The question of why the square root of the hexadecimal value B064 is not a rational number is a fascinating one. To address this, we first need to convert B064 from hexadecimal to decimal form, allowing us to better understand its properties.

Conversion of B064 to Decimal

The hexadecimal number B064 can be expressed in decimal notation as follows:

The hexadecimal digit B is equivalent to 11 in decimal. Therefore, the conversion can be represented as:

B06416 11 × 163 0 × 162 6 × 161 4 × 160

Breaking it down:

11 × 4096 (163) 45056

6 × 16 (161) 96

4 (160) 4

Add these values together:

45056 96 4 45156 (B06416)

Thus, B06416 in decimal is 45156.

Understanding the Square Root of 45156

The square root of 45156 can be simplified using factors:

sqrt(45156) sqrt(4 × 11299) 2 × sqrt(11299)

Now, we need to determine if sqrt(11299) is a rational number. A rational number can be expressed as a fraction p/q where p and q are integers with no common factors other than 1.

However, sqrt(11299) does not simplify to such a form. This is because 11299 is a prime number. Prime numbers have no divisors other than 1 and itself, which means its square root cannot be expressed as a ratio of two integers. Therefore, sqrt(11299) is an irrational number.

Thus, 2 × sqrt(11299) is also an irrational number.

Conclusion

The square root of B064 (45156 in decimal) is not a rational number because 45156 can be simplified to 2 × sqrt(11299), and sqrt(11299) is an irrational number. This makes sqrt(45156) an irrational number as well.

Understanding the conversion between hexadecimal and decimal, and recognizing the properties of numbers, particularly whether they are rational or irrational, is crucial in many mathematical and computational applications.

For further exploration, you might want to practice more conversions between different numeral systems and delve deeper into the properties of rational and irrational numbers.

Keywords: hexadecimal, rational number, square root, irrational number, conversion