Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The concept of square roots is used in various fields such as engineering, physics, and complex number analysis. Here, we will discuss the square root of -37.
The square root is the inverse of the square of the number. Since -37 is a negative number, its square root is not a real number. Instead, we express the square root of -37 using imaginary numbers. The square root of -37 can be expressed as √(-37), which equals i√37, where i is the imaginary unit and equals √(-1).
When dealing with negative numbers under a square root, we enter the realm of complex numbers. The square root of a negative number involves the imaginary unit 'i'. Let's explore this:
1. Imaginary Unit: The imaginary unit 'i' is defined as √(-1).
2. Expression: For -37, we express it as √(-37) = i√37.
3. Real and Imaginary Parts:
The expression i√37 has a real part of 0 and an imaginary part of √37.
To express the square root of -37 in complex form, we use the properties of complex numbers. Here's how it works:
Step 1: Identify the negative number and the imaginary unit.
Step 2: √(-37) is expressed as i√37.
Step 3: Recognize that i√37 is in the form a + bi, where a = 0 and b = √37.
Imaginary numbers, such as the square root of -37, have practical applications in various fields:
1. Electrical Engineering: Used in the analysis of AC circuits.
2. Control Systems: Utilized in system stability and response analysis.
3. Signal Processing: Employed in Fourier transforms and filters.
Complex numbers can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Here's how to visualize i√37:
1. Plot the point (0, √37) on the complex plane.
2. The point lies on the imaginary axis since the real part is 0.
Students often make mistakes while dealing with square roots of negative numbers. Let's discuss some common errors and how to avoid them.
If z = √(-37), what is the modulus of z?
The modulus of z is √37.
The modulus of a complex number a + bi is given by √(a^2 + b^2).
For z = 0 + i√37, the modulus is √(0^2 + (√37)^2) = √37.
Express (√(-37))^2 in terms of real numbers.
The result is -37.
Using the property of square roots: (√(-37))^2 = -37.
This verifies that squaring the square root of a negative number returns the original negative number.
Simplify the expression: 2i√37 + 3i√37.
The simplified expression is 5i√37.
Combine like terms: (2i√37 + 3i√37 = (2 + 3)i√37 = 5i√37).
What are the real and imaginary parts of √(-37)?
The real part is 0, and the imaginary part is √37.
The expression √(-37) = i√37 has a real part of 0 and an imaginary part of √37.
Calculate the argument of the complex number √(-37).
The argument is π/2 or 90 degrees.
The argument of a complex number in the form 0 + bi is π/2 (or 90 degrees) since it lies on the positive imaginary axis.
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.
: He loves to play the quiz with kids through algebra to make kids love it.