Home | Irrational Numbers

Consider an isosceles right-angled triangle whose legs are each 1 unit long.

**Irrational Numbers**

You saw that each rational number is assigned to a
point on the number line and learnt about the denseness property of the
rational numbers. Does that mean the
line is entirely filled with the rational numbers and there are no more numbers
on the number line? Let us explore.

Consider an isosceles right-angled triangle whose
legs are each 1 unit long. Using Pythagoras
theorem, the hypotenuse can be seen having a length √(1^{2}+1^{2}) (see
Fig. 2.6 ). Greeks found that this √2 is neither a whole number nor an ordinary
fraction. The belief of relationship between points on the number line and all
numbers was shattered! √2 was
called an irrational number.

An irrational number is a number that
cannot be expressed as an ordinary ratio of two integers.

A natural question is how one knows that √2 is irrational. It is not
difficult to justify it.

If √2 is
really rational, let it be equal to p/q where p, q are integers without any
common factors (so that p/q will be in its simplest form) and q ≠ 0.

which means p^{2} is even … (2)

As a result, p is even. …. (3)

(Can you prove this?)

Let p = 2m (How?); you get p^{2} = 4m^{2}
;

This, because of (1)

=> 2q^{2} = 4m^{2} or q^{2}=2m^{2}.

As a result q is even ` … (4).

(3) and (4) show that p and q have a common factor
2.

This contradicts our assumption that p and q have
no common factors and hence our assumption
that √2 can be written as p/q is wrong.
That is, √2 is not
rational.

1. Apart from √2 , one can produce a number of examples for such
irrational numbers.

Here are a few: √5 , √7, 2√ 3……..

2. π , the ratio of the circumference of a circle
to the diameter of that same circle, is
another example for an irrational number.

3. e, also known as Euler’s number, is another
common irrational number.

4. Φ , the golden ratio, also known as golden mean,
or golden section, is a number often stumbled upon when taking the ratios of
distances in simple geometric figures such as the pentagon, the pentagram,
decagon and dodecahedron, etc., it is an irrational number.

Where are the points on the number line that
correspond to the irrational numbers?

As an example, let us locate √2 on the number line. This is
easy.

Remember that √2 is the length of the diagonal of the square whose
side is 1 unit (How?)Simply construct a
square and transfer the length of one of its diagonals to our number line. (see
Fig.2.7).

We draw a circle with centre at 0 on the number
line,with a radius equal to that of diagonal of the square. This circle cuts
the number line in two points, locating √2 on the right of 0 and -√2 on its left. (You wanted to
locate √2 ; you have also got a bonus in -√2 )

You started with Natural numbers and extended it to
rational numbers and then irrational numbers. You may wonder if further
extension on the number line waits for us. Fortunately it stops and you can
learn about it in higher classes.

Representation of a Rational number as terminating
and non terminating decimal helps us to understand irrational numbers. Let us
see the decimal expansion of rational numbers.

If you have a rational number written as a
fraction, you get the decimal representation by long division. Study the
following examples where the remainder is 0 always:

Consider the examples,

Can the decimal representation of a rational number
lead to forms of decimals that do not terminate? The following examples (with
non-zero remainder) throw some light on this point.

Represent the following as decimal
form (i) -4/11 (ii)11/75

*Solution*

A rational number can be expressed by

1. **either **a
terminating

2. **or **a
non-terminating and recurring** **(repeating)
decimal expansion.

The converse of this statement is also true.

That is, if the decimal expansion of a number is
terminating or non-terminating and recurring, then the number is a rational
number.

In the decimal expansion of the rational numbers,
the number of repeating decimals is called the length of the period of
decimals.

For example,

Let us now try to convert a terminating decimal,
say 2.945 as rational number in the fraction form.

That is, in any decimal number, each digit after
the decimal point is a fraction with a denominator in increasing powers of 10.
Thus,

Convert the following decimal numbers
in the form of p/q , where p and q are integers and q **≠** 0: (i) 0.35 (ii) 2.176 (iii)
– 0.0028

**Solution **

It was very easy to handle a terminating decimal.
When we come across a decimal such as 2.4, we get rid of the decimal point, by
just using division by 10.

Thus 2.4 = 24/10 , which is simplified as 12/5. 12
. But, when we have a decimal such as the problem is that we have infinite number of 4s and hence will need infinite
number of 0s in the denominator. For example,

How tough it is to have infinite 4’s and work with
them. We need to get rid of the infinite sequence in some way. The good thing
about the infinite sequence is that even if we pull away one , two or more 4 out
of it, the sequence still remains infinite.

We use the same exact logic to convert any number
with a non terminating repeating part into a fraction.

Convert the following decimal numbers
in the form of p/q (p,q∈Z and q≠0)

Without actual division, classify the
decimal expansion of the following numbers as terminating or non – terminating
and recurring.

It can be shown that irrational numbers, when
expressed as decimal numbers, do not terminate, nor do they repeat. For
example, the decimal representation of the number π starts with
3.14159265358979, but no finite number of digits can represent π exactly, nor
does it repeat.

Consider the following decimal expansions:

(i) 0.1011001110001111…

(ii) 3.012012120121212…

(iii) 12.230223300222333000…

(iv) √2 = 1.4142135624…

Are the above numbers terminating (or) recurring
and non- terminating? No… They are neither terminating, nor non–terminating and
recurring. Hence they are not rational numbers. They cannot be written in the
form of p/q, where p,q, ∈ Z and q≠0. They are irrational numbers.

A number having non- terminating and
non- recurring decimal expansion is an irrational number.

Find the decimal expansion of √3

**Solution**

Thus, by division method, √3 = 1.7320508…

It is found that the square root of every positive
non perfect square number is an irrational number. √2, √3, √5, √6, √7 , … are all irrational numbers.

Classify the numbers as rational or
irrational:

Locate an irrational number between
two rational numbers 23/10 and 12/5.

You need an irrational number greater than 2.3 but
less than 2.4

One such irrational number is

2.301001000100001000001000000100000001……..

How do you write it?

See the given box for details.

Is this newly found number irrational?

Yes, observe that it is non-terminating and
non-recurring.

Can you find an irrational number between 63 and 64
now?

Could it be 63.01001000100001000010000001… … … ?

Between the rational numbers 7.568903 and 7.568904,
is it possible that there is this irrational number 7.568903 01 001 0001 00001
000001 0000001… … …?

Find any 4 irrational numbers between 1/4 and 1/3.

Find any 3 irrational numbers between
0.12 and 0.13 .

*Solution*

Three irrational numbers between 0.12 and 0.13 are
0.12010010001…, 0.12040040004…, 0.12070070007…

Give any two rational numbers lying
between 0.5151151115…. and 0.5353353335…

*Solution*

Two rational numbers between the given two
irrational numbers are 0.5152 and 0.5352

For example, when you consider the rational number
4 and the irrational number √5 , then 4 + √5 , 4 – √5 , 4√5 , 4/√5,
√5/4........ ------- are all irrational
numbers.

Tags : Solved Example Problems | Mathematics , 9th EM Mathematics : Real Numbers

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