Question

Let mm be a rational number and nn be an irrational number, (m)(n) - c(m)(n)−c, assume cc is rational:

mnc/m
The proof shows that the product of an irrational and rational number is:
Explain.
A) Rational, since an irrational number cannot equal a rational number.
B) Irrational, since an irrational number cannot equal a rational number.
C) Rational, since you can write it as the division of two rational numbers.
D) Irrational, since you can write it as the division of two rational numbers.
Choose the correct option and provide a concise and clear explanation for choice.

Answer 1

The correct option is B) Irrational, because multiplying a rational number by an irrational number results in an irrational number, and subtracting a rational number does not change this.

Step-by-step explanation:

The question is asking whether the product of a rational number (m) and an irrational number (n), minus a rational number (c) is rational or irrational, and we are to choose from the provided options. The correct answer to this question is:

B) Irrational, since an irrational number cannot equal a rational number.

To understand why this is the correct answer, it helps to recognize what makes a number rational or irrational. A rational number is one that can be expressed as the division of two integers, where the denominator is not zero. An irrational number cannot be written as a simple fraction; it has an infinite, non-repeating decimal expansion.

When you multiply a rational number (m) by an irrational number (n), the result is an irrational number. Subtracting a rational number (c) from the product of these two numbers (mn) will not change the nature of this number. It will remain irrational because the subtraction of a rational number cannot turn an irrational number into a rational one.

Therefore, the expression (m)(n) - c, given that m is rational, n is irrational, and c is rational, will result in an irrational number because the irrationality is maintained through multiplication and is not negated by the subsequent subtraction of a rational number.