Is the square root of 17 a rational number?
Understand the Problem
The question is asking whether the square root of 17 can be expressed as a fraction of two integers, which would classify it as a rational number. To answer this, we need to determine if the square root of 17 is a whole number or can be simplified into a ratio of integers.
Answer
The square root of 17, $\sqrt{17}$, is irrational.
Answer for screen readers
The square root of 17, $\sqrt{17}$, cannot be expressed as a fraction of two integers, which means it is not a rational number.
Steps to Solve
- Determine if $\sqrt{17}$ is a whole number
To find out if $\sqrt{17}$ is a whole number, we can calculate the square root. Since $16 < 17 < 25$, we can conclude that $\sqrt{17}$ is between 4 and 5, thus it is not a whole number.
- Check if $\sqrt{17}$ can be expressed as a fraction
For a number to be rational, it must be expressible as a fraction of two integers. Since we established that $\sqrt{17}$ is not a whole number, we should check to see if it can be expressed in fractional form.
- Evaluate $\sqrt{17}$ for rationality
The square root of a non-perfect square, like 17, is irrational. An irrational number cannot be expressed as a simple fraction $\frac{a}{b}$ where $a$ and $b$ are integers.
The square root of 17, $\sqrt{17}$, cannot be expressed as a fraction of two integers, which means it is not a rational number.
More Information
The square root of 17 is approximately 4.123, which confirms that it is not a whole number and further emphasizes its irrationality. This is an important concept in number theory.
Tips
Common mistakes include assuming that any square root is rational without checking if the number is a perfect square, like how people might mistakenly think $\sqrt{17}$ is rational because it seems "close" to an integer.
