When we multiply a number by itself, we call the result the *square* of the original number. The origin of the name is clearly geometrical.
+----------+
| |
| |
| 5² | 5
| |
+----------+
5
We can see how, in the square above, each side has length 5, while the area, the *content* of the square, is 5².
What about the *square root*? It is the inverse process:
+----------+
| |
| |
| 25 | √(25)
| |
+----------+
√(25)
We are given the content (area) of the square as 25, and we are asked about the sides of the square. The answer is √(25), which is the *root* of the square. Perhaps a better name could have been the *side* of the square, so that the symbol √ would refer to the square side.
A square number is obtained my multiplying the root by itself.
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
and so on. Be sure to remember all these squares by heart. Don't be fooled by those who want to deprecate memory in favour of understanding. It is nonsense. First, understand, and then memorise, memorise everything.
If take the list above and subtract each square from its previous square, we get the following differences:
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11
49 - 36 = 13
64 - 49 = 15
81 - 64 = 17
100 - 81 = 19
121 - 100 = 21
144 - 121 = 23
169 - 144 = 25
196 - 169 = 27
225 - 196 = 29
256 - 225 = 31
289 - 256 = 33
324 - 289 = 35
361 - 324 = 37
400 - 361 = 39
We obtain the list of all odd numbers beginning from 3.
You can see how
(n+1)² - n² = 2n + 1
The squares of fractions are calculated by independently calculating the squares of the numerator and the denominator:
(3/4)² = 3² / 4² = 9/16
Geometrically, we can interpret this as the following. Consider a unit square:
._______.
|_|_|_|_|
|_|_|_|_| 1
|_|_|_|_|
|_|_|_|_|
1
Now, since we have divided each side in 4 parts, each little square is 1/4² = 1/16. But we want to square 3/4, so we can take 3/4 as base and then build its square, which we will fill now:
._______.
|_|_|_|_|
|x|x|x|_| 1
|x|x|x|_|
|x|x|x|_|
1
See how the square filled by x's is 9/16.
If the fraction is greater than 1, then we need to expand beyond the unit square. For example, for the square of 5/4 we need to draw a 5⨯5 grid where each square has a side 1/4:
._________.
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
Its area is 25 (the number of little squares) times 1/4. For the square root, the process is exactly the inverse.119
For mixed numbers, like 2[1/2], we can also find the square. But we previously need to convert the mixed expression into a regular expression. In this case, 2[1/2] = 2 + 1/2 = 4/2 + 1/2 = 5/2. Then, the square (2[1/2])² is equal to (5/2)² = 25/4. And we can go back to a mixed expression as 25/4 = (24 + 1)/4 = 6 + 1/4 = 6[1/4]. Is it possible that a mixed number, when square, becomes non-mixed, without a fraction? In other words, can we have (n[a/b])² = m² ? (See that a ≠ b and a < b.)
n[a/b] = n + a/b = (nb + a)/b
(nb+a)² / b² = n² + n(2a/b) + (a/b)²
On the last expression, n² is an integer, n(2a/b) could sometimes be an integer, but (a/b) could never be an integer. So the answer is no: the square of a mixed number is always a mixed number.120
Euler introduces here the square of a generic number, like a. Its square is a², although Euler writes it as it was customary then: as aa. Curiously, the square was not explicitly written as an exponent, but higher powers were. Here we use powers even for squares. So, the square of 2a is (2a)² = 2²a² = 4a². The square of abc is a²b²c². The square root of c²d² is just cd.121
We are asked for the square root of 2304. At first, it may seem quite a task, perhaps by using a general method for square roots. But we can examine the factors of this number first: 2304 = 1152·2 = 576·4 = 288·4·2 = 144·4² = 12²·4². Conclusion: √(2304) = 12·4 = 48. Sometimes, analysing the factors of a number can reveal its square root in a quick way. This is especially true when we are asked about the square root of an integer and we suspect there must be an integer solution to it.122
What about signs? Without consider complex numbers first, we can say that the square of a positive number is always positive. And the square of a negative number is also positive. In other words: +² = + and -² = +. For square roots, being the inverse process, we can only consider square roots of positive numbers. Since
+² → +
-² → +
or
+² --.
---> +
-² --'
now the inverse process is
+ <--.
---- √(+)
- <--'
This means that, for example, the square root of +25 is either +5 or -5, since both +5 and -5, when squared, give +25. Euler does not deal here with imaginary numbers, but we can discuss them anyway. The imaginary number i is defined as the square root of -1. This means we can expand our diagram of signs as
(+1)² --.
---> +1
(-1)² --'
(+i)² --.
---> -1
(-i)² --'
And, for square roots, the diagram is the same but with inverse arrows:
+1 <--.
--- √(+1)
-1 <--'
+i <--.
--- √(-1)
-i <--'
So, as √(25) is ±5, √(-25) = ±5i. You may be surprised with this notation, being accustomed to explicitly consider that √(25) is strictly +5, and then, to consider the negative solution, write -√(25). Choose whatever you find more elegant! Another thing we can mention here is the notation √, which can also be expressed as a fractional power. For example, √(25) = 25¹⸍². But beware of the Unicode symbol ⸍, which is not a standard symbol for a superscript division sign. It is a symbol with another purpose, but, lacking a superscript bar for division, I take it as a make-up. Moreover, some clients will not render such symbol properly.