Latin1 was the early default character set for encoding documents delivered via HTTP for MIME types beginning with /text . Today, only around only 1.1% of websites on the internet use the encoding, along with some older appplications. However, it is still the most popular single-byte character encoding scheme in use today. A funny thing about Latin1 encoding is that it maps every byte from 0 to 255 to a valid character. This means that literally any sequence of bytes can be interpreted as a valid string. The main drawback is that it only supports characters from Western European languages. The same is not true for UTF8. Unlike Latin1, UTF8 supports a vastly broader range of characters from different languages and scripts. But as a consequence, not every byte sequence is valid. This fact is due to UTF8's added complexity, using multi-byte sequences for characters beyond the general ASCII range. This is also why you can't just throw any sequence of bytes at it and e...
Sometimes, we have to unlearn the things we initially learned. And I don't mean this in the sense of having been deliberately deceived. Rather, I mean that to some extent, there are actually many situations in life that involve necessary lies—or believing things that are wrong for perfectly rational reasons. Sometimes it is only after we have consumed and digested such a falsehood that we can see the truth at all. Really, this form of learning is not unlike some parts of math.
Consider a mathematical proof in which we begin by assuming that something is one way. But by the end of the proof, we may realize, through contradiction, that it's actually another way.
Let us take the number 2 and generously hypothesize that the square root of 2 is actually rational. If this assumption were true, we should be able to prove it with an equation. Let the square root of 2 be the lowest form of $\frac{p}{q}$.
Since squares of even numbers are even, and squares of odd numbers are odd, it follows that in order to get back to two, we would have to raise both p and q to the power of 2, like this:
\[ 2 = \frac{p^2}{q^2} \]Now we get $ p^2 = 2q^2 $. From here, we must infer that $p^2$ is an even number. After all, it is equivalent to $2q^2$. With our generous assumption that $p^2$ is even, let us substitute $p^2$ for $2r$, where r is an integer. Finally, let us test our hypothesis with the equation:
\[ (2r)^2 = 2q^2 \] \[ 4r^2 = 2q^2 \]Uh oh. Now we've hit a snag. From here, if we attempt to divide by two on both sides, we get:
\[ 2r^2 = q^2 \]How can this be? Remember, our initial hypothesis that the square root of 2 was rational rested on the assumption that $\frac{p}{q}$ was in its lowest form. But now here we see that $2r^2$ is equal to $q^2$. In other words, both sides are even!
This means they still share a common factor. Thus, neither side is in its lowest form. Proof by contradiction that the square root of two is not rational after all.
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