WHAT IS IRRATIONAL NUMBERS?

Math Adda

The invention of irrational numbers

We have seen  that when two squares are in the same ratio as two square numbers, then their sides will have the same ratio as the square roots.  Thus if CD were 1 meter, then we would like to say that AB is "Square root of 2" meters -- but there is no such rational number. For if there were, then again, AB, CD would have a common measure, which they do not.

If we insist, however, that there be a number to indicate the ratio of AB to CD, then we keep the name "Square root of 2," and we call it an irrational number.  Its numeral is .  That is,

 ×  = 2.

 is that number which when multiplied by itself -- when squared -- is 2.  "Square root of 2" is its name.  It is not any whole number, any fraction, any mixed number or any decimal -- it is not any rational number.  It is a pure creation.

Now,

7 5

is close to , because

7 5

×

7 5

=

49 25

-- which is almost 2.  But to prove that there is no rational number whose

square is 2, 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.  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.  It will be impossible to divide n × n intom × m and get 2

There is no rational number whose square is 2.

Naturally, we wonder, "How much is ?"  We can only answer

approximately.  As a fraction, we saw it is almost

7 5

.   As a decimal, it is

approximately 1.414.  How would anyone know that?  Again, because 1.414 squared is almost 2.

1.414 × 1.414 = 1.999396.

We could come closer to  by approximating it to more decimal places.  But no decimal squared will ever be exactly 2.   is irrational.

It was argued for many centuries whether  "really" is a number. Mathematicians thought of  as a convenient symbol, but as for its being a number, that was something else.  Because again, if we ask, How much is it?. . . We cannot say.

In the following Topic, we will investigate in what sense irrational numbers "exist."  And we will return to our original inquiry:

If AB, CD are lengths, will there always be a number n -- rational or irrational -- such that, proportionally,

AB is to CD  as  1 is to n?

In any event,  is not the only irrational number.  Let us illustrate that by asking:

The square roots of which natural numbers are rational?

Answer.   Only the square roots of square numbers.  Thus,

 = 1  is rational.

,  are irrational.

 = 2  is rational.

,  ,  ,   are irrational.

 = 3  is rational.

And so on.

Problem 1.   Say the name of each number.

a)	Square root of 3.		b)	Square root of 5.
c)	Two. It's rational.		d)	Square root of 3/5.
e)	Three-fifths.		f)	One.

Problem 2.   Which of the following are rational numbers and which are irrational?

a)	Irrational.		b)	Rational.
c)	Rational.		d)	Irrational.

A common measure with 1

We have seen that every number has a ratio to 1.  A rational number is to 1 in the same ratio as two natural numbers.  That is, every rational number has a common measure with 1 (Topic 8).  We can say, then, that an irrational number is a number that has no common measure with 1.

1 and  are incommensurable.

Problem 3.   What number is a common measure of each pair?

a)   1 and 5   1

b)   1 and

5 8

1 8

c)   1 and 2.617    .001

d)   1 and     None.

Problem 4.   Why have irrational numbers been invented?

To express the ratios of incommensurable magnitudes; in particular, the ratio of a magnitude incommensurable with 1.

Problem 5.   The squares on the sides of triangle ABC are in the ratio 1 : 4 : 3.  Express the ratio of each pair of sides as a ratio of numbers, whether rational or irrational.

a)  AB : BC = 1 : 2         b)  BC : CA = 2 :          c)  CA : AB =  : 1

d)   is approximately 1

5 7

, or

12  7

.  How could you know that?

12  7

·

12  7

=

144  49

-- which is almost 3.

e)   Use that approximation to approximate the ratio of BC to CA as a
e)   ratio of natural numbers.

BC : CA  =  2 :

12  7

=  14 : 12 = 7 : 6

Problem 6.   One square is four fifths of another.

a)  Are the sides commensurable?   No. 4 and 5 are not both square
a) numbers.

b)  Express the ratio of the sides as a ratio of numbers.

2 : 

c)  Are the squares commensurable?

Yes. They are in the same ratio as natural numbers.