Last updated on May 26th, 2025
The square of a number is obtained by multiplying the number by itself. The inverse operation of squaring a number is finding its square root. Square roots are applicable in various fields, such as engineering, physics, and finance. In this discussion, we will explore the concept of the square root of -143.
The square root of a number is a value that, when multiplied by itself, gives the original number. However, -143 is a negative number, and the square root of a negative number is not a real number. In mathematics, the square root of a negative number is expressed using the imaginary unit, denoted as 'i', where i = √-1. Therefore, the square root of -143 is expressed as √-143 = √143 * √-1 = √143 * i. The value √143 is approximately 11.958, and thus √-143 is approximately 11.958i, an imaginary number.
To understand the square root of a negative number, we use the concept of imaginary numbers. The square root of any negative number can be expressed in terms of 'i'. For example, √-143 can be rewritten as √143 * i. Imaginary numbers extend the real number system and are used in advanced fields like electrical engineering and quantum physics.
The square root of -143 can be expressed in both radical and exponential forms: - Radical form: √-143 = √143 * i - Exponential form: (-143)^(1/2) = (143^(1/2)) * i Since -143 is not a perfect square, its square root involves an irrational component (√143) and an imaginary component (i).
Since -143 is negative, its square root is not a real number, and we use imaginary numbers to express it. Here's how you calculate it: 1. Find the square root of the positive component: √143 ≈ 11.958 2. Combine with the imaginary unit: √-143 = 11.958i
Imaginary numbers, including the square roots of negative numbers, have practical applications in various fields:
1. Electrical Engineering: Used in analyzing AC circuits.
2. Control Systems: Imaginary numbers help in the stability analysis of systems.
3. Quantum Mechanics: Used to describe wave functions.
4. Signal Processing: Complex numbers simplify the mathematical analysis of signals.
Students often make mistakes when dealing with the square roots of negative numbers, such as ignoring the imaginary unit or misapplying square root properties. Here are some common mistakes and how to avoid them.
What is the result of multiplying the square root of -143 by 2?
The result is 23.916i.
First, find the square root of -143, which is approximately 11.958i.
Then multiply by 2: 11.958i * 2 = 23.916i.
If x = √-143, what is x^2?
x^2 = -143.
By definition, if x = √-143, then x^2 = (√-143)^2 = -143.
Express the square root of -143 in terms of real and imaginary components.
The square root of -143 can be expressed as 0 + 11.958i.
The expression √-143 = √143 * i, where √143 ≈ 11.958.
Therefore, the real part is 0 and the imaginary part is 11.958i.
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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