**Define Irrational Numbers
with Examples.**

__Definition__: A decimal number which is non-terminating and non-recurring is
called an **irrational number**. Irrational
numbers cannot be expressed in the form of p/q where p and q are integers and q≠0. The square root of all non-square numbers, cube root of all non-cube numbers, etc.
are all non-terminating and non-recurring decimal numbers, so they are
irrational numbers. π(pi) also is an irrational number.

Here are some examples:

√2 = 1.4142135…

√3 = 1.7320508…

√5 = 2.2360679…

√7 = 2.6457513…

π = 3.14159265…

are non-terminating and non-recurring decimal numbers, so they
are irrational numbers.

Some other examples of irrational numbers are:

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**Properties of
irrational numbers:**

Here are some of the properties of irrational numbers:

1. Irrational numbers are non-terminating and non-recurring
decimal numbers.

2. Irrational numbers are closed under the operation of addition
i.e. the sum of two irrational numbers is always an irrational number.

3. Irrational numbers are closed under the operation of
subtraction i.e. the difference of two irrational numbers is always an
irrational number.

4. Irrational numbers are not closed under the operation of
multiplication i.e. the product of two irrational numbers may not be an
irrational number.

5. Irrational numbers are not closed under the operation of
division. i.e. the quotient of two irrational numbers may not be an irrational
number.

**What are the differences
between rational and irrational numbers? **

Following are the differences between rational and
irrational numbers:

1. Rational numbers can be expressed in the form of p/q where q ≠
0 and p and q are integers, whereas irrational numbers cannot be expressed in
such form.

2. Rational numbers may be terminating or non-terminating but recurring
decimal numbers whereas irrational numbers are always non-terminating and non-recurring
decimal numbers.

3. Rational numbers are closed under the operation of
multiplication and division, but irrational numbers are not closed under the
operation of multiplication and division.

**Some Question and
Answers on Irrational Numbers**

Question: How is π(pi) an
irrational number? Because it can be expressed in the form of p/q which is
22/7.

Answer: 22/7 is not the exact
value of π(pi). This is an approximate value only. The exact value of π(pi) is 3.141592653589793238...
which is non-terminating and non-recurring decimals. So, the exact value of π(pi)
is an irrational number.

Question: Is the sum of two
irrational numbers always irrational?

Answer: Yes, the sum of two
irrational numbers is always an irrational number.

Question: Are all square roots
irrational numbers?

Answer: Square roots of
non-square numbers are all irrational numbers.

Question: Are irrational numbers
closed under multiplication?

Answer: No, irrational numbers
are not closed under multiplication.

Question: Are every irrational
number is a real number?

Answer: Yes, every irrational
number is a real number.

Question: What about the between
rational and irrational numbers?

Answer: The difference between
rational and irrational numbers is irrational numbers.

Question: What about the sum of
rational and irrational numbers?

Answer: The sum of a rational and
irrational number is always an irrational number.

Question: How many irrational
numbers are there between 1 and 6?

Answer: There are infinitely many
irrational numbers between any two integers. So there are infinitely many
rational numbers between 1 and 6.

Question: Find at least two
irrational numbers between 2 and 3.

Answer: √5 and √6 are two
irrational numbers between 2 and 3.

Question: Find an irrational number
between 3 and 4.

Answer: √10 is an irrational
number between 3 and 4.

Question: Find an irrational number
between 1 and 2.

Answer: √3 is an irrational
number between 1 and 2.

Question: Find two irrational
numbers between 0.5 and 0.55.

Answer: √0.26 and √0.27 are two
irrational numbers between 0.5 and 0.55.

Question: Find two irrational
between 0.7 and 0.77.

Answer: √50 and √51 are two
irrational numbers between 0.7 and 0.77.

Question: What about the sum of a
rational number and an irrational number?

Answer: The sum of a rational
number and an irrational number is an irrational number.

Question: What about the difference
between rational and irrational numbers?

Answer: The difference between
rational and irrational numbers is an irrational number.

Question: Is the product of two
irrational numbers always irrational?

Answer: No, the product of two
irrational numbers is not always irrational. For example: √2 × √8 = √16 = 4
which is a rational number.

If you have any questions or problems regarding the **Irrational Numbers**, you can ask here,
in the comment section below.

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