Tiger

WHAT IS A SQUARE ROOT?

IRRATIONAL
NUMBER
When the Tiger demands the square root of 99, we are supposed to provide a decimal approximation of sqrt(99). It would be much easier to satisfy this demand if the Tiger would ask for the square root of numbers such as 100 or 49. The difficult thing about the square root of 99 is that it is an irrational number. The effect of this is that a decimal approximation is really the best we can provide.

OK, so we're interested in square roots. Let's begin at the beginning. What is a square root? Here is a true statement.

**Existence of Square Roots Theorem (ESRT)**
Given any nonnegative real number x, there is a unique nonnegative real number y such that y² = x.

For example, there is only one nonnegative real number whose square is 100, namely 10. Of course, this single example is not a complete justification of the ESRT, which is a statement about all nonnegative real numbers. When confronted with a statement such as ESRT, it is perfectly natural to ask: "Why is THAT true?" Let's avoid this excellent question for now; we will return to it later. The ESRT allows us to define what is meant by a square root.

[What?]

Given any nonnegative real number x (such as x = 99),
the square root of x is the unique real number y
with the following two properties.

y 0 and y² = x

We will denote the square root of x as follows:

y = sqrt(x).

Notice that we only define square roots of nonnegative real numbers. Negative numbers do not have real square roots! Here are two chances to check your understanding of the definition of the square root function.

Intuition and experience suggest that the square root of 99 should be a little less than 10. The Lesson continues with an exploration of Question 2 [Less Than?] and Question 3 [Close To?].

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