Solving an Inequality Question Involving Ceiling Function

• jack1234
In summary, the conversation involves a question about inequality with the use of the Ceiling function. The reference book states that B must be large enough to satisfy the equation B*(B-1) >= N when B-1 >= Ceiling(N/B). The conversation also includes a hint to write out the definition of Ceiling (N/B) and then multiply it by B to understand the implication.
jack1234
Inequality question involving Ceiling

I have seen the following sentence in the reference book,
B-1 >= Ceiling(N/B) implies that B must at least be large enough to satisfy
B*(B-1) >= N

but how does
B-1 >= Ceiling(N/B)
implies
B*(B-1) >= N
?

Note that B and N are natural numbers.

Last edited:
jack1234 said:
I have seen the following sentence in the reference book,
B-1 >= Ceiling(N/B) implies that B must at least be large enough to satisfy
B*(B-1) >= N

but how does
B-1 >= Ceiling(N/B)
implies
B*(B-1) >= N
?

Note that B and N are natural numbers.

Hi jack!

Hint: Write out the definition of Ceiling (N/B), in full.

Then multiply that by B - what happens?

1. What is a ceiling function in mathematics?

A ceiling function is a mathematical function that rounds a number up to the next integer. It is denoted by the symbol ⌈x⌉ and is read as "ceiling of x". For example, ⌈3.2⌉ = 4 and ⌈5.8⌉ = 6.

2. How is a ceiling function used in solving inequalities?

A ceiling function is commonly used to solve inequalities involving non-integer values. By applying the ceiling function to both sides of the inequality, the non-integer values are rounded up to the next integer, making it easier to determine the solution set.

3. Can a ceiling function change the value of an inequality?

Yes, a ceiling function can change the value of an inequality. Since the ceiling function rounds up to the next integer, it can change the value of a non-integer number to an integer, thus changing the value of the inequality.

4. What are the rules for using a ceiling function in solving inequalities?

The following are the rules for using a ceiling function in solving inequalities:

• If the inequality is x > a, then applying the ceiling function to both sides results in ⌈x⌉ > a.
• If the inequality is x < a, then applying the ceiling function to both sides results in ⌈x⌉ < a+1.
• If the inequality is x ≥ a, then applying the ceiling function to both sides results in ⌈x⌉ ≥ a.
• If the inequality is x ≤ a, then applying the ceiling function to both sides results in ⌈x⌉ ≤ a+1.

5. Are there any other functions that can be used to solve inequalities?

Yes, there are other functions that can be used to solve inequalities, such as the floor function, which rounds a number down to the previous integer. The absolute value function can also be used to solve certain types of inequalities. It is important to choose the appropriate function based on the given inequality to accurately solve the problem.

• Calculus and Beyond Homework Help
Replies
1
Views
636
• Calculus and Beyond Homework Help
Replies
1
Views
591
• Calculus and Beyond Homework Help
Replies
3
Views
870
• Calculus and Beyond Homework Help
Replies
3
Views
139
• Calculus and Beyond Homework Help
Replies
1
Views
636
• Calculus and Beyond Homework Help
Replies
3
Views
600
• Calculus and Beyond Homework Help
Replies
6
Views
488
• Calculus and Beyond Homework Help
Replies
4
Views
888
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
632