Which of the following numbers will have an irrational square root?
1825
Understanding Irrational Square Roots
An irrational number is a number that cannot be expressed as a simple fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. The square root of a number is irrational if the number is not a perfect square. A perfect square is an integer that is the square of another integer (e.g., $9$ is a perfect square because $9 = 3^2$).
To determine which of the given numbers has an irrational square root, we need to check if each number is a perfect square. If a number is a perfect square, its square root will be a rational number (an integer). If it's not a perfect square, its square root will be irrational.
Checking Each Option for Perfect Squares
Let's examine each number provided:
- 1225
- 625
- 1825
- 3025
We can calculate the square root of each number:
| Number | Square Root Calculation | Is it a Perfect Square? | Nature of Square Root |
|---|---|---|---|
| 1225 | $\sqrt{1225} = 35$ | Yes ($35^2 = 1225$) | Rational (35 is an integer) |
| 625 | $\sqrt{625} = 25$ | Yes ($25^2 = 625$) | Rational (25 is an integer) |
| 1825 | $\sqrt{1825} \approx 42.72$ | No | Irrational (not an integer, not a simple fraction) |
| 3025 | $\sqrt{3025} = 55$ | Yes ($55^2 = 3025$) | Rational (55 is an integer) |
Identifying the Irrational Square Root
From the calculations above, we see that:
- The square root of 1225 is 35, which is a rational number.
- The square root of 625 is 25, which is a rational number.
- The square root of 1825 is approximately 42.72... It is not an integer, and 1825 is not a perfect square, so its square root is an irrational number.
- The square root of 3025 is 55, which is a rational number.
Therefore, the number that will have an irrational square root is 1825 because it is not a perfect square.
Revision Table: Rational vs. Irrational Square Roots
| Concept | Description | Example |
|---|---|---|
| Rational Number | Can be written as $\frac{p}{q}$ where $p, q$ are integers, $q \neq 0$. Includes integers, fractions, terminating and repeating decimals. | $5$, $\frac{1}{2}$, $0.75$, $0.333...$ |
| Irrational Number | Cannot be written as $\frac{p}{q}$. Non-terminating, non-repeating decimals. | $\sqrt{2}$, $\pi$, $e$, $\sqrt{1825}$ |
| Perfect Square | An integer that is the square of another integer. | $4 (=2^2)$, $9 (=3^2)$, $16 (=4^2)$ |
| Square Root of Perfect Square | Always a rational number (specifically, an integer). | $\sqrt{16} = 4$ |
| Square Root of Non-Perfect Square | Always an irrational number (for positive integers). | $\sqrt{3} \approx 1.732...$ |
Additional Information on Irrational Numbers
Irrational numbers are a significant part of the real number system. They arise in many mathematical contexts. For example:
- The diagonal of a square with side length 1 is $\sqrt{2}$, an irrational number.
- The ratio of a circle's circumference to its diameter, $\pi$ (pi), is an irrational number.
- The base of the natural logarithm, $e$, is an irrational number.
Identifying whether a number is a perfect square is a common way to determine if its square root is rational or irrational. For large numbers, checking the last digit can sometimes give a clue (e.g., a perfect square cannot end in 2, 3, 7, or 8), but it's not definitive. Calculating or approximating the square root is the most reliable method.
Similar Questions
Which of the following numbers will have an irrational square root?
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Which of the numbers given below is NOT rational?
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Which of the following is a rational number?
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Important Questions from Rational or Irrational Numbers
Which of the following is the smallest fraction?
4/5, 7/8, 6/7, 5/6
When 0.232323.... is converted into a fraction, then it is equal to:
Which of the following number is irrational?
What is the square root of 16 + 6√7?
A non-terminating but recurring decimal is:
