Here Euler briefly reviews the last chapter. We have square numbers, square roots and a concern for the signs of the square roots.
Euler recommends us here that we memorise as many squares as we can, so that we can use them to obtain square roots of these numbers.
What I find interesting here is that, although the processes of squaring and finding the square root are inverse of each other, they are not symmetrical. Like hiding a small paper in a book in a library is not symmetrical to finding it, even if it is its inverse.
The same happens with multiplication and division: multiplication has very clear and direct rules, while division feels more indirect. Again, integration is way harder and subtler than derivation.
In John Colson's annotations for Newton's fluxions book, we can read:
"...all operations are of two kinds, either Compositive and Resolutative. The Compositive or Synthetic Operations proceed necessarily and directly, ..., and not tentatively or by way of tryal. Such are Addition, Multiplication, Raising of Powers, and taking of Fluxions. But the Resolutative or Analytical Operations, as Subtraction, Division, Extraction of Roots and finding the Fluents, are forced to proceed indirectly and tentatively, by long deductions, ..., and suppose or require the contrary Synthetic Operations, to prove and confirm every step of the Process. "
When the number given for root extraction is not a square, like 12, we will not find an exact answer.
But we can see that it is greater than 3 because 3² < 12 and smaller than 4 bevause 4² > 12.
We can also try (3[1/2])² = (7/2)² = 49/4 = 12[1/4], which is greater than 12.
Wen can also try (3[7/15])² = (52/15)² = 2704/225 = 12[4/225].
See how nice is the mixed representation of a number for approximating our target.
We see that 3[7/15] is quite a good approximation for square root of 12.
We showed in the previous chapter that if a number is mixed, its square must be mixed as well, so there is no hope for finding a mixed number whose square is exactly 12.
You can, however, bound your results between two values. For example,
(3[6/13])² < 12 < (3[7/15])²
But the square root of 12 is definitely a number: only one that cannot be written as an exact fraction.
We call such a number an *irrational* number.
They are also called *surd quantities* or *incommensurables*.
We can continually approximate the square root of 12 with fractions, with as much precision as we want, so that we have no doubt that the root exists. And, if we would raise this solution to the power of 2, we should find 12 again.
Here is where Euler actually introduces the symbol √ as "square root".
So, the *result* of the square root of 12 can be *exactly* written as √(12). It is, of course, a symbolic way of expressing it, but it is a shorthand way of saying that we are considering the exact value of the root.
We can also apply the symbol to a fraction, like √(2/3), or to any general expression like √(2 + a/b).
Although we cannot write all the decimal places for √a, we can use the rules of algebra for doing operations like √a·√a = a or √(2/3)·√(2/3) = 2/3.
If we multiply √a by √b, we can write √(ab).
If we want the square root of c·d, we can write √(cd).
If we want to divide √a by √b, we write √(a/b).
Sometimes, the irrationality can be cancelled. For example, dividing the two irrational numbers √18 by √8, we get √(18/8) = √(9/4) = 3/2. The reason is that up and down we had an irrational factor of √2 that cancelled out.
Nothing prevents us to write √9, but this does not mean this is an irrational number, because it can also written as 3. So don't take the √ symbol as a sign that the number inside it is irrational. Euler calls this "apparent irrationality".
We can multiply irrational by rational numbers. If we want 2 multiplied by √5 we can simply write 2√5 and that's our result.
We have the choice of taking 2 as √4 and then rewrite the result as √(4·5) = √20.
Division also follows these rules. √8 / √2 = √(8/2) = 2.
More interesting is when having 2 / √2. Many people don't like having square roots in the denominator, so they prefer to rewrite this as √4 / √2 = √2.
In this case, it seems justified, but what about 1 / √2 ? Would you prefer to multiply up and down by √2 to obtain √2 / 2 ? Why?
For addition and subtraction of square roots, we simply add √2 + √3 or √5 - 2.
Numbers are either irrational or rational. Rational include integers and fractions.