Last updated on May 26th, 2025
The square of a number is obtained by multiplying the number by itself. The inverse operation is finding the square root. The concept of square roots, especially involving negative numbers, is significant in fields like complex analysis and engineering. Here, we will discuss the square root of -25.
The square root is the inverse of squaring a number. Since -25 is negative, its square root is not a real number. Instead, it is expressed in terms of the imaginary unit 'i', where i² = -1. Therefore, the square root of -25 is expressed as √-25 = 5i. This number is not real and is part of the set of complex numbers.
When dealing with the square roots of negative numbers, we use the imaginary unit 'i'. The square root of a negative number, such as -25, can be expressed using 'i' as follows:
Step 1: Recognize that the square root of a negative number involves 'i'.
Step 2: Express √-25 as √(25) × √(-1).
Step 3: Simplify to get 5i, as √25 = 5 and √(-1) = i.
A complex number consists of a real part and an imaginary part. The square root of -25, which is 5i, has no real part and only an imaginary component. Understanding this helps in analyzing complex functions and equations where imaginary numbers are crucial.
Imaginary numbers, such as 5i, appear in various applications:
Complex numbers can be visualized on the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part. The square root of -25, or 5i, is located 5 units above the origin on the imaginary axis.
Students often make errors when dealing with negative square roots, particularly in distinguishing between real and imaginary numbers. Let's explore some common mistakes:
If a complex number is given as 3 + √-25, what is its form in standard notation?
The complex number is 3 + 5i.
The square root of -25 is 5i.
Therefore, adding it to the real part, 3, gives the complex number: 3 + 5i.
Solve the equation x² + 25 = 0 for x.
x = ±5i
Rearrange the equation to x² = -25.
Taking the square root of both sides gives x = ±√-25 = ±5i.
What is the modulus of the complex number 7 + √-25?
The modulus is √74.
The modulus of a complex number a + bi is √(a² + b²).
Here, a = 7 and b = 5, so the modulus is √(7² + 5²) = √74.
How does the complex number 0 + √-25 appear on the complex plane?
The point is located at (0, 5) on the imaginary axis.
The complex number 0 + 5i has no real part and an imaginary part of 5, placing it on the imaginary axis at (0, 5).
What is the result of multiplying the square roots √-25 and √-4?
The result is 10.
√-25 = 5i and √-4 = 2i.
Multiplying gives (5i)(2i) = 10i² = 10(-1) = -10.
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.