# Square Roots and Irrational Numbers

## Key Questions

• Guess what the square root of the irrational number is. For example, if your irrational number is 2, you might guess 1.2.

Divide the initial irrational number by the guessed number. For example, 2 divided by 1.2 is 1.67.

Add the resulting sum to the original guessed number. For example, 1.67 plus 1.2 is 2.87.

Divide the new result by 2. In our example, 2.87 divided by 2 is 1.435.

• An irrational number is a real number that cannot be written as a ratio of integers.

I hope that this was helpful.

• Given a positive real number a, there are two solutions to the equation ${x}^{2} = a$, one is positive, and the other is negative. We denote the positive root (which we often call the square root) by $\setminus \sqrt{a}$. The negative solution of ${x}^{2} = a$ is −\sqrt{a} (we know that if $x$ satisfies ${x}^{2} = a$, then (−x)^2=x^2=a, therefore, because $\setminus \sqrt{a}$ is a solution, so is −\sqrt{a}). So, for $a > 0 , \setminus \sqrt{a} > 0$, but there are two solutions to the equation ${x}^{2} = a$, one positive $\left(\setminus \sqrt{a}\right)$ and one negative (−\sqrt{a}). For $a = 0$, the two solutions coincide with $\setminus \sqrt{a} = 0$.

As we all know a square root is occurrence when an integer n is multiplied to itself to give us an integer n* n. We also know when 2 integers with the same signs are multiplies it gives a positive integer .

with a these facts in mind we can say that n can be negative or positive and still give us the same perfect square.
PS . note that something like $\sqrt{- 1}$ wouldn't exist as we know that 2 integers with opposite symbols will not give a negative number.And for it to be a square number both the nos . have to be same.

Hopefully this helps

An operation that when executed on a number returns the value that when multiplied by itself returns the number given.

#### Explanation:

An operation that when executed on a number returns the value that when multiplied by itself returns the number given.

They have the form $\setminus \sqrt{x}$ where x is the number you are executing the operation on.

Note that if you are constrained to values in the real numbers, the number you are taking the square root of must be positive as there are no real numbers that when multiplied together will give you a negative number.