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 can state 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

The rational numbers are the ordinary 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

integers (b ≠ 0).

Problem 4.   Which of the following numbers are rational?

All of them!

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.

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.

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.

a)

Square root of 3

b)

Square root of 8

c)

3

d)

2 5

e)

Square root of 10

As for the decimal representation of both irrational and rational numbers, see Topic 2 of Precalculus.

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

Problem 6.   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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