To determine which given pair of numbers supports the idea that irrational numbers are dense in the real numbers, let's examine each pair and identify the irrational number in each.
Pair 1: 3.14 and [tex]\(\pi\)[/tex]
- [tex]\(3.14\)[/tex] is a rational approximation of [tex]\(\pi\)[/tex], which is irrational.
- Thus, this pair contains one rational (3.14) and one irrational number ([tex]\(\pi\)[/tex]).
Pair 2: 3.33 and [tex]\(\frac{1}{3}\)[/tex]
- [tex]\(3.33\)[/tex] is a rational number (a finite decimal representation).
- [tex]\(\frac{1}{3}\)[/tex] is also a rational number.
Hence, this pair contains two rational numbers.
Pair 3: [tex]\(e^2\)[/tex] and [tex]\(\sqrt{54}\)[/tex]
- [tex]\(e^2\)[/tex] is an irrational number, as [tex]\(e\)[/tex] (Euler's number) is irrational, and the square of an irrational number is also irrational.
- [tex]\(\sqrt{54}\)[/tex] (which can be written as [tex]\(\sqrt{3 \cdot 18}\)[/tex]) is also irrational because [tex]\(\sqrt{54} = 3\sqrt{6}\)[/tex] involves the square root of 6, an irrational number.
Both numbers in this pair are irrational.
Pair 4: [tex]\(\frac{\sqrt{64}}{2}\)[/tex] and [tex]\(\sqrt{16}\)[/tex]
- [tex]\(\frac{\sqrt{64}}{2} = \frac{8}{2} = 4\)[/tex] is a rational number.
- [tex]\(\sqrt{16} = 4\)[/tex] is also a rational number.
Hence, this pair contains two rational numbers.
After examining all pairs, we need to find which pair best supports the idea that irrational numbers are dense in the real numbers. The key is to identify a pair containing irrational numbers close to each other within the real number line.
From the Python code provided earlier, we know that the correct result is:
[tex]\[ (7.3890560989306495, 7.3484692283495345) \][/tex]
This fits the number pair [tex]\(e^2\)[/tex] and [tex]\(\sqrt{54}\)[/tex], both of which are irrational and closely situated.
Therefore, Pair 3: [tex]\(e^2\)[/tex] and [tex]\(\sqrt{54}\)[/tex] supports the idea that irrational numbers are dense in the real numbers.
