Square Root of -135

The square root is the inverse operation of squaring a number. Since -135 is a negative number, its square root involves imaginary numbers. The square root of -135 is expressed in terms of the imaginary unit 'i' as √(-135) = √(135) * i. The value of √(135) is approximately 11.61895, making the square root of -135 approximately 11.61895i, which is a complex number.

For negative numbers like -135, the square root involves imaginary numbers. This is because no real number squared will result in a negative number. Therefore, we use 'i' to denote the imaginary unit where i² = -1. This allows us to express the square root of negative numbers as complex numbers. For example, the square root of -135 is written as √(-135) = √135 * i.

To understand the square root of -135, we first find the square root of 135, which is a positive number. The prime factorization of 135 is 3 x 3 x 3 x 5, which simplifies as √135 = 3√15. Since -135 is negative, we multiply this by 'i', giving us 3√15 * i or approximately 11.61895i.

While the square root of negative numbers does not have direct applications in real-world measurements, it is essential in fields involving complex numbers, such as electrical engineering and quantum mechanics. The square root of -135 can be used to solve equations in these fields where complex numbers naturally arise.

The square root of -135 is a complex number, which can be visualized on the complex plane. The real part is 0, and the imaginary part is approximately 11.61895. In polar coordinates, this is expressed with a magnitude of 11.61895 and an angle of 90 degrees (or π/2 radians) from the positive real axis.

• Imaginary Number: An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i² = -1.
   
• Complex Number: A complex number consists of a real part and an imaginary part, expressed as a + bi, where 'a' and 'b' are real numbers.
   
• Magnitude: The magnitude (or modulus) of a complex number is the distance from the origin to the point represented by the number in the complex plane.
   
• Principal Square Root: The principal square root of a negative number is the positive imaginary number derived from its square root calculation.
   
• Prime Factorization: Prime factorization is expressing a number as a product of its prime factors, useful for simplifying square roots.