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When we solve irrational numbers, we must know about irrational numbers. What are their properties? What is the product of two irrational numbers? What is the product of rational and irrational numbers? And, when the irrational numbers were discovered? So, let’s know the answer to all these questions. The irrational numbers were discovered in the 5th century BC. It was discovered by Hippasus, who was a Pythagorean philosopher. His theory was not accepted at that time. But today, we all know, irrational numbers exist. We can’t imagine mathematics without rational and irrational numbers. So, here we explain the difference between rational and irrational numbers.
What are irrational numbers?
From the above discussion, we get an overview of irrational numbers. So, know about it in some detail. Irrational numbers are real numbers that can not be represented in a fraction form like rational numbers. It means it can not be expressed in p/q form. Here p and q are integers where q is not equal to zero.
Examples- √2, π, Euler’s number e, Golden ratio φ.
Properties of irrational numbers
There are some properties of irrational numbers. It helps you to find irrational numbers from the real numbers.
1 . Irrational numbers include non-terminating and non-recurring decimals.
- Irrational numbers are always real numbers.
- When we add rational and irrational numbers, we always get irrational numbers as their result. Suppose ‘a’ is a rational number and ‘b’ is an irrational number. So, a + b will always be an irrational number.
- When we multiply an irrational number with a non-zero rational number. Then, we get an irrational number as a result.
- We can’t solve the LCM (Least Common factor) of any two irrational numbers. Because it doesn’t exist.
- The product of two irrational numbers is always a rational number.
- The product of rational numbers and irrational numbers is always an irrational number.
Solved example of irrational numbers
Here are some solved examples of irrational numbers.
1 . Prove that 3√2 is an irrational number.
Suppose 3√2 is a rational number. Then, we have to find co-prime ‘p’ and ‘q,’ where ‘q’ is not equal to 0. Then, 3√2 = a/b.
We rearrange it like, √2= a/3b.
Then, we know 3, a, and b are integers. And, a/3b is a rational number. But we know √2 is an irrational number.
Hence, proving 3√2 is an irrational number.
- What is the irrational number between 1/7 and 2/7?
We calculate 1/7= 0.142857 and 2/7= 0.285717.
So, if we find an irrational between 1/7 and 2/7, we get the number that is Non-terminating and non-recurring, lying between 1/7 and 2/7. So, we get infinite irrational numbers. Like, 0.150150015000….
- What will we get by adding 2√2 + 5√3 and √2 – 3√3.
(2√2 + 5√3) + (√2 – 3√3)
= (2√2 + √2) + (5√3 – 3√3)
=(2+1) √2 + (5 – 3)√3
= 3√2 + 2√3.
- What will we get by multiplying 4√5 and 2√5?
4√5 × 2√5
=4 × 2 × √5 × √5
= 8 × 5
= 40.
Hence, proving the product of two irrational numbers is always a rational number.
Steps to identify irrational numbers
If you want to know if the real number is rational or irrational, you need to know whether it can be expressed or not in fraction form. If it can be represented as a fraction, it is a rational number. Or it can not be represented as a fraction; it is an irrational number.
Rational vs Irrational numbers
Rational numbers can be expressed as fractions and ratios, whereas irrational numbers can not be expressed as fractions and ratios.
The decimal value of rational numbers is terminating or non-terminating recurring, whereas the decimal value of irrational numbers is non-terminating and non-recurring.
Conclusion
After reading the above paragraph, we get a clear understanding of irrational numbers. Irrational numbers can never be expressed as a ratio like rational numbers. But the product of two irrational numbers is rational numbers. Besides, the product of a rational number and an irrational number is always an irrational number. Also, we know all square roots of a prime number are irrational numbers. And Euler’s number e, Golden ratio φ, and Pi π are some important irrational numbers. Because the decimal value of these numbers or ratios never stops at any point. Moreover, the solved example makes the concept crystal clear.
