26 ## Radicals: Rational and Irrational NumbersThe radical sign and the radicand Rational and irrational numbers Which square roots are rational? An equation The definition of the square root radical HERE ARE THE FIRST TEN square numbers and their roots:
We write, for example, = 5. "The square root of 25 is 5." This mark is called the radical sign (after the Latin Problem 1. Evaluate the following. To see the answer, pass your mouse over the colored area.
Example 1. Evaluate .
For, 13 If Problem 2. Evaluate the following.
We can state the following theorem: A square number times a square number is itself a square number. For example, 36 Problem 3. Without multiplying the given square numbers, each product of square numbers is equal to what square number? a) 25 b) 16 c) 4 Rational and irrational numbers The rational numbers are the everyday numbers of arithmetic: the whole numbers, fractions, mixed numbers, and decimals; together with their negative images. A rational number has the same ratio to 1 as two natural numbers. That is what a rational number
integers ( Problem 4. Which of the following numbers are rational?
All of them. At this point, the student might wonder, What is a number that is not rational? An example of such a number is ("Square root of 2"). is not a number of arithmetic. We cannot name any whole number, any fraction
-- which is But to prove that there is
terms. That is, suppose
no common divisors except 1. Therefore, There is no rational number whose square is 2. Therefore we call an irrational number.
= 1 Rational Irrational Irrational = 2 Rational , , , Irrational = 3 Rational And so on. The square roots of the square numbers are the only square roots that we can name. The existence of irrationals was first realized by Pythagoras in the 6th century B.C. He called them "without a name." For if we ask, " How much
Problem 5. Say the
As for the decimal representation of both irrational and rational numbers, see Topic 2 of Precalculus. An equation Example 2. Solve this equation:
We say however that the = 5. As for −5, it is "the negative of the square root of 25." − = −5. Thus the symbol refers to one non-negative number. Example 3. Solve this equation:
Always, if an equation looks like this,
Problem 6. Solve for
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