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Number Systems

Irrational Numbers Between Two Numbers

What is an irrational number? Imagine a square having side 1. The diagonal of that square is exactly the square root of two, which is an irrational number. π, e, √3 are examples of irrational numbers. Let’s study what is an irrational number between any two numbers.

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What is an Irrational Number?

An irrational number is a number that is not rational that means it is a number that cannot be written in the form \( \frac{p}{q} \). It cannot be written as the ratio of two integers. Representation of irrational numbers on a number line. From the below figure, we can see the irrational number is √2

What is an Irrational Number

Irrational Number between Two Rational Numbers

Suppose we have two rational numbers a and b, then the irrational numbers between those two will be, √ab. Now let us find two irrational numbers between two given rational numbers.

1. Find an irrational number between two rational numbers 2 – √3 and 5 – √3

Let x be the irrational number between two rational numbers  2 – √3 and 5 – √3. Then we get,
2 – √3 < x < 5 – √3
⇒ 2 < x + < √3 < 5
We see that  x + √3 is an irrational number between 2 – √3 and 5 – √3 where 2 – √3 < x < 5 – √3.

2. Find two irrational numbers between two given rational numbers.

Now let us take any two numbers, say a and b. Let x be any number between a and b. Then,
We have a < x < b….. let this be equation (1)
Now, subtract √2 from both the sides of equation (1)
So, a – √2 < x < b – √2……equation (2)
= a  < x + √2 < b
Addition of irrational number with any number results into an irrational number. So, x + √2 is an irrational number which exists between two rational numbers a and b.

Irrational Number between Two Irrational Numbers

The easiest way to find the number of two rational numbers is to square both the irrational numbers and take the square root of their average. If the square root is irrational, then we get the number we want. If you do not the number you are looking for,  then repeat the procedure using one of the original numbers and the newly generated number.

An irrational number between any two irrational numbers a and b is given by √ab. For example,

1. Find the rational numbers between √2 and √3

Let us first find the difference between √2 and √3. Since the difference lies between \( \frac{3}{10} \) and \( \frac{1}{3} \). There exist an integer between 4√2 and 4√3 that is 6, such that \( \frac{6}{4} \) = \( \frac{3}{2} \) is between √2 and √3. So now we can find other rationals by taking another multiple than 4.

2. We can also find many rationals between any two irrational numbers.

Let us take two irrational numbers a and b. To find the difference between a and b that is b – a, take n ∈ N and n > 1. Now, there exists some integer m between na and nb. Then, \( \frac{m}{n} \) is an irrational number between a and b.

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Rational Number between Two Rational Numbers

If m and n are the two ration numbers such that m < n then, \( \frac{1}{2} \) ( m + n ) is the rational number between m and n. Let us see common denominator method to find the rational number between two rational numbers. Rational numbers between two rational numbers can be found out by using common denominator method. For example,

Let us assume two rational numbers as \( \frac{-3}{2} \) and \( \frac{5}{3} \)
\( \frac{-3}{2} \)  = \( \frac{-3 × 3}{2 × 3} \)   = \( \frac{-9}{6} \)
\( \frac{5}{3} \) = \( \frac{5 × 2}{3× 2} \) = \( \frac{10}{6} \)
Rational numbers between these numbers are \( \frac{-8}{6} \), \( \frac{-7}{6} \),…, \( \frac{9}{6} \)

Solved Examples for You

Question 1: Which of the following irrational numbers lies between  \( \frac{3}{5} \) and  \( \frac{9}{10} \)

  1.  \( \frac{√80}{10} \)
  2. \( \frac{√85}{10} \)
  3. \( \frac{√82}{10} \)
  4. \( \frac{√83}{10} \)

Answer : A. √36 < √80 < √81. On dividing with 10 we get, \( \frac{6}{10} \) < \( \frac{√80}{10} \) < \( \frac{9}{10} \)

Question 2: How many irrational numbers lie between √2 and √3?

  1. One
  2. Zero
  3. Ten
  4. Infinite

Answer : D. Infinite irrational numbers lie between √2 and √3. For example, √2.1, √2.11, √2.101 and so on.

Question 3: What is an irrational number and give examples?

Answer: Irrational numbers are ones which we can’t write as a ratio of two integers. In other words, can’t express them as fractions. For instance, the square root of 2 will be an irrational number as we cannot write it as a ratio of two integers.

Question 4: Is 0 an irrational number?

Answer: A number that does not fulfil the mentioned criteria above won’t be irrational. Thus, as we can represent zero as a ratio of two integers plus its ratio is also an irrational number like it isn’t dividend in any case, so zero will be rational due to being an integer.

Question 5: Are irrational numbers infinite?

Answer: There is an infinite number of irrational numbers much similar to how there is an infinite number of integers, rational numbers plus real numbers. But, as reals are uncountable and rationals are countable thus, irrationals will be uncountable. In other words, there will be many more irrationals than rationals.

Question 6: Is the square root of an irrational number irrational?

Answer: If a square root is not a perfect square, then we consider it an irrational number. We cannot write these numbers as fractions as the decimals won’t end and won’t repeat a pattern. In other words, they are non-terminating and non-repeating.

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Very clear. ..thanks

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