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Prove that if æ is a positive real number, then LV=]] = LVE] %3D

Prove that if æ is a positive real number, then LV=]] = LVE] %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematical Proof on Educational Website** **Statement:** Prove that if \( x \) is a positive real number, then \[ \lfloor \sqrt{\lfloor x \rfloor} \rfloor = \lfloor \sqrt{x} \rfloor \] **Explanation:** Here, \( \lfloor \cdot \rfloor \) denotes the floor function, which outputs the greatest integer less than or equal to a given number. The task is to prove that applying the floor function to both the square root of a floored number and the square root of the original number yields the same result when \( x \) is any positive real number.
Transcribed Image Text:**Mathematical Proof on Educational Website** **Statement:** Prove that if \( x \) is a positive real number, then \[ \lfloor \sqrt{\lfloor x \rfloor} \rfloor = \lfloor \sqrt{x} \rfloor \] **Explanation:** Here, \( \lfloor \cdot \rfloor \) denotes the floor function, which outputs the greatest integer less than or equal to a given number. The task is to prove that applying the floor function to both the square root of a floored number and the square root of the original number yields the same result when \( x \) is any positive real number.
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Step 1

Solution:

Definition:

The Floor function f is a function that takes any real number x and gives f(x), a real number which is the greatest integer less than or equal to x.

The function f is denoted by, fx=x  ∀x∈ℝ.

Example:

1> If n∈ℤ, then f(n)=n ∀n∈ℤ.

2> If a∈ℝ be any real, then ∃an unique integer n with n<a<n+1. Then a=n.

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