Question

Here are two expressions with a missing number: Expression 1: −5/9⋅____________ Expression 2: ____________+4 Choose the same number to put in both blanks so that both expressions would result in an irrational answer. You can choose your number from the following list: 3.2, square root 3, 2/3, and 0.232323... Explain your reasoning.

a) 3.2
b) square root 3
c) 2/3
d) 0.232323...

Answer 1

The number that makes both expressions result in an irrational number is the square root of 3, because the product and sum of a rational and an irrational number are always irrational.

Step-by-step explanation:

To answer the student's question, we are looking for a number that, when multiplied by −5/9 for Expression 1 and added to 4 for Expression 2, results in an irrational number in both cases. We can quickly eliminate 3.2 and 2/3 because these are rational numbers, and their resulting products and sums would also be rational. The number 0.232323... is also rational because it is a repeating decimal, a representation of the fraction 23/99. The remaining number, the square root of 3 (√3), is an irrational number because it cannot be expressed as a simple fraction and its decimal form does not repeat or terminate. Thus, −5/9 times √3 will be irrational because the product of a rational and an irrational number is irrational. Similarly, √3 + 4 is irrational, because the sum of a rational number and an irrational number is irrational. Therefore, the correct number to fill in both blanks is square root of 3.