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26

Radicals:  Rational and Irrational Numbers


The square numbers

2nd level:

Equations  (x + a)² = b

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,

radical  =  5.

"The square root of 25 is 5."

This mark radical 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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Do the problem yourself first!

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

Example 1.   Evaluate radical.

Solution.   radical = 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

radical

Problem 2.   Evaluate the following.

   a)   radical  =  28.   b)   radical  =  135.  
  c)   radical  =  · 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 · · 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 = · 8 · 5 · 8 = 40 · 40

b)  16 · 49 = · 7 · 4 · 7 = 28 · 28

c)  4 · 9 · 25 = · 3 · 5 · 2 · 3 · 5 = 30 · 30

Rational and irrational numbers

A rational number is simply a number of arithmetic: a whole number, fraction, mixed number, or decimal; together with its 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 of a fraction a/b, where a and b are integers (b ≠ 0).

Problem 4.   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 radical ("Square root of 2"). radical is not a number of arithmetic.  7-5 is close because

7
5
 ·   7
5
  =   49
25

-- 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.

squares

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 Square root of 2 is an irrational number.

As a decimal approximation,

radical1.414

(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.

radical is irrational.

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

Answer.   Only the square roots of square numbers.

radical = 1  Rational

radical  Irrational

radical  Irrational

radical = 2  Rational

radical,  radical,  radical, radical  Irrational

radical = 3  Rational

And so on.

Problem 5.   Say the name of each number.

   a)  radical  Square root of 3.   b)  radical   Square root of 8.   c)  radical  3.
   d)   radical   2
5
  e)   radical   Square root of 10

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

a)  radical  Irrational          b)  radical  Rational

c)  radical  Rational            d)  radical 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  =  radical  or  −radical.

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.

radical = 5.

As for −5, it is "the negative of the square root of 25."

radical = −5.

Thus the symbol radical refers to one non-negative number.

Example 3.   Solve this equation:

  x²  =  10.
 
   Solution. x  =  radical  or  −radical.

Always, if an equation looks like this,

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

Problem 7.   Solve for x.

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

2nd Level

end

Next Lesson:  Simplifying radicals

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