Question
according to the closure property, which of these sums results in an irrational number?
A. 0.668357 + 0.488712197
B. √4 + 5
C. 1 + √2
D. 9/7 + 5/3
A. 0.668357 + 0.488712197
B. √4 + 5
C. 1 + √2
D. 9/7 + 5/3
Answers
Answered by
GPT-4o mini
To determine which of the sums results in an irrational number, we need to analyze each option:
A. \(0.668357 + 0.488712197\): Both numbers are rational (they can be expressed as fractions), and the sum of two rational numbers is rational. So this sum is rational.
B. \(\sqrt{4} + 5\): \(\sqrt{4} = 2\), which is a rational number. Therefore, \(2 + 5 = 7\), which is also rational. So this sum is rational.
C. \(1 + \sqrt{2}\): Here, \(1\) is rational, but \(\sqrt{2}\) is irrational. The sum of a rational number and an irrational number is irrational. Thus, this sum is irrational.
D. \(9/7 + 5/3\): Both \(9/7\) and \(5/3\) are rational numbers. The sum of two rational numbers is rational. So this sum is rational.
Based on this analysis, the correct answer is:
**C. \(1 + \sqrt{2}\)**, which results in an irrational number.
A. \(0.668357 + 0.488712197\): Both numbers are rational (they can be expressed as fractions), and the sum of two rational numbers is rational. So this sum is rational.
B. \(\sqrt{4} + 5\): \(\sqrt{4} = 2\), which is a rational number. Therefore, \(2 + 5 = 7\), which is also rational. So this sum is rational.
C. \(1 + \sqrt{2}\): Here, \(1\) is rational, but \(\sqrt{2}\) is irrational. The sum of a rational number and an irrational number is irrational. Thus, this sum is irrational.
D. \(9/7 + 5/3\): Both \(9/7\) and \(5/3\) are rational numbers. The sum of two rational numbers is rational. So this sum is rational.
Based on this analysis, the correct answer is:
**C. \(1 + \sqrt{2}\)**, which results in an irrational number.