Radicals: Rational and Irrational Numbers
We write, for example,
"The square root of 25 is 5."
This mark is called the radical sign (after the Latin radix = root). The number under the radical sign is called the radicand. In the example, 25 is the radicand.
Problem 1. Evaluate the following.
To see the answer, pass your mouse over the colored area.
Example 1. Evaluate .
Solution. = 13.
For, 13· 13 is a square number. And the square root of 13· 13 is 13
If a is any whole number, then a· a is a square number, and
Problem 2. Evaluate the following.
We can state the following theorem:
A square number times a square number is itself a square number.
36· 81 = 6· 6· 9· 9 = 6· 9· 6· 9 = 54· 54
Problem 3. Without multiplying the given square numbers, each product of square numbers is equal to what square number?
a) 25· 64 = 5· 8· 5· 8 = 40· 40
b) 16· 49 = 4· 7· 4· 7 = 28· 28
c) 4· 9· 25 = 2· 3· 5· 2· 3· 5 = 30· 30
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 is. As for what it looks like, it can take the form , where a and b are integers (b ≠ 0).
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 almost 2.
But to prove that there is no rational number whose square is 2, then suppose there were. Then we could express it as a fraction in lowest terms. That is, suppose
· = 2.
But that is impossible. Since is in lowest terms, then m and n have no common divisors except 1. Therefore, m· m and n· n also have no common divisors -- they are relatively prime -- and it will be impossible to divide n· n into m· m and get 2.
There is no rational number whose square is 2. Therefore we call an irrational number.
Answer. Only the square roots of square numbers.
= 1 Rational
= 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 is ? -- we cannot say. We can only call it, "Square root of 2."
Problem 5. Say the name of each number.
As for the decimal representation of both irrational and rational numbers, see Topic 2 of Precalculus.
An equation x² = a, and the principal square root
Example 2. Solve this equation:
We say however that the positive value 5 is the principal square root. That is, we say that "the square root of 25" is 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 x.
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