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Chapter 9: Problem 12

\(k(n)=\left\lfloor n^{1 / 2}\right\rfloor\) for each integer \(n \geq 0\)

Short Answer

Expert verified
The function \(k(n) = \left\lfloor n^{1/2} \right\rfloor\) is the floor function applied to the square root of a given integer n. To find the value of k(n) for a specific integer, first calculate the square root of n and then apply the floor function, which rounds down the number to the nearest integer. For example, if n=5, the square root of 5 is approximately 2.236, and the floor function of this value is 2, so \(k(5) = 2\).

Step by step solution

01

Understand the floor function

The floor function, denoted by \(\left\lfloor x \right\rfloor\), is defined as the greatest integer that is less than or equal to \(x\). In other words, it "rounds down" x to the nearest integer. Examples of the floor function are: - \(\left\lfloor 3.7 \right\rfloor = 3\) - \(\left\lfloor -1.5 \right\rfloor = -2\) - \(\left\lfloor 5 \right\rfloor = 5\)
02

Calculate \(n^{1/2}\) for a given integer \(n\)

To find the value of \(k(n)\), first calculate \(n^{1/2}\). This can be done using a calculator or by finding the square root of the given integer n manually if it's a perfect square. Remember that if n is not a perfect square, the result will be an irrational number. For example, suppose we want to find k(9). First, we calculate \(9^{1/2}\), which is equal to: \(9^{1/2} = 3\)
03

Apply the floor function to \(n^{1/2}\)

Now that we have calculated the square root of n, apply the floor function to this result. This means rounding down the square root value to the nearest integer. In our example with n=9, we have: \(k(n) = k(9) = \left\lfloor 3 \right\rfloor = 3\)
04

Interpret the result

The result we obtained in the previous step gives us the value of the function k(n) for the given integer n. For example, when n=9: The value of the function k(n) is 3, so we can say that \(k(9) = 3\).
05

Example: Apply the process to a different value of n

Now let us find the value of k(n) for n=5. Step 1: Calculate \(n^{1/2}\) \(5^{1/2} = 2.236...\) Step 2: Apply the floor function to \(n^{1/2}\) \(k(n) = k(5) = \left\lfloor 2.236... \right\rfloor = 2\) Step 3: Interpret the result The value of the function k(n) for n=5 is 2. Therefore, \(k(5) = 2\).

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Key Concepts

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

Integer Square Roots
When we talk about integer square roots, we're discussing a specific type of mathematical operation. The square root of a number, such as 9, is a number which, when multiplied by itself, gives the original number, in this case, 3. An integer square root simply means that the square root of a number is an integer. This is only true for numbers that are perfect squares, like 4, 9, or 16.

In many mathematical problems, we're tasked with finding the largest integer less than or equal to the square root of any given number, which is where the floor function comes into play. If the number is not a perfect square, its square root will not be an integer; thus, taking the floor of that square root gives us the highest integer value less than the square root.
Perfect Squares
A perfect square is an integer that is the square of another integer. For example, 16 is a perfect square because it can be expressed as the square of 4. Perfect squares have particularly neat properties in mathematics, such as always having an odd number of factors and producing integer square roots.

Understanding perfect squares is essential when working with square roots since the square root of a perfect square is always an integer. This becomes useful when dealing with floor functions in exercises like the one above, as the floor of an integer is the integer itself.
Mathematical Functions
A mathematical function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. The function we see in our problem, described by the floor symbol \(\left\lfloor x \right\rfloor\), is just one of many types of functions.

Functions like the floor function are fundamental in math because they enable us to model and solve real-world problems, encapsulate a process, and provide clarity and precision in mathematical reasoning.
Discrete Mathematics
The realm of discrete mathematics involves study areas such as graph theory, counting, and, importantly for our problem, number theory, where the concept of the floor function usually finds its use. Discrete mathematics deals with distinct and separate values; it's about objects that can be enumerated. This domain of math underpins computer science, cryptography, and combinatorics.

It's within discrete mathematics that we encounter functions like the floor function, which assigns to each real number the greatest integer less than or equal to that number. Floor functions produce discrete outputs, making them essential tools in problems involving integer-valued results.

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Most popular questions from this chapter

Define binary relations \(R\) and \(S\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: $$ \begin{aligned} &R=\\{(x, y) \in \mathbf{R} \times \mathbf{R}|y=| x \mid\\} \text { and } \\ &S=\\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid y=1\\} \end{aligned} $$ Graph \(R, S, R \cup S\), and \(R \cap S\) in the Cartesian plane,

Exercises \(36-39\) refer to the following algorithm to compute the value of a real polynomial. Algorithm 9.3.3 Term-by-Term Polynomial Evaluation [This algorithm computes the value of the real polynomial \(a[n] x^{n}+a[n-1] x^{n-1}+\cdots+a[2] x^{2}+a[1] x+a[0]\) by computing each term separately, starting with \(a[0]\), and adding it on to an accumulating sum.] Input: \(n\) [a nonnegative integer], \(a[0], a[1], a[2], \ldots, a[n]\) [an array of real numbers], \(x[\) a real number \(]\) Algorithm Body: polyval := \(a[0]\) for \(i:=1\) to \(n\) term : = \(a[i]\) for \(j:=1\) to \(i\) term \(:=\) term \(\cdot x\) next \(j\) polyval := polyval + term next \(i\) [At this point polyval \(=a[n] x^{n}+a[n-1] x^{n-1}\) \(\left.+\cdots+a[2] x^{2}+a[1] x+a[0] .\right]\) Output: polyval [a real number] Trace Algorithm \(9.3 .3\) for the input \(n=3, a[0]=2, a[1]=\) \(1, a[2]=-1, a[3]=3\), and \(x=2\).

Graph each function defined in 1-8 below. \(f(x)=3^{x}\) for all real numbers \(x\)

Define a binary relation \(P\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: For all real numbers \(x\) and \(y_{+}\) $$ (x, y) \in P \quad \Leftrightarrow \quad x=y^{2} . $$ Is \(P\) a function? Explain.

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