Prove that if x is a positive real number, then a) ⌊√(⌊x⌋)⌋=⌊√(x)⌋b) ⌈√( | x ⌉) |=⌈√(x)⌉

Name: Problem 77, Chapter 2: Discrete Mathematics and its Applications
Uploaded: 2019-10-01T02:02:50-08:00
Duration: 3 min 42 s
Channel: James Chok
Description: Prove that if x is a positive real number, then a) ⌊√(⌊x⌋)⌋=⌊√(x)⌋b) ⌈√( | x ⌉) |=⌈√(x)⌉

step 1 Okay, first up we want to prove this inequality is correct. So if x is a positive real number, t

Prove that if x is a positive real number, then a)...

Key Concepts

Floor Function

The floor function, typically denoted by ?x?, maps any real number x to the greatest integer less than or equal to x. This function is fundamental in number theory and analysis, as it helps in approximating and simplifying expressions by isolating the integer part of a number. Understanding its properties, such as its step-wise constancy and its behavior under addition or multiplication with other functions, is essential for many proofs involving discrete approximations of continuous values.

Ceiling Function

The ceiling function, denoted by ?x?, maps any real number x to the smallest integer greater than or equal to x. Like the floor function, it is crucial in problems that involve rounding or approximating numbers to the nearest integer. Its properties are especially important in combinatorial arguments and proofs where controlling the upper bound of a value is necessary.

Square Root Function

The square root function assigns to each non-negative real number its non-negative square root, and it is strictly increasing. This means that if a < b then ?a < ?b. This monotonicity is a key property when composing the square root function with the floor or ceiling functions, because it ensures that the order of numbers is preserved. Being able to apply inequalities reliably to ?x is important in proving identities that mix integer-valued and continuous functions.

Monotonicity of Functions

Monotonic functions, which either never decrease or never increase as their input increases, allow for the preservation of inequalities under function composition. In the context of this problem, the monotonically increasing nature of the square root function means that applying it to both sides of an inequality preserves the inequality. This concept is vital in proofs that compare transformed versions of x, such as when taking the floor or ceiling after applying a continuous transformation like the square root.

Handling Compositions of Functions with Rounding Operations

When working with compositions that involve rounding functions (like floor or ceiling) and continuous functions (like square roots), one must carefully analyze how the operations interact. This includes understanding how rounding after a transformation compares with applying the transformation after rounding. Such analysis often involves establishing bounds using inequalities and exploring the interplay between the integer and fractional parts of numbers, which is a common technique in real analysis and number theory.

Transcript

00:01 Okay, first up we want to prove this inequality is correct.

00:05 So if x is a positive real number, then there exists n in the integers such that n squared is going to be less than or equal to x is less than equal to n plus one square.

00:23 So first up, we have n is going to be less than equal to square of x, x less than equal to n.

00:33 Plus one, just squaring everything.

00:36 No, not lessen.

00:37 This is strictly less than.

00:40 So clearly, n is going to be equal to the floor of square of x.

00:45 Because we have n and n plus one, we know that square of x is going to be somewhere in this region.

00:53 So we're going to take the bottom number, which is n.

00:56 So now for this side, the floor of n squared is going to be less than or equal to the floor of n squared is going to be less than or equal to the floor of x, which is going to be less than the floor of n plus 1 square.

01:16 Then what we'll do is the floor of n squared is just n squared.

01:21 So that's less than equal to its flow of x.

01:24 So that's going to be less than the floor of an integer is still an integer.

01:27 That'll be n plus 1 square.

01:30 Then taking square root of everything, we have n is less than equal to square root of floor of x.

01:36 So that would be less than n plus 1.

01:38 So once again, we are bound once again between n and n plus one.

01:43 So we just take the bottom one...