Question

Which number can you multiply by any irrational number to obtain an irrational number?

A) 2
B) 8
C) 16
D) 32

Answer 1

Thus the correct answer is D 32.

Step-by-step explanation:

To determine which number can be multiplied by any irrational number to obtain an irrational number, we can analyze the properties of irrational numbers. An irrational number is a number that cannot be expressed as a simple fraction and has an infinite, non-repeating decimal expansion. When we multiply an irrational number by a rational number, the result is still irrational.

Let's consider option D) 32. Suppose we have an irrational number represented by 'π' (pi). When we multiply π by 32, the result is 32π, which is still irrational. This is because π is irrational, and the product of an irrational number and a non-zero rational number remains irrational.

Mathematically, let x be an irrational number, and y be the number we want to find. If y * x = z, where z is irrational, then y = z/x. Therefore, for y to be irrational, x must not be a factor of z. In the case of 32, it is not a factor of common irrational numbers like π or √2, making it a suitable choice.

In conclusion, multiplying any irrational number by 32 results in an irrational number, satisfying the given condition.

Answer 2

Number 2 can be multiplied by any irrational number to obtain an irrational number. So the correct option is A) 2

Step-by-step explanation:

Multiplying an irrational number by an irrational number does not always result in an irrational number. However, multiplying an irrational number by a rational number can yield an irrational number under certain conditions. To identify the rational number that, when multiplied by any irrational number, results in an irrational number, we need to find a rational number that is not zero and doesn't have a finite decimal representation when divided into 1.

Consider the rational number 2. When any irrational number, such as π or √2, is multiplied by 2, the product remains irrational. This is because irrational numbers multiplied by non-zero rational numbers retain their irrational nature. Therefore, option A) 2 is the correct answer.

Step-by-step explanation:

Irrational numbers multiplied by non-zero rational numbers can result in irrational numbers due to the nature of irrational numbers' infinite and non-repeating decimal expansions.

Among the given choices, only 2 fits the criteria as it is a non-zero rational number.

Multiplying any irrational number (like √2 or π) by 2 results in an irrational number due to the property that the product retains the irrational nature of the initial number.

Hence, the rational number 2 is the only option that can be multiplied by any irrational number to obtain another irrational number.