26

Radicals:  Rational and Irrational Numbers

The square numbers

The radical sign and the radicand

Rational and irrational numbers

Which square roots are rational?

An equation  x² = a, and the principal square root

2nd level:

Equations  (x + a)² = b

The definition of the square root radical

Rationalizing a denominator

Real numbers

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 radix = root). The number under the radical sign is called the radicand.  In the example, 25 is the radicand.

Problem 1.   Evaluate the following.

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a)	=  8		b)	=  12		c)	=  20
d)	=  17		e)	=  1		f)		=	7 9

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.

a)

=  28.

b)

=  135.

c)

=  2 · 3 · 5 = 30.

We have the following theorem:

A square number times a square number is itself a square number.

For example,

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

A rational number is any number of arithmetic: any whole number, fraction, mixed number, or decimal; together with its negative image. 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 of a fraction , 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.   is close because

-- which is almost 2.

To see that there is no rational number whose square is 2, suppose there were. Obviously, it is not a whole number. It will be in the form of a fraction in lowest terms. But the square of a fraction in lowest terms is also in lowest terms.

No new factors are introduced and the denominator will never divide into the numerator to give 2—or any whole number.

There is no rational number whose square is 2  or any number that is not a perfect square. We say therefore that is an irrational number.

As a decimal approximation,

1.414

(The wavy equal sign means "is approximately".)

How could we know that?  By multiplying 1.414 by itself.  If we do, we get 1.999396, which is almost 2.  But it should be clear that no decimal multiplied by itself can ever be exactly 2.000000000000000000. If the decimal ends in 1, then its square will end in 1. If the decimal ends it 2, its square will end in 4. And so on.  No decimal—no number of arithmetic—multiplied by itself can ever produce 2.

is irrational.

Question.   The square roots of which natural numbers are rational?

Answer.   Only the square roots of square numbers.

= 1  Rational

 Irrational

 Irrational

= 2  Rational

,  ,  ,  Irrational

= 3  Rational

And so on.

Problem 5.   Say the name of each number.

a)

Square root of 3.

b)

Square root of 8.

c)

3.

d)

2 5

e)

Square root of 10

Problem 6.   Which of the following numbers are rational and which are irrational?

a)    Irrational          b)   Rational

c)    Rational            d)   Irrational

Only a rational number can we know and name exactly. An irrational number we can know only as a rational approximation.

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:

	x²	=	25.
Solution.	x	=	5  or  −5, because (−5)² = 25, also.
In other words,
	x	=	or  −.

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.

= 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:

	x²	=	10.
Solution.	x	=	or  −.

Always, if an equation looks like this,

	x²	=	a,
then the solution will look like this:
	x	=	or  −.
We often use the double sign  ± ("plus or minus")  and write:
	x	=	±.

Problem 7.   Solve for x.

a)	x² = 9  implies x = ±3		b)	x² = 144  implies x = ±12
c)	x² = 5  implies x = ±		d)	x² = 3  implies x = ±
e)	x² = a − b  implies x = ±

2nd Level

Next Lesson:  Simplifying radicals

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