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Chapter 2: Problem 58

Let \(a\) and \(b\) be real numbers with \(a

Short Answer

Expert verified
The number of integers is \floor{b} - \ceil{a} + 1 \.

Step by step solution

01

- Understand the inequality

First, recognize that the given inequality represents the range of integers bounded by the real numbers \(a\) and \(b\), inclusive. We need to find how many integers fall within this interval.
02

- Apply the floor function to the upper bound

To determine the largest integer within the interval \(a \leq x \leq b\), apply the floor function to the upper bound \(b\). This finds the greatest integer less than or equal to \(b\): \floor{b}\.
03

- Apply the ceiling function to the lower bound

To determine the smallest integer within the interval \(a \leq x \leq b\), apply the ceiling function to the lower bound \(a\). This finds the smallest integer greater than or equal to \(a\): \ceil{a}\.
04

- Calculate the number of integers

The number of integers \(n\) that satisfy the inequality \(a \leq n \leq b\) is the difference between these two values plus one (because the range is inclusive): \floor{b} - \ceil{a} + 1\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers are all the numbers you encounter in everyday mathematics. They include all the integers (like -2, 0, and 4), fractional numbers (like -1.5 and 3/4), and irrational numbers (like 鈭�2 or 蟺). They can be positive or negative, and they fill up a continuous number line without any gaps. Understanding real numbers is crucial because the floor and ceiling functions we will discuss apply to any real number, not just whole numbers.
Integers within an Interval
An interval between two real numbers represents all the values between those numbers. For example, if you have an interval from 2 to 5, it includes 2, 2.1, 2.2, all the way to 5. When we talk about integers within an interval, we specifically mean the whole numbers (like 2, 3, and 4) that fall between the two chosen real numbers. To find these integers, we identify the smallest and largest integers within the interval. This is where the floor and ceiling functions come into play.
Inclusive Inequalities
Inequalities help us understand the range of values a variable can take. Inclusive inequalities, like in our exercise (step-by-step solution), mean the endpoints are included in the range. For example, the inequality \(2 \leq x \leq 5\) includes the numbers 2 and 5 themselves. To express the total count of integers within such an interval, we use the floor and ceiling functions to clearly define the range's boundaries and count the values between them.
Floor Function
The floor function is used to round a real number down to the nearest integer. For example, \(\lfloor 2.7 \rfloor = 2\) and \(\lfloor -1.2 \rfloor = -2\). In our exercise's context, applying the floor function to the upper bound \(b\) of the interval gives us the largest integer within the interval. This is crucial for determining the inclusive set of integers between the real number bounds.
Ceiling Function
The ceiling function rounds a real number up to the nearest integer. For example, \(\lceil 2.3 \rceil = 3\) and \(\lceil -1.7 \rceil = -1\). In the exercise, applying the ceiling function to the lower bound \(a\) of the interval gives us the smallest integer within the interval. Combined with the floor function's result, this helps us calculate how many integers are within the given range, ensuring we include both endpoints if the inequality is inclusive.

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Most popular questions from this chapter

An employee joined a company in 2017 with a starting salary of \(\$ 50,000\) . Every year this employee receives a raise of \(\$ 1000\) plus 5\(\%\) of the salary of the previous year. a) Set up a recurrence relation for the salary of this employee \(n\) years after 2017 . b) What will the salary of this employee be in 2025\(?\) c) What will the salary of this employee be in 2025\(?\) Find an explicit formula for the salary of this employee \(n\) years after 2017 .

Let \(A\) and \(B\) be the multisets \(\\{3 \cdot a, 2 \cdot b, 1 \cdot c\\}\) and \(\\{2 \cdot a, 3 \cdot b, 4 \cdot d\\},\) respectively. Find a) \(A \cup B\) b) \(A \cap B\) c) \(A-B\) d) \(B-A\) e) \(A+B\)

Show that a set \(S\) is infinite if and only if there is a proper subset \(A\) of \(S\) such that there is a one-to-one correspondence between \(A\) and \(S .\)

Assume that \(a \in A,\) where \(A\) is a set. Which of these statements are true and which are false, where all sets shown are ordinary sets, and not multisets. Explain each answer. a) \(\\{a, a\\} \cup\\{a, a, a\\}=\\{a, a, a, a, a\\}\) b) \(\\{a, a\\} \cup\\{a, a, a\\}=\\{a\\}\) c) \(\\{a, a\\} \cap\\{a, a, a\\}=\\{a, a\\}\) d) \(\\{a, a\\} \cap\\{a, a, a\\}=\\{a\\}\) e) \(\\{a, a, a\\}-\\{a, a\\}=\\{a\\}\)

Data are transmitted over a particular Ethernet network in blocks of 1500 octets (blocks of 8 bits). How many blocks are required to transmit the following amounts of data over this Ethernet network? (Note that a byte is a synonym for an octet, a kilobyte is 1000 bytes, and a megabyte is \(1,000,000\) bytes.) a) 150 kilobytes of data b) 384 kilobytes of data c) 1.544 megabytes of data d) 45.3 megabytes of data
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