To know what the square root of 3 is, it is important to know the definition of the square root of a number.
Given a positive number "a", the square root of "a", denoted by √a, is a positive number "b" such that when "b" is multiplied by it, the result is "a".
The mathematical definition says: √a = b if, and only if, b² = b * b = a.
Therefore, to know what the square root of 3 is, that is, the value of √3, a number "b" must be found such that b² = b * b = √3.
In addition, √3 is an irrational number, so it consists of an infinite non-periodic number of decimal places. For this reason, it is difficult to calculate the square root of 3 manually.
Square root of 3
If you use a calculator you can see that the square root of 3 is 1.73205080756887…
Now, you could manually try to approximate this number as follows:
-1 * 1 = 1 and 2 * 2 = 4, this says that the square root of 3 is a number between 1 and 2.
-1.7 * 1.7 = 2.89 and 1.8 * 1.8 = 3.24, therefore the first decimal place is 7.
-1.73 * 1.73 = 2.99 and 1.74 * 1.74 = 3.02, so the second decimal place is 3.
-1.732 * 1.732 = 2.99 and 1.733 * 1.733 = 3.003, therefore the third decimal place is 2.
And so on you can continue. This is a manual way to calculate the square root of 3.
There are also other much more advanced techniques, such as the Newton-Raphson method, which is a numerical method to calculate approximations.
Where can we find the number √3?
Due to the complexity of the number, it might be thought that it does not appear in everyday objects, but this is false. If we have a cube (square box), such that the length of its sides is 1, then the diagonals of the cube will have a measure of √3.
To check this, the Pythagorean Theorem is used, which says: given a right triangle, the hypotenuse squared is equal to the sum of the squares of the legs (c² = a² + b²).
By having a cube with side 1, we have that the diagonal of the square of its base is equal to the sum of the squares of the legs, that is, c² = 1² + 1² = 2, therefore the diagonal of the base measures √2.
Now, to calculate the diagonal of the cube, the following figure can be observed.
The new right triangle has legs of lengths 1 and √2, therefore, when using the Pythagorean theorem to calculate the length of its diagonal, we obtain: C² = 1² + (√2) ² = 1 + 2 = 3, that is say, C = √3.
Thus, the length of the diagonal of a cube with side 1 is equal to √3.
√3 an irrational number
At the beginning it was said that √3 is an irrational number. To check this, it is assumed by the absurdity that it is a rational number, with which there are two numbers "a" and "b", relative primes, such that a / b = √3.
Squaring the last equality and solving for "a²", the following equation is obtained: a² = 3 * b². This says that "a²" is a multiple of 3, which leads to the conclusion that "a" is a multiple of 3.
Since "a" is a multiple of 3, there is an integer "k" such that a = 3 * k. Therefore, by replacing in the second equation, we obtain: (3 * k) ² = 9 * k² = 3 * b², which is the same as b² = 3 * k².
As before, this last equality leads to the conclusion that "b" is a multiple of 3.
In conclusion, "a" and "b" are both multiples of 3, which is a contradiction, since they were originally assumed to be relative primes.
Therefore, √3 is an irrational number.
References
- Bails, B. (1839). Arismatic principles. Printed by Ignacio Cumplido.
- Bernadet, JO (1843). Complete elementary treatise on linear drawing with applications to the arts. José Matas.
- Herranz, DN, & Quirós. (1818). Universal, pure, testamentary, ecclesiastical and commercial arithmetic. printing house that was from Fuentenebro.
- Preciado, CT (2005). Mathematics Course 3rd. Editorial Progreso.
- Szecsei, D. (2006). Basic Math and Pre-Algebra (illustrated ed.). Career Press.
- Vallejo, JM (1824). Children's arithmetic… Imp. That was from García.