Finding the Square Root of 0.9

The question asks us to find the square root of 0.9. The square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, we are looking for a value $x$ such that $x^2 = 0.9$. This is written as $x = \sqrt{0.9}$.

Understanding Square Roots

For a positive number $N$, its square root, denoted as $\sqrt{N}$, is a positive number $x$ such that $x \times x = N$. For example, $\sqrt{25} = 5$ because $5 \times 5 = 25$. When dealing with decimals, the concept remains the same.

Calculating the Square Root of 0.9

Finding the exact value of $\sqrt{0.9}$ by hand can be complicated as 0.9 is not a perfect square. The result will be an irrational number, meaning its decimal representation is non-terminating and non-repeating. We can use a calculator to find an approximate value or check the given options by squaring them to see which one is closest to 0.9.

Let's examine the options provided:

1. 0.9487
2. 0.3
3. 0.9463
4. 0.03

We can square each option to see which value is closest to 0.9:

• Squaring 0.9487: $(0.9487)^2 = 0.9487 \times 0.9487 = 0.89993169$
• Squaring 0.3: $(0.3)^2 = 0.3 \times 0.3 = 0.09$
• Squaring 0.9463: $(0.9463)^2 = 0.9463 \times 0.9463 = 0.89558269$
• Squaring 0.03: $(0.03)^2 = 0.03 \times 0.03 = 0.0009$

Comparing the Results

Now let's compare these squared values to 0.9:

• $0.89993169$ is very close to $0.9$. The difference is $|0.89993169 - 0.9| = 0.00006831$.
• $0.09$ is much smaller than $0.9$. The difference is $|0.09 - 0.9| = 0.81$.
• $0.89558269$ is close to $0.9$, but not as close as the first option. The difference is $|0.89558269 - 0.9| = 0.00441731$.
• $0.0009$ is much smaller than $0.9$. The difference is $|0.0009 - 0.9| = 0.8991$.

Comparing the differences, $0.00006831$ is the smallest difference. This means that $0.9487^2$ is the value closest to $0.9$ among the squares of the given options.

Conclusion

Based on our calculations, the square root of 0.9 is approximately 0.9487 because $(0.9487)^2$ is the closest value to 0.9 among the options.

The option with the squared value closest to 0.9 is 0.9487.

Revision Table: Square Roots of Decimals

Additional Information: Numbers and Their Properties

Understanding different types of numbers is helpful when dealing with square roots.

• Rational Numbers: Numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include 0.5 ($\frac{1}{2}$), 3 ($\frac{3}{1}$), 0.333... ($\frac{1}{3}$).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating. Examples include $\sqrt{2}$, $\pi$, and $\sqrt{0.9}$ (since 0.9 is not a perfect square).
• Perfect Squares: Numbers that are the square of an integer or a rational number. For example, 4 is a perfect square ($2^2$), 0.25 is a perfect square ($0.5^2$). The square root of a perfect square is a rational number.

Since 0.9 is not a perfect square (its factors are 0.9 = 9/10, and neither 9 nor 10 results in integer or simple decimal square roots that combine to give $\sqrt{0.9}$ as a rational number), its square root, $\sqrt{0.9}$, is an irrational number. The options given are decimal approximations of this irrational number.