuracy. The result will be really close to 1, but not 1 exactly. A computer has no idea that, for instance, $\sqrt{3} \times{\sqrt{3}}$ is equal to $3$. This is only one of the uncountable examples out there.

Did people already find a solution, as it seems elementary for mathematics and computations? If they didn't, is this because it didn't serve any purpose in the real world?

Sage is an open source computer algebra system. Let's see if it can handle your basic example:

sage: sqrt(3) * (4/sqrt(3) - sqrt(3))

1

What is happening under the hood? Sage is storing everything as a symbolic expression, which it is able to manipulate and simplify using some basic rules.

Here is another example:

sage: 1 + exp(pi*i)

0

So sage can also handle complex numbers.

Computers never handle real numbers, since real numbers cannot be represented exactly on a computer. Instead, they either handle approximate representations of real numbers (usually floating point numbers but sometimes fixed point numbers), or they represent real numbers symbolically, as in the example above. Sage can convert between the two representations (in one direction!), and it can handle floating point numbers of arbitrary accuracy. For example,

sage: RealField(100)(pi^2/6 - sum(1/n^2 for n in range(1,10001)))

0.000099995000166666666333333336072

This computes $\pi^2/6 - \sum_{n=1}^{10^4} 1/n^2$ to 100 bits of accuracy (in the mantissa).

Another approach worth mentioning is interval arithmetic, which is a way of computing expressions with a guaranteed level of accuracy, using provable error brackets. Interval arithmetic is used in computational geometry, together with exact representation of rational numbers.

In theoretical computer science there are several other notions of real computation, but they are mostly of theoretical interest. See the answers to this question.

StackExchange Computer Science Q#134846, answer score: 38