To determine which numbers could have been Miko's other number when he multiplied [tex]\(\frac{4}{3}\)[/tex] by it and the product was an irrational number, we need to consider the properties of rational and irrational numbers.
1. Rational Numbers: Rational numbers are numbers that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
2. Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They have non-terminating, non-repeating decimal parts.
A key fact we need to use is:
- When a rational number is multiplied by another rational number, the result is always a rational number.
- When a rational number is multiplied by an irrational number, the result is an irrational number.
Given that the product was irrational and [tex]\(\frac{4}{3}\)[/tex] is rational, Miko's other number must have been irrational.
Let’s go through the given options one by one to identify which are irrational numbers:
1. [tex]\(\frac{3}{4}\)[/tex]: This is a rational number because it can be expressed as a fraction of two integers (3 and 4).
2. [tex]\(\frac{\pi}{4}\)[/tex]: This is an irrational number because [tex]\(\pi\)[/tex] is irrational, and dividing it by 4 (a rational number) still results in an irrational number.
3. [tex]\(\frac{5}{4}\)[/tex]: This is a rational number because it can be expressed as a fraction of two integers (5 and 4).
4. [tex]\(\sqrt{3}\)[/tex]: This is an irrational number because the square root of a non-perfect square is irrational.
5. [tex]\(\pi\)[/tex]: This is an irrational number.
6. [tex]\(\sqrt{4}\)[/tex]: This simplifies to 2, which is a rational number because it can be expressed as a ratio of integers (2/1).
So, the numbers that could have been Miko's other number are:
- [tex]\(\frac{\pi}{4}\)[/tex]
- [tex]\(\sqrt{3}\)[/tex]
- [tex]\(\pi\)[/tex]
Thus, the possible options are:
[tex]\[
\boxed{\frac{\pi}{4}, \sqrt{3}, \pi}
\][/tex]
