Square Root of -143
2026-02-28 23:32 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-143

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Last updated on August 5, 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.

What is 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.

Understanding the Square Root of Negative Numbers

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.

Square Root of -143 in Radical and Exponential Form

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).

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Calculating the Square Root of -143

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

Applications of Imaginary Numbers

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.

Common Mistakes and How to Avoid Them when Dealing with Square Roots of Negative Numbers

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.

Problem 1

What is the result of multiplying the square root of -143 by 2?

Okay, lets begin

The result is 23.916i.

Explanation

First, find the square root of -143, which is approximately 11.958i.

Then multiply by 2: 11.958i * 2 = 23.916i.

Well explained 👍

Problem 2

If x = √-143, what is x^2?

Okay, lets begin

x^2 = -143.

Explanation

By definition, if x = √-143, then x^2 = (√-143)^2 = -143.

Well explained 👍

Problem 3

Express the square root of -143 in terms of real and imaginary components.

Okay, lets begin

The square root of -143 can be expressed as 0 + 11.958i.

Explanation

The expression √-143 = √143 * i, where √143 ≈ 11.958.

Therefore, the real part is 0 and the imaginary part is 11.958i.

Well explained 👍

FAQ on the Square Root of -143

1.What is the square root of -143 in its simplest form?

The square root of -143 in its simplest form is √143 * i.

2.Can the square root of a negative number be a real number?

No, the square root of a negative number is not a real number.

It is an imaginary number, expressed using the imaginary unit 'i'.

3.Is the square root of -143 a rational number?

No, the square root of -143 is an irrational number because it involves the square root of 143, which is not a perfect square, and includes the imaginary unit.

4.What is the imaginary unit 'i'?

The imaginary unit 'i' is defined as the square root of -1. It is the basis for imaginary numbers.

5.How do you represent the square root of a negative number?

The square root of a negative number is represented as a product of the square root of its positive counterpart and the imaginary unit 'i'. For example, √-143 = √143 * i.

Important Glossaries for the Square Root of -143

  • Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit 'i', which is defined as √-1.
  • Complex Number: A number comprising both a real and an imaginary part, generally expressed as a + bi, where a and b are real numbers.
  • Radical: A symbol (√) representing the root of a number. For negative numbers, it involves the imaginary unit 'i'.
  • Irrational Number: A number that cannot be expressed as a simple fraction, often involving non-terminating, non-repeating decimals.
  • Exponential Form: A way to express numbers using powers, often used with complex numbers to represent roots, such as (-143)^(1/2).

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Jaskaran Singh Saluja

About the Author

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.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.