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26

Radicals:  Rational and Irrational Numbers


The square numbers

The radical sign and the radicand

Rational and irrational numbers

An equation  x² = a

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:

Square numbers 1 4 9 16 25 36 49 64 81 100
Square roots 1 2 3 4 5 6 7 8 9 10

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		  

Rational and irrational numbers

The rational numbers are the numbers of arithmetic:  the whole numbers, fractions, mixed numbers, and decimals; together with their negative images.

That is what a rational number is. As for what it looks

like, it will take the form   a
b		  , where a and b are

integers (b ≠ 0).

Problem 3.   Which of the following numbers are rational?

1     −6     3½     4
5		     −  13
 5		     0     7.38609

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.  There is no whole number, no fraction, and no decimal whose square is 2.  (1.414 is close, because (1.414)² = 1.999396 -- 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  m
n		  in lowest

terms. That is, suppose

m
n		 ·   m
n		  =  m· m
 n· n		  = 2.
But that is impossible.  Because  since  m
n		 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.

Question.   Which square roots are rational?

Answer.   Only the square roots of square numbers.

= 1  Rational

 Irrational

 Irrational

= 2  Rational

,  ,  ,  Irrational

= 3  Rational

And so on.

Only the square roots of square numbers are rational.

The existence of these irrationals was first realized by Pythagoras in the 6th century B.C.  He called them "unnameable" or "speechless" numbers.  For, if we ask, "How much is ? -- we cannot say.   We can only call it, "Square root of 2."

Problem 4.   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.

An equation  x² = a

Example 1.   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 is "the negative of the square root of 25."

Example 2.   Solve this equation:

  x²  =  10
 
   Solution. x  =   or  −.

In general, 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 5.   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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