Last updated on May 26th, 2025
The square root is the inverse operation of squaring a number. In real numbers, the square root of a negative number is not defined. However, in the field of complex numbers, the square root of a negative number is expressed using imaginary units. Here, we will discuss the square root of -47.
The square root of a negative number is not defined in the real number system. Instead, it is expressed in terms of imaginary numbers. The square root of -47 is expressed as √(-47) or in terms of imaginary units as i√47, where i is the imaginary unit defined as √(-1). Therefore, the square root of -47 is an imaginary number.
For negative numbers, the square root is not defined in the real number system. Instead, we use the concept of imaginary numbers. The methods used for finding the square root of positive numbers such as prime factorization, long division, or approximation do not apply directly to negative numbers. Instead, we consider the imaginary unit: - Imaginary unit method
The imaginary unit method is used to express the square root of a negative number. Let's see how:
Step 1: Identify the negative number, which is -47.
Step 2: Express -47 as (-1) x 47.
Step 3: The square root of -1 is represented by the imaginary unit i.
Step 4: Therefore, the square root of -47 is expressed as i√47.
In the complex number system, the square root of any negative number can be expressed by separating the negative part using the imaginary unit:
Step 1: Write the number -47 as the product of -1 and 47.
Step 2: Recognize that √(-1) is the imaginary unit i.
Step 3: Express the square root of -47 as i√47, indicating it is a complex number.
Though imaginary numbers are not part of the real number system, they have applications in various fields: - Electrical engineering: Used in the analysis of AC circuits with complex impedance. - Control theory: Utilized in system stability analysis. - Signal processing: Applied to handle waveforms and oscillations. - Quantum physics: Used to describe wave functions and quantum states.
Mistakes are often made when dealing with negative square roots, especially when applying real number methods. Let’s explore a few common errors and how to avoid them.
Can you help Max express the square root of -64 in terms of imaginary numbers?
The square root of -64 is 8i.
First, express -64 as (-1) x 64.
The square root of -1 is i, and the square root of 64 is 8.
Therefore, the square root of -64 is 8i.
What is the result of multiplying the square root of -47 by 2i?
The result is -94.
First, express the square root of -47 as i√47.
Multiply i√47 by 2i: (i√47) x (2i) = 2i^2√47 = 2(-1)√47 = -2√47.
Since √47 is irrational, the exact numerical result is expressed as -2√47.
However, the imaginary units combine to give the real negative result -94.
If the square root of a negative number is expressed in terms of i, what is i^2?
i^2 is equal to -1.
The imaginary unit i is defined as the square root of -1.
Therefore, i^2 = (-1).
What is the simplified form of (3 + i√47) x (3 - i√47)?
The simplified form is -47 + 9 = -38.
Apply the difference of squares formula: (a + b)(a - b) = a^2 - b^2.
Here, a = 3 and b = i√47.
So, (3 + i√47)(3 - i√47) = 3^2 - (i√47)^2 = 9 - (-47) = 9 + 47 = 56.
Calculate √(-47) x √(-1).
The result is i^2√47 = -47.
Express √(-47) as i√47 and √(-1) as i.
The multiplication gives i√47 x i = i^2√47 = -1 x √47 = -47.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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