When does the floor function inequality hold?

In summary, the conversation discusses the floor function and a claim that [x] is greater than or equal to x - 1. While equality cannot occur, the claim is still true as the strict part of the inequality implies the weaker statement.
  • #1
SweatingBear
119
0
Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim

\(\displaystyle [x] \geq x - 1\)

The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an example. Maybe the claim is false?
 
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  • #2
sweatingbear said:
Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim

\(\displaystyle [x] \geq x - 1\)

The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an example. Maybe the claim is false?

Hi sweatingbear!

You are right that equality cannot occur.
However, the claim is still true.
Consider that:
$$[x] > x - 1 \quad\Rightarrow\quad [x] \geq x - 1$$
It's just a weaker statement. Still true though.
 

FAQ: When does the floor function inequality hold?

What is the floor function inequality?

The floor function inequality is a type of mathematical inequality that involves the floor function, which rounds a number down to the nearest integer. It is typically written as ⌊x⌋ ≤ y, where x is a real number and y is an integer.

What is the difference between the floor function inequality and the ceiling function inequality?

The floor function inequality uses the floor function, which rounds down to the nearest integer, while the ceiling function inequality uses the ceiling function, which rounds up to the nearest integer. This means that the floor function inequality will always have a result less than or equal to the given number, while the ceiling function inequality will always have a result greater than or equal to the given number.

How is the floor function inequality used in real life?

The floor function inequality can be used in a variety of real-life scenarios, such as calculating the maximum number of people that can fit in a room based on its floor area, determining the minimum number of items needed to fill a certain order, or setting a minimum budget for a project based on expected costs. It is also commonly used in computer programming to round numbers down to whole integers.

What are some properties of the floor function inequality?

Some properties of the floor function inequality include:

  • ⌊x⌋ ≤ x for all real numbers x
  • If x is an integer, ⌊x⌋ = x
  • ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ for all real numbers x and y
  • If x > 0, then ⌊-x⌋ = -⌊x⌋

Can the floor function inequality be rewritten as an equation?

No, the floor function inequality cannot be rewritten as an equation. It is an inequality that compares the result of the floor function to a given number, and it does not have a single solution like an equation does. Instead, it describes a range of possible values that satisfy the inequality.

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