Subscribe to our βΆοΈYouTube channelπ΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
As we know that the numbers which canβt be written in pq form or fraction form are known as irrational numbers. These are non- recurring decimal numbers. The square roots, cute roots of numbers which are not perfect roots are examples of irrational numbers. In such cases in which perfect square roots or cube roots canβt be found out, it is difficult to compare them without knowing their approximate or actual value.
For comparing them, we should always keep in mind that if square or cube roots of two numbers (βaβ and βbβ) are to be compared, such that βaβ is greater than βbβ, then a2 will be greater than b2 and a3 will be greater than b3 and so on, i.e., nth power of βaβ will be greater than nth power of βbβ.
1. Compare β2 and β3
Solution:
We know that if βaβ and βbβ are two numbers such that βaβ is greater than βbβ, then a2 will be greater than b2. Hence, for β2 and β3, let us square both the numbers and then compare them:
(β2)2 = β2 Γ β2 = 2,
(β3)2 = β3 Γ β3 = 3
Since, 2 is less than 3.
Hence, β2 will be less than β3.
2. Compare β17 and β15.
Solution:
Let us find out the square of both the numbers and then compare them. So,
(β17)2 = β17 Γ β17 = 17,
(β15)2 = β15 Γ β15 = 15
Since, 17 is greater than 15.
So, β17 will be greater than β15.
3. Compare 2β3 and β5.
Solution:
To compare the given numbers let us first find the square of both the numbers and then carry out the comparison process. So,
2(β3)2 = 2β3 x 2β3 = 2 Γ 2 Γ β3 Γ β3 = 4 Γ 3 = 12,
(β5)2 = β5 Γ β5 = 5
Since, 12 is greater than 5.
So, 2β3 is greater than β5.
4. Arrange the following in ascending order:
β5, β3, β11, β21, β13.
Solution:
Arranging in ascending order stands for arrangement of series from smaller value to the larger value. To arrange the given series in ascending order let us find the square of every element of the series. So,
(β5)2 = β5 Γ β5 = 5.
(β3)2 = β3 Γ β3 = 3.
(β11)2 = β11 Γ β11 = 11.
(β21)2 = β21 Γ β21 = 21.
(β13)2 = β13 Γ β13 = 13.
Since, 3 < 5 < 11 < 13 < 21. Hence, the required order of the series is:
β3 < β5 < β11 < β13 < β21.
5. Arrange the following in descending order:
3β5, 3β7, 3β15, 3β2, 3β39.
Solution:
Descending order stands for arrangement of given series in larger value to the smaller value. To find the required series, let us find the cube of each element of the series. So,
(3β5)3 = 3β5 Γ 3β5 Γ 3β5 = 5.
(3β7)3 = 3β7 Γ 3β7 Γ 3β7 = 7.
(3β15)3 = 3β15 Γ 3β15 Γ 3β15 = 15.
(3β2)3 = 3β2 Γ 3β2 x 3β2 = 2.
(3β39)3 = 3β39 Γ 3β39 Γ 3β39 = 39.
Since, 39 > 15 > 7 > 5 > 2.
So, the required order of the series is:
3β39 > 3β15 > 3β7 > 3β5 > 3β2
Irrational Numbers
Definition of Irrational Numbers
Representation of Irrational Numbers on The Number Line
Comparison between Two Irrational Numbers
Comparison between Rational and Irrational Numbers
Problems on Irrational Numbers
Problems on Rationalizing the Denominator
Worksheet on Irrational Numbers
From Comparison between Two Irrational Numbers to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 17, 25 01:06 AM
Jul 17, 25 12:40 AM
Jul 16, 25 11:46 PM
Jul 16, 25 02:33 AM
Jul 15, 25 11:46 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.