Problem 5

Question

Let $x=3.456$ and $y=2.789 .$ Compute each. $$\lfloor x+y\rfloor$$

Step-by-Step Solution

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Answer

The value of $\lfloor x+y\rfloor$ for the given values of $x$ and $y$ is 6.

1Step 1: Add the given numbers

Add the given numbers $x$ and $y$ as follows: $$ x+y = 3.456 + 2.789 $$

2Step 2: Simplify the sum

Now, calculate the sum of this addition: $$ x+y = 3.456 + 2.789 = 6.245 $$

3Step 3: Apply the floor function

Now that we have the sum of $x$ and $y$, we will find the greatest integer less than or equal to this sum by applying the floor function: $$ \lfloor x+y\rfloor = \lfloor 6.245\rfloor $$

4Step 4: Evaluate the floor function

Finally, determine the greatest integer less than or equal to the calculated sum: $$ \lfloor x+y\rfloor = \lfloor 6.245\rfloor = 6 $$ The value of $\lfloor x+y\rfloor$ for the given values of $x$ and $y$ is 6.

Key Concepts

Floor FunctionAdditionGreatest Integer Function

Floor Function

The floor function, represented by $ \lfloor x \rfloor $, is a mathematical concept that deals with the rounding down of real numbers. Essentially, the floor function refers to finding the greatest integer that is less than or equal to a given number. For instance, if the number is 6.245, applying the floor function will yield 6. This is because 6 is the largest integer not greater than 6.245.

Key characteristics of the floor function include:

It can handle both positive and negative numbers. For example, $ \lfloor -2.3 \rfloor = -3 $ as $-3$ is less than $-2.3$ but still an integer.
It always rounds towards negative infinity, meaning it's not about typical rounding rules.
It is different from the ceiling function, which rounds up to the nearest integer.

Understanding this function is crucial in discrete mathematics as it helps in decision-making scenarios, such as computing maxima or simplifying expressions.

Addition

Addition is one of the most basic operations in arithmetic and mathematics, involving the calculation of the total of two or more numbers or amounts. The numbers added together are called "addends", and the result is the "sum" of the addition. For the problem of adding $x = 3.456$ and $y = 2.789$, the operation is performed as follows:

Add the decimal parts together: 0.456 + 0.789 = 1.245. Here, carry over the 1 to the whole number part.
Add the whole number parts together, including the carry from the decimal part: 3 + 2 + 1 = 6.
The sum is thus $6.245$.

When performing addition, pay attention to aligning decimal points for accuracy. This practice ensures that each digit is added to its corresponding place value.

Greatest Integer Function

The greatest integer function is often synonymous with the floor function in many contexts, and it represents the same idea of rounding down to the nearest integer. The notation $ \lfloor x \rfloor $ conveys this.

A useful property of the greatest integer function is $ x - 1 < \lfloor x \rfloor \leq x $. This means the greatest integer is always less than or equal to $x$ but absolutely closest to it below.
For the expression $ \lfloor 6.245 \rfloor $, it simplifies directly to 6, as 6 is the nearest integer below 6.245.
In contrast, if you deal with negative values, it gets more interesting: $ \lfloor -1.5 \rfloor = -2 $.

The greatest integer function is a vital part of functions involving division and modular arithmetic, making it a tool of great importance in discrete math.

Problem 5

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