Square Root of -100
2026-02-28 13:34 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-100

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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The concept of square roots extends to complex numbers when dealing with negative values under the square root. Here, we will discuss the square root of -100.

What is the Square Root of -100?

The square root of a negative number involves complex numbers because there is no real number whose square is negative. The square root of -100 is expressed in terms of the imaginary unit 'i', where i² = -1. Therefore, the square root of -100 can be expressed as ±10i.

Understanding Complex Numbers

Complex numbers include a real part and an imaginary part. They are usually expressed in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part. The imaginary unit 'i' is defined as the square root of -1. Thus, for -100, its square roots are 10i and -10i, which are purely imaginary numbers.

Square Root of -100 Using Exponential Form

The exponential form helps to express complex numbers. Since i is the square root of -1, the square root of -100 can be written as: (-100)^(1/2) = 100^(1/2) * (-1)^(1/2) = 10 * i = 10i. Thus, the square root of -100 is ±10i in exponential form.

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Graphical Representation of Complex Numbers

Complex numbers can be represented graphically on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The square roots of -100, which are ±10i, lie on the imaginary axis, 10 units above and below the origin.

Applications of Complex Numbers

Complex numbers are used in various fields such as engineering, quantum physics, applied mathematics, and signal processing. They are particularly useful in problems involving oscillations, waves, and alternating current (AC) circuits. The concept of imaginary numbers allows for solutions to equations that do not have real solutions.

Common Mistakes and How to Avoid Them in Understanding the Square Root of -100

Students often make mistakes when working with the square root of negative numbers, especially when transitioning from real to complex numbers. Let's explore a few common errors and how to avoid them.

Problem 1

What is the square of the square root of -100?

Okay, lets begin

The square of the square root of -100 is -100.

Explanation

The square root of -100 is ±10i.

Squaring this gives: (10i)² = 100 * i² = 100 * (-1) = -100.

Similarly, (-10i)² = 100 * (-1) = -100.

Thus, squaring the square root returns the original number, -100.

Well explained 👍

Problem 2

If z = √-100, what is the modulus of z?

Okay, lets begin

The modulus of z is 10.

Explanation

The modulus of a complex number a + bi is √(a² + b²). For z = ±10i, the modulus is: |10i| = √(0² + 10²) = √100 = 10.

Well explained 👍

Problem 3

Calculate the sum of (3 + √-100) and (5 - √-100).

Okay, lets begin

The sum is 8.

Explanation

Let √-100 = 10i. Then (3 + 10i) + (5 - 10i) = 3 + 5 + 10i - 10i = 8.

The imaginary parts cancel out, leaving the real part as the sum.

Well explained 👍

Problem 4

What is the result of multiplying √-100 by √-1?

Okay, lets begin

The result is -10.

Explanation

Let √-100 = 10i.

Then multiplying by √-1 (which is i) gives: 10i * i = 10i² = 10(-1) = -10.

Well explained 👍

Problem 5

If f(x) = √-100, for what value of x is f(x) defined?

Okay, lets begin

f(x) is defined for all x, as it is a constant function.

Explanation

The function f(x) = √-100 is a constant function with a value of ±10i, indicating it is defined for any input x, as it doesn't depend on x.

Well explained 👍

FAQ on Square Root of -100

1.What is the simplest form of √-100?

The simplest form of √-100 is ±10i, where 'i' is the imaginary unit.

2.What does the imaginary unit 'i' represent?

The imaginary unit 'i' represents √-1. It is used to express the square roots of negative numbers in the complex number system.

3.Are there real numbers as square roots for -100?

No, there are no real numbers whose square is -100. The square roots of -100 are complex numbers, ±10i.

4.What is the significance of complex numbers?

Complex numbers are significant in various fields such as engineering and physics, particularly in contexts involving oscillations, waves, and AC circuits, where they simplify the mathematics and provide solutions that real numbers alone cannot.

5.Can complex numbers be graphed?

Yes, complex numbers can be graphed on the complex plane, with the x-axis representing the real part and the y-axis representing the imaginary part.

Important Glossaries for the Square Root of -100

  • Complex Number: A number that has both a real part and an imaginary part, expressed as a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.
     
  • Imaginary Unit: The imaginary unit 'i' is defined as √-1, enabling the expression of square roots of negative numbers.
     
  • Modulus: The modulus of a complex number a + bi is the distance from the origin on the complex plane, calculated as √(a² + b²).
     
  • Imaginary Number: A number in the form of bi, where 'b' is a real number and 'i' is the imaginary unit.
     
  • Complex Plane: A graphical representation of complex numbers where the x-axis represents the real part and the y-axis represents the imaginary part.

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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.