Problem 7 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The answer is 9.

Step 1: Multiply the given numbers

First, we need to multiply the given numbers \(x\) and \(y\): \[x * y = 3.456 * 2.789\] To find the product, you can use a calculator or long multiplication: \[3.456 * 2.789 = 9.635664\]

Step 2: Apply the floor function

Now we need to find the greatest integer less than or equal to the product \(x * y\). The floor function, denoted as \(\lfloor x \rfloor\), is used for this purpose. Thus, \[\lfloor x * y \rfloor = \lfloor 9.635664 \rfloor\] The greatest integer less than or equal to \(9.635664\) is 9. So, \[\lfloor x * y \rfloor = 9.\] So, the answer to the exercise is 9.

Key Concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with distinct and separate values, often focusing on integers and combinatorial structures. Unlike continuous mathematics, which deals with objects that can vary smoothly, discrete mathematics studies things that can be counted individually.
Understanding discrete mathematics is essential for computer science and related fields because it's involved in algorithms, data structures, and cryptography among other areas.

• Topics include set theory, logic, graph theory, and number theory.
• Practical applications are found in coding, networks, and problem-solving strategies.

With ceiling and floor functions being a part of discrete mathematics, they help in rounding numbers to the nearest integers, thus bridging continuous numbers to discrete ones.

Real Numbers

Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers. They're essential in continuous mathematics and are the basis for calculus and analysis.
Real numbers can be:

• Rational: These are numbers that can be expressed as a fraction or ratio of two integers, such as 1/2 or 0.75.
• Irrational: These numbers cannot be expressed as a simple fraction, like π or √2.

In our exercise, both 3.456 and 2.789 are real numbers. They illustrate how real numbers are manipulated to find integer solutions using the floor function, maintaining the essence of discrete mathematics.

Multiplication

Multiplication is one of the basic arithmetic operations, and it's used to find the product of two numbers. When multiplying real numbers, the result is often a real number as well.
In this exercise:

• We multiply 3.456 by 2.789 to find their product.
• The result is 9.635664, which showcases how numbers can grow and change significantly through multiplication.

Understanding multiplication is crucial as it forms the foundation for more complex topics in mathematics, such as algebra and calculus. It helps bridge discrete and continuous ideas, particularly when dealing with functions like the floor function.

Floor Function

The floor function, represented by \(\lfloor x \rfloor\), rounds a real number down to the nearest integer. It's widely used in discrete mathematics and programming.
Important aspects of the floor function include:

• It always moves a number to the nearest lesser integer.
• Useful in scenarios where whole numbers are required, like in counting objects or steps.

In the exercise example, after multiplying 3.456 and 2.789, the product was 9.635664. The floor function was applied to round this down to 9. This illustrates how real numbers are translated into discrete numbers, essential for digital computations and logical processes.