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Geometry/Irrational Numbers

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Question
Hello sir

Sir where are these irrational numbers location on number line are used in our daily life.For eg locate root 2 on number line


Sir in our textbook one question is there sir ie find rational numbers between 3 and 4 . Sir how we can find sir because after 3 four comes then how its possible.How there can be infinity of rational numbers between 3 and 4

Answer
Hi Apsara,

Unlike natural numbers, we don't actively recognize our use of irrational numbers in day-to-day life.  Suppose you are asked to build a fence to enclose an area of one metre by one metre.  Afterwards, you are asked to divide that square into two equal triangles.  If you did not have irrational numbers, it would not be possible to construct that divider.

Of course in practice, nobody would tell you to "build a fence with length of root 2".  You would always round it somewhat, e.g. "build a fence with length of 1.41."

You might learn later that the existence of irrational numbers allows for many interesting properties of real numbers.  For the moment, being able to define an irrational number and knowing they exist should be enough.


Now for the second part of your question: making an irrational number between 3 and 4.  "After 3 comes 4".  This is not true when it comes to rational numbers.  There is never a "next" rational number, because we can always create a number in between.  You wish to find a rational number between 3 and 4?  Take (3+4)/2.  Done.  You want to create another one?  Use the same procedure to find a number between 3 and the new number you just found.  You can repeat that process ad infinitum to create rational numbers in any range you desire.  (a+b)/2 will always be between a and b.

I hope this is clear.

Best regards,
Azeem

Geometry

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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