Irrational number
A number which we cannot express in the fractional form i.e. p/q where p & q are some integers & q≠0.
Example:- √(15), √(3), √(2), π, 0.202200222....., etc.
Symbol of an irrational number is P.
How to Identify an irrational number?
It is very simple to identify any number if it is irrational. It should be non-terminating & non-recurring decimal expansion. Let's discuss-
Non-terminating
Non-terminating means when we divide numerator from denominator & we do not get zero as a reminder so it will be non-terminating decimal expansion.
Such as (22÷7) = 3.14159265358....
You noticed when we divide numerator 22 from denominator 7 so we do not get zero as remainder.
We continue solving but not get remainder zero so it will be non-terminating. Remember if we get zero as remainder so it will be terminating.
Non-recurring(non-repeating)
Non-recurring means when we divide numerator from the denominator & we do not get repeating numbers in quotient so it will be non-recurring decimal expansion.
Such as (22÷7) = 3.14159265358....
You noticed when we divide numerator 22 from denominator 7 so we get non-repeating numbers in quotient 3.14159265358.....
We continue solving but we do not get a repeating number in quotient so it will be non- recurring. Remember if we get repeating numbers in quotient so it will be recurring.
Questions of an irrational number
1. Why √2 is an irrational number?
Because when we solve √(2) as decimal expansion so we get 1.4142135624.....
And here we get non-terminating(not getting zero in the remainder).
And Non-recurring(continue getting the different number in quotient) decimal expansion so √(2) will be an irrational number.
2. What is the product of rational & irrational number?
Suppose, we have a rational number (1/2) & irrational number √(3).
After the product of (1/2) & √(3), we get √(3)/2.
We get it's a decimal expansion (1.73205080...)/2 & after solving we get 0.8660254038....
We may notice here we are getting non-terminating & non-recurring decimal expansion. So the product of rational & irrational number will be irrational.
3. Write irrational number without root.
We may write many irrational numbers without root such as-
1.23463100112...(non-terminating & non-recurring decimal expansion).
5.31267340012...(non-terminating & non-recurring decimal expansion).
In this way, we may write different irrational numbers.
4. Why π is an irrational number?
Because when we find it's decimal expansion then we get 3.14159265358....
And here we get non-terminating & non-recurring decimal expansion so π is an irrational number.
5. Why we write π=(22/7) while an irrational number is not in fractional form.
Yes, π is an irrational number & also irrational number not is in fractional form.
Remember, we use the approximate value of π as (22/7) or 3.14. Because we cannot show it's overall value so we do it.
6. What is the sum of two irrational number?
Suppose we have two irrational number √2 & √3.
After adding both numbers we get (√2+√3). If we add it's decimal expansion so we get the irrational number.
Now we can say that the sum of two irrational numbers is irrational.
6. Write the list of irrational numbers.
There is a different irrational number-
√2, √5, π, √6, 1.101100011011...., etc. ( All numbers have non-terminating & non-recurring decimal expansion.
7. Difference between rational & irrational number.
We write a rational number in fractional form or in the form of p/q where q ≠ 0.
While the irrational number is not in fractional form or not in form of p/q.
While the irrational number is not in fractional form or not in form of p/q.
All-natural numbers are a rational number. Because we can show natural numbers as p/q.
While a natural number is not an irrational number.
8. Find irrational between 3.5 & 4.5.
We can write an uncountable irrational number between 3.5 & 4.5.
3.6912789543....(non-terminating & non-recurring decimal expansion).
4.01278953472....(non-terminating & non-recurring decimal expansion).
3.901276541299...(non-terminating & non-recurring decimal expansion).
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