Definition of a square root

We know that $$$2\cdot2=4$$$ Therefore we know that the number $$2$$ multiplied by itself is $$4$$.

In this unit we will learn how to find a number that, multiplied by itself, results in a certain number in particular.

The square root of any number $$x$$ is the number that, when multiplied by itself, equals $$x$$. It is expressed as $$\sqrt{x}$$.

$$\sqrt{25}=5$$ since $$5\cdot5=25$$. In this case it is said that $$5$$ is the square root of $$25$$.

Or also $$\sqrt{9}=3$$ since $$3\cdot3=9$$. $$3$$ is the square root of $$9$$.

But let's see now what happens with $$-5$$.

If we multiply it by itself we obtain: $$(-5)\cdot(-5)=25$$ since if we multiply two negative numbers it always gives a positive number, that is to say: $$(-)\cdot(-)=+$$.

And so, $$-5$$ is also a square root of $$25$$ since multiplying it by itself it gives $$25$$.

In this way, if we calculate the square root of any number $$x$$, we will obtain two solutions: The positive solution and the negative solution. These are called the positive root and negative root, respectively. In this way, in general, we can write:

$$\sqrt{x}= \left\{\begin{array}{c} \mbox{positive root} =a \\ \mbox{negative root} =-a \end{array} \right. \mbox{so, it is satisfied that }$$

$$ a\cdot a=x,(-a)\cdot(-a)=x$$

Let's calculate the square root of $$36$$.

Since we know that $$6\cdot6=36$$ we can already say which are the square roots of such a number.

The positive square root of $$36$$ is $$6$$, since $$6\cdot6=36$$.

The negative square root of $$36$$ is $$-6$$, since $$(-6)\cdot(-6)=36$$.

It is usual to write only the positive root and it is implied that the same number, in its negative form, is also a root. From now on we will only write the positive root of a number to give the solution to a square root.

Until now we have calculated square roots which result was integers. When that happens it is said that the number whichS root we are calculating is a perfect square. In other words, if the root of a number $$x$$ is an integer it is said that $$x$$ is a perfect square.

$$25$$ is a perfect square, since $$\sqrt{25}=5$$ and $$5$$ is an integer.

And also $$16$$ is a perfect square because $$\sqrt{16}=4$$ because $$4$$ is the number that satisfies that $$4\cdot4=16$$.

When we calculate square roots of numbers that are not perfect squares the result is no longer an integer, but an irrational number. This means that it is a number that it is not possible to be written as the quotient of two integers.

In conclusion, the square root of an integer will always be an integer or an irrational number.

$$\sqrt{2}=1,414213562\ldots$$ is an irrational number and has infinite decimals.

The square root can also be calculated using non-integers. The only indispensable requisite for calculating the root of a number is for this one to be positive.

The square root of a negative number does not exist for, using the rule of multiplication, when we multiply two positives the result is positive and when we multiply two negatives it is also positive.

In this way, there is no possibility for a number multiplied by itself (product of two numbers with the same sign) to give a negative result.

In short, for any positive number, integer or not, it is possible to calculate the square root.

$$\sqrt{17,2}=4,1473$$ since if we calculate $$4,1473\cdot4,1473=17,2$$

These are the roots of the most frequent integers that are perfect squares.

$$\sqrt{4}=2 \ \ \ \ $$ $$\sqrt{36}=6 \ \ \ \ $$ $$\sqrt{100}=10$$

$$\sqrt{9}=3 \ \ \ \ $$ $$\sqrt{49}=7 \ \ \ \ $$ $$\sqrt{121}=11$$

$$\sqrt{16}=4 \ \ \ $$ $$\sqrt{64}=8 \ \ \ $$ $$\sqrt{144}=12$$

$$\sqrt{25}=5 \ \ \ $$ $$\sqrt{81}=9 \ \ \ $$ $$\sqrt{169}=13$$