Problem 11 - Discrete Mathematics with Applications

Question

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x+y\rceil$$

Step-by-Step Solution

Answer

The short answer is: "\(\lceil x+y \rceil = \lceil 3.456 + 2.789 \rceil = \lceil 6.245 \rceil = 7\)".

Step 1: Calculate the sum of x and y

First, we need to compute the sum of \(x\) and \(y\): \(x + y = 3.456 + 2.789\)

Step 2: Perform the addition

Now, let's add the two values: \(3.456 + 2.789 = 6.245\)

Step 3: Determine the ceiling function

Next, we need to find the smallest integer greater than or equal to the result from step 2. The ceiling function of 6.245 can be written as: \(\lceil 6.245 \rceil\)

Step 4: Compute the ceiling function

Finally, we need to determine the value of the ceiling function: \(\lceil 6.245 \rceil = 7\) So, the output is \(7\).

Key Concepts

Addition of Decimals

When adding decimals, it's important to line up the decimal points. This alignment ensures that each digit is in the correct place value, whether it's the units, tenths, hundredths, etc. This is analogous to how you might add whole numbers. Let's see this in action:

• Align the decimal numbers vertically so that the decimal points are one above the other.
• Add each column starting from the rightmost digits (the smallest place value) and move left.
• If the sum of a column exceeds 9, carry over the extra value to the next left column, just like in whole number addition.

For instance, while adding 3.456 and 2.789, we add the numbers vertically:

• In the thousandths place, 6 + 9 equals 15, write 5 and carry over 1.
• In the hundredths place, 5 + 8 equals 13, plus the carry-over 1 equals 14, write 4 and carry over 1.
• In the tenths place, 4 + 7 equals 11, plus the carry-over 1 equals 12, write 2 and carry over 1.
• Finally, in the units place, 3 + 2 equals 5, and we add the carry-over 1 to get 6.

This results in 6.245. This precision makes decimal addition unique compared to whole numbers.

Rounding Up

Rounding up is a method of approximating numbers where the value is increased to the nearest whole number. In the context of the ceiling function, rounding up involves finding the smallest whole number greater than or equal to a given decimal number. Here’s how it works:

• Inspect the decimal part of the number.
• If any non-zero digit exists in the decimal part, round up to the next whole number.
• If the number is already a whole number, then it stays the same.

For example, given the decimal 6.245, the criteria for rounding up based on the ceiling function dictates you must "round up" to the smallest integer that is not less than 6.245. Here that smallest integer is 7. This method of rounding ensures no part of a decimal value is lost through truncation or rounding down.

Integer Functions

Integer functions, such as the ceiling function, play a crucial role in mathematical computations where whole number results are required despite starting with decimal inputs. The ceiling function specifically returns the smallest integer that is greater than or equal to a given number.

• The function is denoted by the symbol \( \lceil x \rceil \).
• Its primary use is when decimal numbers must be rounded up to perform further integer-only operations.
• This function contrasts with the floor function, which rounds down to the nearest integer.

Consider the decimal 6.245; applying the ceiling function gives us 7. This specific integer ensures that any calculations requiring whole numbers subsequently have a clear and precise value to work with. Integer functions are especially useful in scenarios where precision in whole numbers is necessary, such as in rounding off currency or ensuring discrete units in digital computations.